Networks of Queues Plan for today (lecture 6): Last time / Questions? Product form preserving blocking Interpretation traffic equations Kelly / Whittle.

Networks of Queues Plan for today (lecture 6): Last time / Questions? Product form preserving blocking Interpretation traffic equations Kelly / Whittle network Optimal design of a Kelly / Whittle network: optimisation problem Intermezzo: mathematical programming Optimal design of a Kelly / Whittle network: Lagrangian and interpretation Optimal design of a Kelly / Whittle network: Solution optimisation problem Optimal design of a Kelly / Whittle network: network structure Exercises Questions??

Blocking in tandem networks of simple queues (1) Simple queues, exponential service queue j, j=1,…,J state move depart arrive Transition rates Traffic equations Solution

Blocking in tandem networks of simple queues (2) Simple queues, exponential service queue j, j=1,…,J Transition rates Traffic equations Solution Equilibrium distribution Partial balance PICTURE J=2

Blocking in tandem networks of simple queues (3) Simple queues, exponential service queue j, j=1,…,J Suppose queue 2 has capacity constraint: n2<N2 Transition rates Partial balance? PICTURE J=2 Stop protocol, repeat protocol, jump-over protocol

Kelly / Whittle network Transition rates for some functions ψ:S  (0,∞), Traffic equations Open network Partial balance equations: Theorem: Assume then satisfies partial balance, and is equilibrium distribution Kelly / Whittle network

Interpretation traffic equations Transition rates for some functions ψ:S  (0,∞), Traffic equations Open network Theorem: Suppose that the equilibrium distribution is then and rate j  k PROOF

Source How to route jobs, and how to allocate capacity over the nodes? sink

Optimal design of Kelly / Whittle network (1) Transition rates for some functions ψ:S  (0,∞), Routing rules for open network to clear input traffic as efficiently as possible Cost per time unit in state n : a(n) Cost for routing j  k : Design : b_j0=+  : cannot leave from j; sequence of queues Expected cost rate

Optimal design of Kelly / Whittle network (2) Transition rates Given: input traffic Maximal service rate Optimization problem : minimize costs Under constraints

Intermezzo: mathematical programming Optimisation problem Lagrangian Lagrangian optimization problem Theorem : Under regularity conditions: any point that satisfies Lagrangian optimization problem yields optimal solution of Optimisation problem

Intermezzo: mathematical programming (2) Optimisation problem Introduce slack variables Kuhn-Tucker conditions: Theorem : Under regularity conditions: any point that satisfies Lagrangian optimization problem yields optimal solution of Optimisation problem Interpretation multipliers: shadow price for constraint. If RHS constraint increased by , then optimal objective value increases by  i 

Optimal design of Kelly / Whittle network (3) Optimisation problem Lagrangian form Interpretation Lagrange multipliers :

Optimal design of Kelly / Whittle network (4) KT-conditions Computing derivatives:

Optimal design of Kelly / Whittle network (5) Theorem : (i) the marginal costs of input satisfy with equality for those nodes j which are used in the optimal design. (ii) If the routing j  k is used in the optimal design the equality holds in (i) and the minimum in the rhs is attained at given k. (iii) If node j is not used in the optimal design then α j =0. If it is used but at less that full capacity then c j =0. Dynamic programming equations for nodes that are used

Optimal design of Kelly / Whittle network (6) PROOF: Kuhn-Tucker conditions :

Exercise 1: insensitivity and Norton Consider a network consisting of subnetworks. Each subnetwork is quasi reversible. The subnetworks are linked via Markov routing (as in the transparancy in class, but now state independent routing). 1.Formulate the transition rates at local and global level 2.Formulate and prove the decomposition theorem 3.Suppose that we want to aggregate a component into a single queue. Give the service rates for this queue. 4.Now assume that all queues in the network are infinite server queues. Give the service rate for the aggregated components. Show the relation between Norton’s theorem and the insensitivity property. 5.Investigate the relation between Norton’s theorem and insensitivity

Exercise 2: Optimal design of Jackson network (1) Consider an open Jackson network with transition rates Assume that the service rates and arrival rates are given Let the costs per time unit for a job residing at queue j be Let the costs for routing a job from station i to station j be (i) Formulate the design problem (allocation of routing probabilities) as an optimisation problem. (ii) Provide the solution to this problem

