Zakynthos Island, Greece May 23, 2005 Performance Analysis of a Multiple Vehicle Tandem System with Inter-vehicle Buffers and Blocking Matthew E. H. Petering.

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Zakynthos Island, Greece May 23, 2005 Performance Analysis of a Multiple Vehicle Tandem System with Inter-vehicle Buffers and Blocking Matthew E. H. Petering 1,2, Jaeyoung Seo 2, and Chulung Lee 2 1 IOE Department, University of Michigan—Ann Arbor 2 ISE Department, National University of Singapore

Zakynthos Island, Greece May 23, 2005 Port of Hong Kong

Zakynthos Island, Greece May 23, 2005 Quay Cranes Single trolley quay crane

Zakynthos Island, Greece May 23, 2005 The Double Trolley Quay Crane Intermediate platform Trolley #2 Trolley #1

Zakynthos Island, Greece May 23, 2005 The Double Trolley Quay Crane Intermediate platform Trolley #2 Trolley #1 Trolley #2

Zakynthos Island, Greece May 23, 2005 The Double Trolley Quay Crane Intermediate platform Trolley #2 Trolley #1

Zakynthos Island, Greece May 23, 2005 Literature Review

Zakynthos Island, Greece May 23, 2005 Literature Review Few exact analytical results for general types of manufacturing systems or material handling systems

Zakynthos Island, Greece May 23, 2005 Literature Review Production lines with blocking phenomena: No buffers ; Exponential service time Buffers with positive, non-identical capacities ; Erlang service time Unreliable stations ; Exponential time to failure ; Erlang repair time Coxian-2 service time Papadopoulos and O’Kelly 1989 Papadopoulos et al. 1989, 1990 Heavey et al Vidalis and Papadopoulos 1999 Few exact analytical results for general types of manufacturing systems or material handling systems

Zakynthos Island, Greece May 23, 2005 System Description

Zakynthos Island, Greece May 23, 2005  N stations (buffers), origin, destination  N+1 vehicles  Poisson rates of travel: R 1 +, R 1 -, R 2 +, R 2 -, …, R N+1 +, R N+1 -  Buffer capacities: C 1, C 2, …, C N  Blocking System Description

Zakynthos Island, Greece May 23, 2005 Markovian State Description

Zakynthos Island, Greece May 23, 2005  Buffer status = number of items occupying the buffer  Vehicle status = M +, M -, I +, I - Markovian State Description

Zakynthos Island, Greece May 23, 2005 Markovian State Description  Buffer status = number of items occupying the buffer  Vehicle status = M +, M -, I +, I -

Zakynthos Island, Greece May 23, 2005 Markovian State Description State = ( M -, 2, I +, 3, M + )  Buffer status = number of items occupying the buffer  Vehicle status = M +, M -, I +, I -

Zakynthos Island, Greece May 23, 2005 Number of States

Zakynthos Island, Greece May 23, 2005 Theorem. Let be the number of states in the CTMC representation of a system with n buffers (n+1 vehicles) with capacities C 1,C 2,...,C n from the origin to the destination respectively. Assuming S 0 = 0 and S 1 = 2, the following is true for all n ≥ 1: Number of States

Zakynthos Island, Greece May 23, 2005 Number of States

Zakynthos Island, Greece May 23, 2005 Computing System Performance

Zakynthos Island, Greece May 23, 2005 Computing System Performance avg. throughput rate avg. number of items in system avg. sojourn time In steady state:

Zakynthos Island, Greece May 23, 2005 Computing System Performance avg. throughput rate avg. number of items in system avg. sojourn time In steady state: A = transpose of state transition matrix x = column vector of steady state probabilities A x = 0, e T x = 1

Zakynthos Island, Greece May 23, 2005 Generating the State Transition Matrix

Zakynthos Island, Greece May 23, 2005 Generating the State Transition Matrix  Recursive Method  Finds the transition matrix for successively larger subsystems of original system

Zakynthos Island, Greece May 23, 2005 Generating the State Transition Matrix Definition. A B-buffer subsystem of an original system is a system with B buffers and B+1 vehicles whose vehicle rates are equal to the rates of the final B+1 vehicles in the original system (from origin to destination respectively) and whose buffer capacities are equal to the capacities of the final B buffers in the original system (from origin to destination respectively).  Recursive Method  Finds the transition matrix for successively larger subsystems of original system

