The G/M/1 queue

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.