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Queuing Theory Operational Research-Level 4

Queuing Theory Operational Research-Level 4. Prepared by T.M.J.A. Cooray Department of Mathematics. Introduction. Markov processes in continuous time was considered in the previous section.

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Queuing Theory Operational Research-Level 4

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  1. Queuing TheoryOperational Research-Level 4 Prepared by T.M.J.A. Cooray Department of Mathematics

  2. Introduction • Markov processes in continuous time was considered in the previous section. • There we considered the Poisson processes for which the new events occur entirely at random and quite independently of the (current) present state of the system. • However, there are processes where the occurrence/non-occurrence of the new events depend in some degree to the present state of the system. MA(4020) Operational Research

  3. Examples • Appearance of new individuals : births, • disappearance of existing individuals: deaths MA(4020) Operational Research

  4. The simple birth and death process • Usual notations : • X(t)-population size of individuals at time t; notation pn(t)=Pr[X(t)=n]; • Assume that : • All individuals are capable of giving birth to a new individual • Chance that any individual giving birth to a (producing )new one in time Δt, Δt+ o(Δt) . where  is the chance of a new birth per individual per unit time. and all the other assumptions as in the Poisson process. MA(4020) Operational Research

  5. Also assume that : • All individuals are capable of dying • Chance that any individual dying (in time Δt,is Δt+ o(Δt) . where  is the chance of a death of an individual per unit time, and all the other assumptions as in the Poisson process. • The chance of no reproduction or no death (no change ) in time Δt of the individual can be written as 1- (+)Δt + o(Δt) , • The chance of more than one of these events ( a birth and a death in time Δt) is o(Δt) MA(4020) Operational Research

  6. Hence the corresponding probability that the whole population of size X(t) will produce a birth is  X(t)Δt to first order in Δt and probability of one death for the whole population of size X(t) is X(t)Δt to first order in Δt and • probability that the whole population of size X(t) will not change is • 1-(+ )X(t)Δt to first order in Δt MA(4020) Operational Research

  7. If the population size is 0,i.e.. no births or deaths can occur. thus we have to start the process ,with a non zero population size. X(0)=n0. say a. then pa(0)=1 and pa-1(0)=0 MA(4020) Operational Research

  8. Now by writing the probability of a population size X(t+ Δt)= n at time t+ Δt is MA(4020) Operational Research

  9. we use the method of generating functions as mentioned in the Poisson processes MA(4020) Operational Research

  10. The subsidiary equations take the form MA(4020) Operational Research

  11. This expression and integration is bit tedious . • Reference : • (Stochastic Processes -Norman T J Bailey , page 91-95) • Now let us consider the two cases • separately • =0 when >0 (simple birth process) • and =0 when>0 (simple death process) MA(4020) Operational Research

  12. The pure birth process • Assume that : no deaths are possible . =0 • All individuals are capable of giving birth to a new individual • Chance that any individual giving birth to a (producing )new one in time Δt, Δt . where  is average number births per individual. and all the other assumptions as in the Poisson process. • The chance of no reproduction in time Δt of the individual can be written as 1- Δt , MA(4020) Operational Research

  13. Now by writing the probability of a population size n at time t+ Δt is if the population size is 0,i.e.. no births can occur. thus we have to start the process ,with a non zero population size. X(0)=n0. say a. then pa(0)=1 and pa-1(0)=0----(9) MA(4020) Operational Research

  14. now (8) becomes at t=0 : • this can be solved in succession starting with the above. • we use the method of generating functions as mentioned in the Poisson processes MA(4020) Operational Research

  15. this is a partial differential equation of simple linear type already shown in Poisson processes. • the subsidiary equations take the form : MA(4020) Operational Research

  16. the first and the last term gives : • M=constant----(13) • first and second gives : • this is integrated as : MA(4020) Operational Research

  17. this is now in the form shown in solution of partial differential equations • the general solution can now be written as • using the two independent integrals (13) and (14) ,where is an arbitrary function to be determined by the initial conditions. • thus at t=0, MA(4020) Operational Research

  18. thus by writing • we get • hence the arbitrary function is • applying (17) to ( 15) we get • this is the generating function of a negative binomial distribution function MA(4020) Operational Research

  19. It is equivalent to MA(4020) Operational Research

  20. negative binomial distribution • picking out the xn terms we get MA(4020) Operational Research

  21. The mean and the variance for this distribution are given as • Mean value is precisely equal to the value obtained for a deterministically growing population of size n at time t ,in which the increment in time t is nt MA(4020) Operational Research

  22. The simple (pure) death process • With all the assumptions and assuming =0 • We get MA(4020) Operational Research

  23. Then the subsidiary equations are : MA(4020) Operational Research

  24. the first and the last term gives : • M=constant----(25) • first and second gives : • this is integrated as : MA(4020) Operational Research

  25. M(,0)=ea (at t=o , n=a and pa(0)=1) • the general solution can now be written as • using the two independent integrals (25) and (26) ,where  is an arbitrary function to be determined by the initial conditions. • thus at t=0, MA(4020) Operational Research

  26. thus by writing • we get • hence the arbitrary function is • applying (29) to ( 27) we get • this is the generating function of a binomial distribution function MA(4020) Operational Research

  27. Now (30 and (31) can be written as : Let p=e - t and q=1- e - t MA(4020) Operational Research

  28. (34) and (35) can be easily identified as the generating function of the binomial distribution. MA(4020) Operational Research

  29. The population 0(vanishes) as tinfinity. MA(4020) Operational Research

  30. The mean and the variance of the of this distribution are • The stochastic mean decreases according to a negative exponential law. MA(4020) Operational Research

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