1 / 21

The G/M/1 queue

The G/M/1 queue. G. U. Hwang Next Generation Communication Networks Lab. Department of Mathematical Sciences KAIST. Model description. Ref: L. Kleinrock, Queueing systems, vol. I, John Wiley & sons, 1975, chapter 6 (or section 6.4).

shaun
Télécharger la présentation

The G/M/1 queue

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The G/M/1 queue G. U. Hwang Next Generation Communication Networks Lab. Department of Mathematical Sciences KAIST

  2. Model description • Ref: L. Kleinrock, Queueing systems, vol. I, John Wiley & sons, 1975, chapter 6 (or section 6.4). • arrival process: inter-arrival times are independent and generally distributed with distribution function A(x) • service times are independent and exponentially distributed with rate  • we have a single server in the system Next Generation Communication Networks Lab.

  3. To analyze the G/M/1 queue we construct an embedded DTMC where the embedded points are the epochs just before arrivals. server queue Next Generation Communication Networks Lab.

  4. Basic notation • qn : the number of customers in the system just before the arrival instant of the n-th customer Cn • vn+1 : the number of customers served between the arrivals of Cn and Cn+1 • In+1 : the interval between arrival times of Cn and Cn+1, distribution function A(x) • Then, by the structure of the G/M/1 queue we have qn+1 = qn + 1 - vn+1, n¸ 0, q0 = 0. • Further, we see {qn} forms a DTMC. • Note that qn+1· qn +1. Next Generation Communication Networks Lab.

  5. One step transition probability matrix P • Let i+1-j departures ……. Next Generation Communication Networks Lab.

  6. Therefore the one step transition probability matrix P is given by • Since all elements are positive, the DTMC is irreducible and aperiodic. • The chain is skipfree to the right. Next Generation Communication Networks Lab.

  7. The stationary distribution • Let's assume that the DTMC is positive recurrent. • For k¸ 0, first compute P{ n visits to state k+1 between two successive visits to state k} • Observe that once the DTMC goes below state k+1, it should pass through state k before returning to state k+1. Next Generation Communication Networks Lab.

  8. Let k k+1 ………. Next Generation Communication Networks Lab.

  9. Then for n¸ 1, we get • Accordingly, for k¸ 0 which is independent of k Next Generation Communication Networks Lab.

  10. Let • Note that the stationary distribution k+1 is given by Next Generation Communication Networks Lab.

  11. Since • we have k+1 = k for k¸ 0. • Hence, we get Next Generation Communication Networks Lab.

  12. The computation of  •  = (0, 1, ): the stationary probability vector • From  P = , we get i=k-11ii-k+1 = k for k¸ 1 • Putting i = 0i for i¸ 1 and substituting them into the above equations yield  i=k-11i-k+10i = 0k for k¸ 1 Next Generation Communication Networks Lab.

  13. Hence, we get where A*(s) is the LT (LST) of A(x). • Note that, if 0 <  < 1, then we have a stationary distribution . Next Generation Communication Networks Lab.

  14. For ¸ 0, define f() = A*(-). • We know that 0 < f(0) = A*() < 1 and f(1) = A*(0) = 1. Further, where  =1/(E[A] ). Next Generation Communication Networks Lab.

  15. Therefore, if = 1/(E[A] ) = / < 1, then there exists a unique solution * between 0 <  < 1, and in this case, the stationary distribution exists. Next Generation Communication Networks Lab.

  16. Computation of the constant 0 • From the normalizing condition n=01n = 1, we get 0 /(1-) = 1. • Hence, 0 = 1 - . • Therefore, n = (1-) n for all n¸ 0. Next Generation Communication Networks Lab.

  17. The distribution of waiting time in the queue • Consider the distribution of the waiting time Wq in the queue. Observe that • When an arrival sees no customer in the system, the waiting time of the arrival in the queue is 0. • When an arrival sees n (¸ 1) customers in the system, the waiting time in the queue is i=1n Yi where Yi» exp(). Next Generation Communication Networks Lab.

  18. Next Generation Communication Networks Lab.

  19. Conditional distribution of the queue length • Consider the conditional distribution of the queue length, given that a customer must queue. • Then • The conditional queue length distribution, given that an arrival queues, is geometric. Next Generation Communication Networks Lab.

  20. Conditional distribution of waiting time in the queue • Consider the conditional distribution of the waiting time in the queue, given that a customer must queue. • Let Wq*c(s) be the LT of the conditional waiting time in the queue. From the fact that Next Generation Communication Networks Lab.

  21. we obtain • Therefore, the conditional waiting time in the queue for the G/M/1 queue is exponentially distributed with (1-). Next Generation Communication Networks Lab.

More Related