Zakynthos Island, Greece May 23, 2005 Generating the State Transition Matrix  Recursive Method  Finds the transition matrix for successively larger subsystems of original system

Zakynthos Island, Greece May 23, 2005 Generating the State Transition Matrix  Recursive Method  Finds the transition matrix for successively larger subsystems of original system

Zakynthos Island, Greece May 23, 2005 Generating the State Transition Matrix  Recursive Method  Finds the transition matrix for successively larger subsystems of original system

Zakynthos Island, Greece May 23, 2005 Generating the State Transition Matrix  First construct A 1  Recursive method successively constructs A 2, A 3, …, A N A B = transpose of the transition matrix for the B-buffer subsystem

Zakynthos Island, Greece May 23, 2005 Generating the State Transition Matrix DBDB EBEB EBEB EBEB EBEB EBEB EBEB EBEB HBHB GBGB FBFB HBHB DBDB DBDB DBDB DBDB DBDB DBDB... C N-B +1 copies of D B and E B C N-B +2 copies of D B and E B A B+1 is constructed recursively using A B as a building block

Zakynthos Island, Greece May 23, 2005 Computing Steady State Probabilities

Zakynthos Island, Greece May 23, ) set x 1 = 1 2) solve resulting nonsingular system A’ x’ = b using Gauss-Seidel method (Cooper and Gross, 1991) 3) renormalize so sum of x i = 1 Find the unique x satisfying A N x = 0, e T x = 1 Computing Steady State Probabilities

Zakynthos Island, Greece May 23, 2005 Computing System Performance

Zakynthos Island, Greece May 23, 2005 Computing System Performance T = avg. throughput rate = ( R i + ) x Prob( v i = M + ) for any i In steady state:  Buffer status = b i  Vehicle status = v i L = avg. number of items in system =  ( x i ) x (number of items in system for State i ) W = avg. sojourn time for an item = L / T Likelihood vehicle i waits at forward buffer = Prob( v i = M +, b i = C i, v i+1 ≠ I - ) / Prob( v i = M + ) all i

Zakynthos Island, Greece May 23, 2005 Results and Observations

Zakynthos Island, Greece May 23, 2005 Results and Observations Likelihood of Vehicles Waiting System#States#IterT'put#ItemsSojrn1st2nd 2 0 2,1,2, ,1,2, ,1,2, ,1,8, ,1,8, ,1,98, Throughput increases as buffer capacity or vehicle speed is increased 2-vehicle systems

Zakynthos Island, Greece May 23, 2005 Results and Observations Throughput deteriorates as system grows Number of Vehicles in System#States#Iter T'put#ItemsSojrn Zero buffer capacity, all vehicle rates = 1

Zakynthos Island, Greece May 23, 2005 Results and Observations System#States#IterT'put#ItemsSojrn 2 1 2,2,2, ,1 3,3,3,3,3, ,1,1 4,4,4,4,4,4,4, ,1,1,1 5,5,5,5,5,5,5,5,5, Constant pure sojourn time Throughput increases almost linearly

Zakynthos Island, Greece May 23, 2005 Results and Observations System#States#Iter T'put#ItemsSojrn 3 0,0 1,2,1,2,9998, ,0 1,2,9998,9999,1, ,0 9998,9999,1,2,1, Fast vehicle should be placed in the center

Zakynthos Island, Greece May 23, 2005 Results and Observations Likelihood of Vehicles Waiting SystemT'put#ItemsSojrn1st2nd3rd4th 5 1,0,0,0 (ALL = 1) ,1,0,0 (ALL = 1) ,0,1,0 (ALL = 1) ,0,0,1 (ALL = 1) Large buffer should be placed in the center

Zakynthos Island, Greece May 23, 2005 Results and Observations SystemT'put#ItemsSojrn 4 0,0,0 3,3,3,3,3,3,3, ,0,0 2,6,2,6,2,6,2, ,0,0 2,6,2,6,6,2,6, Constant expected pure round-trip travel time for all vehicles Throughput depends on vehicle forward/reverse rates

