Modeling and simulation of systems Introduction to queuing theory Slovak University of Technology Faculty of Material Science and Technology in Trnava

The system of queuing model Service Queue Arriving customer Serviced customer The system of queuing model is an arbitrary service where the service of a particular kind is provided. The customers (demands) who require service arrive into this system. The element of the system which provides the service is called serviced channel or line.

The system of queuing model Important qualities of elements of queuing model :  Arrivals of customers intervals between arrivals are important  run of service one or more serviced channel (links) period of service  behaviour of customers when they cannot be serviced immediately  systems without queue  with limited length of queue  with unlimited length

Qualification of queuing models  Kandal´s qualification which divides queuing models according to three criteria is used the most frequently:  Arrival of customers is described by random process  The division of probabilities of period of service  The number of servicing channels Used shape of qualification X/Y/c

Qualification of queuing models Examples of marking: M/M/nn = 1,2,3.... Arrival – Poisson´s process Service – exponential division n – number of serviced channels D/D/c Arrival – regular intervals Service – constant period G/G/c general case it means no assumptions about arrivals and general division of period of service

Arrivals of customers (stochastic process) The customers arrive individually For each time t we get number N(t) – number of customers who came at time (0,t  The result is non falling function N(t), t  0, which gains the whole non negative values and N(0) = 0 t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 N(t)

Arrivals of customers (stochastic process) Arrivals into the system are random then for given t is N(t) random quality  what is division of probabilities p k (t) = P{N(t) = k}, k = 0,1,2,... Arrival of customer is defined as the system of random variables N(t), t  0 and is edited {N(t)} t  T T  0,  Such system is called stochastic (random) process

Poisson´s homogenous process Is given random process {N(t)} t  0 with independent growths it means random qualities N(b 1 ) - N(a 1 ), N(b 2 ) - N(a 2 ),.., N(b n ) - N(a n ) are independent, if intervals (a 1,b 1 , (a 2,b 2 ,.., (a n,b n  are disjunctive when all random qualities N(t+  ) - N(t) have Poisson´s division of probability. The middle value of number of customers who come in time  is 

Qualities of Poisson´s process Homogeneity The probability of it that in the time  comes to customers does not depend on the beginning of interval (t, t+  It means that arrival of customers is regular for the whole period. Independence the number of customers who came during each period does not depend on the number of customers who came into the system in other disjunctive time period.

Qualities of Poisson´s process Ordinary It means that more than one customer comes in a very short time interval with insignificant probability smaller that the length of this interval. It is not probable that more customers come at the same time.

The relation between Poisson´s division and exponential division The division of random quantity which is the interval between arrival of two customers that is  k = t k - t k-1 k = 1,2,... Distributive function F(  ) of random variable  k F(  ) = P{  k   } = 1 - P{  k   } P{  k   } = P{N(  ) = 0} = e -  Then: F(  ) = 1 - e - ,  0 0,  0 Exponential distribution If is the mean value of the number of customers arriving into the system for time unit, then.... is the mean value of time between arrivals of customers

The number of customers in the system Markovov´s homogeneous process The system is characterized by the number of customers X(t), who are in the time t in the system. If arrivals of service period is random then the number of customers X(t) at the moment t is random. The function X(t) can fall as well as rise in dependence on how customers arrive and how fast are they served

The run of the number of customers in the system t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t X(t) Random value X(t) gains values k = 0,1,2,... with probabilities p k (t) = P  X(t) = k  where for each t  0

Markovov´s random process Definition: the random process  X(t)  t  0 is called Markovov´s, if is in force: P  X(t) = k  X(s 1 ) = j 1, X(s 2 ) = j 2,..., X(s n ) = j n  = P  X(t) = k  X(s 1 ) = j 1  for t  s 1  s 2 ... s n  0 It means that the future X(t) does not depend on the past X(s i ), i  1, but only on the presence X(s 1 )

Homogeneous process If is in force P  X(t+  ) = j  X(t) = i  = = P  X(s+  ) = j  X(s) = i  = = P  X(  ) = j  X(0) = i , then the process is homogeneous. It means that there will be j of customers in the system after the time , if there were i of customers in the time t it depends only on length of time , and not on since when it is monitored.

Probability of transition from the state j into the state i Probability of transition from j into i is a probability by which the system transits from the state i into the state j and is marked as p ij (  ). Then p ij (  ) = P  X(t+  ) = j  X(t) = i  = P  N(t+  ) - N(t) = j - i  Poisson´s process is also Markovov´s and is in force:

Intensity of transition The intensity of transition of Markovov´s homogeneous process  X(t)  t  0 is called number ijij

Intensity of transition SHO is given with one service channel without waiting. The arrivals with Poisson´s division with the middle value, period of service – exponential division with the middle value 1/ . Random process (then the number of customers in the system gains the values 0 a 1. It is necessary to determine

Intensity of transition It is necessary to determine for the calculation p 01. The system transits from the state 0 into the state 1: A:For the time  just one customer arrives and his service after the time  does not end. B:For the time  more that one customer arrives and the service of one of them after the time  does not end It means that p 01 (  ) = P  A  + P  B  a

Intensity of transition Event B: P  N(  )  1  Ordinary of Poisson´s random process

Intensity of transition Event A: Probability A is given by product of probability that for the time  just one customer arrives and probabilities that his service after  does not end. P  A  = P  N(  ) =1 . P  t obs   P  N(  ) =1  =e - .  Distributive function of period of service t obs F(  ) = P  t obs   = 1- e - ,   0 then P  t obs   = 1- P  t obs   = e - 

Intensity of transition Intensity of transition 01 = +0= It is possible to determine in similar way 10 =  Then it is necessary to determine for the system with one service channel without queue probabilities of the fact that there is in the system in the time t k of customers: p k (t)= P  X(t) = k 

Kolmogorov´s differential equations Determination of previous probabilities leads to the following equations: These equations are Kolmogorov´s differential equations. They are not ependent. It is force: p 0 (t) + p 1 (t) = 1

Kolmogorov´s differential equations For transcript of Kolmogorov´s equations is in force: Derivation p ’ k of probability that the system in the time t in the state k equals to summation of probabilities, that in the state k multiplied by sum of negatively outstanding intensities of transition getting out of state k and probabilities of all other states multiplied by intensities of transition which getting out into the state k

Kolmogorov´s differential equations Kolmogorov´s differential equations: p ’ 0 = - 01 p p 1 p ’ 1 = -( ) p p p 2 p ’ 2 = -( ) p p p 3 p ’ 3 = - 32 p p 2 Example of transcript of equations: