(a) Use the birth and death model described in Figure 6.4 to find the steady-state PMF for the number of customers in the system (queue plus service facility) for the following queues:
(i) M/M/1 with arrival probability λδ, service completion probability μδ;
(ii) M/M/m with arrival probability λδ, service completion probability iμδ for i servers busy, 1 ≤ i ≤ m;
(iii) M/M/∞ with arrival probability λδ, service probability iμδ for i servers. Assume δ so small that iμδ 1 for all i of interest. Assume the system is positive recurrent.
(b) For each of the queues above give necessary conditions (if any) for the states in the chain to be (i) transient, (ii) null recurrent, iii) positive recurrent.
(c) For each of the queues find: L = (steady-state) mean number of customers in the system; Lq = (steady-state) mean number of customers in the queue; W = (steady-state) mean waiting time in the system; Wq = (steady-state) mean waiting time in the queue.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.