Zakynthos Island, Greece May 23, 2005 Results and Observations Number of Stations/ Buffer size Vehicles This system has slightly reduced throughput Comparison with Erlang-2 production line (Papadopoulos et al., 1990) (Throughput)  Erlang-2  This system

Zakynthos Island, Greece May 23, 2005 Results and Observations Likelihood of Vehicles Waiting SystemT'put#ItemsSojrn1st2nd3rd 3 0,5 1,6,3,5,4, ,4 1,6,3,5,4, ,3 1,6,3,5,4, ,2 1,6,3,5,4, ,1 1,6,3,5,4, ,0 1,6,3,5,4, A buffer allocation problem…

Zakynthos Island, Greece May 23, 2005 Conclusions  Existing methodology was used to investigate a new kind of material handling system  New numerical results regarding system performance were obtained  Numerical results agree with intuition and with previously obtained results for other systems

Zakynthos Island, Greece May 23, 2005 Further Research  Compare theoretical results to simulation trials on systems with less variable vehicle travel times  Consider more complicated systems  Consider buffer allocation problems

Zakynthos Island, Greece May 23, 2005 Questions?

Zakynthos Island, Greece May 23, 2005 The Double Trolley Quay Crane

Zakynthos Island, Greece May 23, 2005 Generating the List of States

Zakynthos Island, Greece May 23, 2005 Generating the List of States  Recursive Method

Zakynthos Island, Greece May 23, 2005 Generating the List of States  Recursive Method  Find the list of states and transitions for successively larger subsystems of original system

Zakynthos Island, Greece May 23, 2005 Generating the List of States  Recursive Method  Find the list of states and transitions for successively larger subsystems of original system

Zakynthos Island, Greece May 23, 2005 Generating the List of States  Recursive Method  Find the list of states and transitions for successively larger subsystems of original system

Zakynthos Island, Greece May 23, 2005 Generating the List of States  Recursive Method  Find the list of states and transitions for successively larger subsystems of original system

Zakynthos Island, Greece May 23, 2005 Generating the List of States

Zakynthos Island, Greece May 23, 2005 Generating the List of States L B = list of states for the B-buffer subsystem  First construct L 1  Recursive method successively constructs L 2, L 3, …, L N

Zakynthos Island, Greece May 23, 2005 Generating the List of States

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example:

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: 3 Vehicles Rates of travel: R 1 +/-, R 2 +/-, R 3 +/- 2 Intermediate Buffers Buffer capacities: C 1 = 1, C 2 = 1

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: 3 Vehicles Rates of travel: R 1 +/-, R 2 +/-, R 3 +/- 2 Intermediate Buffers Buffer capacities: C 1 = 1, C 2 = 1

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: 3 Vehicles Rates of travel: R 1 +/-, R 2 +/-, R 3 +/- 2 Intermediate Buffers Buffer capacities: C 1 = 1, C 2 = 1 start by making list of states for the 1-buffer subsystem:

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: 3 Vehicles Rates of travel: R 1 +/-, R 2 +/-, R 3 +/- 2 Intermediate Buffers Buffer capacities: C 1 = 1, C 2 = 1 start by making list of states for the 1-buffer subsystem:

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- Example: the 1-buffer subsystem:

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) Example: the 1-buffer subsystem:

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 - L 1 =

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- ( I -, 0, I - ) ( I -, 0, M - ) ( I -, 0, M + ) ( I -, 1, M - ) ( I -, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- ( I -, 0, I - ) ( I -, 0, M - ) ( I -, 0, M + ) ( I -, 1, M - ) ( I -, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States 2 Vehicles 1 Intermediate Buffer Buffer capacity = 1 Rates of travel: R 2 +/-, R 3 +/- ( I -, 0, I - ) ( I -, 0, M - ) ( I -, 0, M + ) ( I -, 1, M - ) ( I -, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) the 1-buffer subsystem: Example: R 2 + R 2 - R 3 + R 3 - Z 1 =

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: Going from L 1 to L 2 :