Exercise 3: Optimal design of Jackson network (2) Consider an open Jackson network with transition rates Assume that the routing probabilities and arrival rates are given Let the costs per time unit for a job residing at queue j be Let the costs for routing a job from station i to station j be Let the total service rate that can be distributed over the queues be, i.e., (i) Formulate the design problem (allocation of service rates) as an optimisation problem. (ii) Provide the solution to this problem (iii) Now consider the case of a tandem network, and provide the solution to the optimisation problem for the case for all j,k

Exercise 4 : routes vs Markovian routing 1. Show that an open Jackson network with transition rates Can equivalently be represented as a network with fixed routing (i.e., the route is determined by the customer type). 2. Now consider an open Jackson network with queues with finite capacity, i.e. n i <N i, i=1,…,J. Consider the network under a product form preserving blocking protocol. Investigate whether or not the network can equivalently be represented as a network with fixed routing as in 1.

Exercise 5 Consider an open tandem network of 3 single server queues (M/M/1 queues). 1.Formulate the transitions rates. 2.Give the equilibrium distribution, and show that this distribution satisfies partial balance 3.Show that a customer arriving to queue j, j=1,2,3 observes the equilibrium distribution 4.Now assume that the total number of customers in the network is restricted not to exceed N. Formulate the transition rates. Give the equilibrium distribution, and the distribution of the state observed by a customer arriving to queue 1 5.Now assume that each queue has a capacity restriction, say the number in queue i is restricted not to exceed Ni. A queue that has reached its capacity constraint is blocked, which means that no extra customer may enter the queue. This means that a customer cannot leave its predecessor, i.e., when queue 3 is blocked a customer cannot leave queues 2 (the server at queue 2 is stopped). Formulate the transition rates, and show that the equilibrium distribution cannot satisfy partial balance. 6.Again consider the tandem network of part 5. Consider the stop protocol, that is the following modification of the transition rates. The arrival process, and each non- blocked queue is stopped until a service completion has occurred at the blocked queue. Give the state space, and formulate the transition rates. Show that the equilibrium distribution has a product form and satisfies partial balance. Give the distribution of the state observed by a customer arriving to (and entering) queue 2. 7.Again consider the tandem network of part 5. Assume that a customer arriving to a queue that has reached its capacity restriction, jumps over the queue and moves to the next queue in the tandem. Formulate the transition rates. Give the equilibrium distribution (proof!).

Exercise 6 Consider a tandem network consisting of 4 queues. Queue 1 is a FCFS queue, queue 2 a PS queue, queue 3 a LCFS queue, and queue 4 an Infinite server queue. Let customers of type 1 and 2 arrive to queue 1 according to a Poisson process with rates λ 1, and λ 2. 1.Let the service requirements at each of the queues be exponentially distributed. Formulation the network as BCMP network. Also give the transition rates. 2.Show that the queues are quasi reversible (you may do this for the queues in isolation, i.e., for each queue consider only that queue [thus part of the network] with Poisson arrival process) 3.Give the equilibrium distribution for the tandem network. 4.Show that this distribution satisfies partial balance. 5.Prove that the equilibrium distribution of queue 2, 3 and 4 is insensitive to the distribution of the service time except for its mean (you may do this for the queue in isolation). 6.Prove that the equilibrium distribution of the tandem network is insensitive to the distribution of the service time at each of the queues 2,3,4 except for their means. 7.Indicate why the distribution cannot be insensitive to the service time distribution in an FCFS queue.

Exercises: rules In all exercises: give proof or derivation of results (you are not allowed to state something like: proof as given in class, or proof as in book…) motivate the steps in derivation hand in exercises week before oral exam, that is based on these exercises All oral exams (30 mins) for this part on the same day, you may suggest the date… Hand in 4 exercises: 1 and $ and select 2 or 3 and 5 or 6 Each group 2 or 3 persons: hand in a single set of answers G1: 1,2,4,5 G2: 1,2,4,6 G3: 1,3,4,5 G4: 1,3,4,6

Networks of Queues Plan for today (lecture 6): Last time / Questions? Product form preserving blocking Interpretation traffic equations Kelly / Whittle.

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Networks of Queues Plan for today (lecture 6): Last time / Questions? Product form preserving blocking Interpretation traffic equations Kelly / Whittle.

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