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ” “ (M -, 1 ) + L 1 ” “ ( I +, 1) + L 1 ” “ (M +, 0) + Z 1 ” “ (M +, 0) + L 1 ” “ (M +, 1) + L 1 ” = L 2 Going from L 1 to L 2 : Make initial list of states with inherited transitions

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ” “ (M -, 1 ) + L 1 ” “ ( I +, 1) + L 1 ” “ (M +, 0) + Z 1 ” “ (M +, 0) + L 1 ” “ (M +, 1) + L 1 ” = L 2 Going from L 1 to L 2 : Make initial list of states with inherited transitions Adjust inherited transitions with rate R 2 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ” “ (M -, 1 ) + L 1 ” “ ( I +, 1) + L 1 ” “ (M +, 0) + Z 1 ” “ (M +, 0) + L 1 ” “ (M +, 1) + L 1 ” = L 2 R 1 + R 1 - Going from L 1 to L 2 : Make initial list of states with inherited transitions Adjust inherited transitions with rate R 2 - Add new transitions with rates R 1 + and R 1 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ” “ (M -, 1 ) + L 1 ” “ ( I +, 1) + L 1 ” “ (M +, 0) + Z 1 ” “ (M +, 0) + L 1 ” “ (M +, 1) + L 1 ” 3 Vehicles Rates of travel: R 1 +/-, R 2 +/-, R 3 +/- 2 Intermediate Buffers Buffer capacities: C 1 = 1, C 2 = 1 = L 2 R 1 + R 1 - Going from L 1 to L 2 : Make initial list of states with inherited transitions Adjust inherited transitions with rate R 2 - Add new transitions with rates R 1 + and R 1 -

Zakynthos Island, Greece May 23, 2005 Generating the List of States The entire list L 2 : Example:

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M +, 0, I - ) (M +, 1, M +, 0, M - ) (M +, 1, M +, 0, M + ) (M +, 1, M +, 1, M - ) (M +, 1, M +, 1, M + ) (M +, 1, M -, 0, I - ) (M +, 1, M -, 0, M - ) (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) The entire list L 2 :

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M +, 0, I - ) (M +, 1, M +, 0, M - ) (M +, 1, M +, 0, M + ) (M +, 1, M +, 1, M - ) (M +, 1, M +, 1, M + ) (M +, 1, M -, 0, I - ) (M +, 1, M -, 0, M - ) (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) The entire list L 2 :

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M +, 0, I - ) (M +, 1, M +, 0, M - ) (M +, 1, M +, 0, M + ) (M +, 1, M +, 1, M - ) (M +, 1, M +, 1, M + ) (M +, 1, M -, 0, I - ) (M +, 1, M -, 0, M - ) (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) The entire list L 2 :

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M +, 0, M - ) (M +, 1, M +, 0, M + ) (M +, 1, M +, 1, M - ) (M +, 1, M +, 1, M + ) (M +, 1, M -, 0, I - ) (M +, 1, M -, 0, M - ) (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) … The entire list L 2 :

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M +, 0, M + ) (M +, 1, M +, 1, M - ) (M +, 1, M +, 1, M + ) (M +, 1, M -, 0, I - ) (M +, 1, M -, 0, M - ) (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) … The entire list L 2 :

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M +, 1, M - ) (M +, 1, M +, 1, M + ) (M +, 1, M -, 0, I - ) (M +, 1, M -, 0, M - ) (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) … The entire list L 2 :

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M +, 1, M + ) (M +, 1, M -, 0, I - ) (M +, 1, M -, 0, M - ) (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) …

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M -, 0, I - ) (M +, 1, M -, 0, M - ) (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) …

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M -, 0, M - ) (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) …

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M -, 0, M + ) (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) …

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M -, 1, M - ) (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) …

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, M -, 1, M + ) (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) …

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, I +, 1, M - ) (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) …

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 1) + L 1 ” “ (M +, 0) + L 1 ” (M +, 1, I +, 1, M + ) (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) … “ (M +, 0) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) … “ (M +, 0) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) … “ (M +, 0) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) … “ (M +, 0) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M +, 0, I - ) (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) … “ (M +, 0) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M +, 0, M - ) (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) … “ (M +, 0) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M +, 0, M + ) (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) … “ (M +, 0) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M +, 1, M - ) (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) … “ (M +, 0) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M +, 1, M + ) (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M -, 0, I - ) (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M -, 0, M - ) (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M -, 0, M + ) (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M -, 1, M - ) (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, M -, 1, M + ) (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, I +, 1, M - ) (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - “ (M +, 0) + L 1 ” (M +, 0, I +, 1, M + ) (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M +, 0, I -, 0, I - ) (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M +, 0, I -, 0, M - ) (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M +, 0, I -, 0, M + ) (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M +, 0, I -, 1, M - ) (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M +, 0, I -, 1, M + ) ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) … “ (M +, 0) + Z 1 ” “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M +, 0, I - ) ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M +, 0, M - ) ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M +, 0, M + ) ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M +, 1, M - ) ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M +, 1, M + ) ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M -, 0, I - ) ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M -, 0, M - ) ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M -, 0, M + ) ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M -, 1, M - ) ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, M -, 1, M + ) ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, I +, 1, M - ) ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - ( I +, 1, I +, 1, M + ) (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) … “ ( I +, 1) + L 1 ” “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M +, 0, I - ) (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M +, 0, M - ) (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M +, 0, M + ) (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M +, 1, M - ) (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M +, 1, M + ) (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M -, 0, I - ) (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M -, 0, M - ) (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M -, 0, M + ) (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M -, 1, M - ) (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, M -, 1, M + ) (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, I +, 1, M - ) (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 1, I +, 1, M + ) (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) (M -, 0, I -, 0, I - ) … “ (M -, 1 ) + L 1 ” “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) (M -, 0, I -, 0, I - ) (M -, 0, I -, 0, M - ) … “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) (M -, 0, I -, 0, I - ) (M -, 0, I -, 0, M - ) … “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) (M -, 0, I -, 0, I - ) (M -, 0, I -, 0, M - ) … “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 0, M +, 0, I - ) (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) (M -, 0, I -, 0, I - ) (M -, 0, I -, 0, M - ) … “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 0, M +, 0, M - ) (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) (M -, 0, I -, 0, I - ) (M -, 0, I -, 0, M - ) (M -, 0, I -, 0, M + ) … “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States (M +, 0, I - ) (M +, 0, M - ) (M +, 0, M + ) (M +, 1, M - ) (M +, 1, M + ) (M -, 0, I - ) (M -, 0, M - ) (M -, 0, M + ) (M -, 1, M - ) (M -, 1, M + ) (I +, 1, M - ) (I +, 1, M + ) (M +, 0, I - ) (M +, 0, M - ) … Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 0, M +, 0, M + ) (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) (M -, 0, I -, 0, I - ) (M -, 0, I -, 0, M - ) (M -, 0, I -, 0, M + ) (M -, 0, I -, 1, M - ) … “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) (M -, 0, I -, 0, I - ) (M -, 0, I -, 0, M - ) (M -, 0, I -, 0, M + ) (M -, 0, I -, 1, M - ) (M -, 0, I -, 1, M + ) … “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States Example: (M +, 1, (M +, 0, R 2 + R 2 - R 3 + R 3 - (M -, 0, M +, 1, M - ) (M -, 0, M +, 1, M + ) (M -, 0, M -, 0, I - ) (M -, 0, M -, 0, M - ) (M -, 0, M -, 0, M + ) (M -, 0, M -, 1, M - ) (M -, 0, M -, 1, M + ) (M -, 0, I +, 1, M - ) (M -, 0, I +, 1, M + ) (M -, 0, I -, 0, I - ) (M -, 0, I -, 0, M - ) (M -, 0, I -, 0, M + ) (M -, 0, I -, 1, M - ) (M -, 0, I -, 1, M + ) … “ (M -, 0 ) + L 1 ” “ (M -, 0 ) + Z 1 ”

Zakynthos Island, Greece May 23, 2005 Generating the List of States L B = list of states for the B-buffer subsystem  First construct L 1  Recursive method successively constructs L 2, L 3, …, L N

Zakynthos Island, Greece May 23, 2005 Example of System Behavior

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1

Zakynthos Island, Greece May 23, 2005 Example of System Behavior  2 Vehicles  1 Intermediate Buffer  Buffer capacity = 1