. Multiserver queue with balking and reneging. Consider a c-server queueing system with Poisson input with rate > and exponential service time with rate u for each of all c-servers. Suppose that (i) an arriving customer who finds all the c-servers busy on arrival may balk (leave without joining the system) with probability q or may join the system with probability p(=1 - q); (ii) after joining the queue, a customer may renege independently of others; he waits for a random length of time for service to begin, the length of time being an exponential random variable with parameter a; otherwise he departs; and (iii) a customer who balks or reneges and decides to return later is considered as a new arrival independent of his previous balking or reneging. Find the differential-difference equations of the state N(t) of the sys- tem. If p,(t) P{N(t)=n) and lim, P(t)= p, then show that Pn = 1 n! (A) Po, (p)"-c (c+)(c + 2a)... [c + (n c )] Per - n > c.

Find p ,, when there is only balking and no reneging. Also deduce Erlang's loss formula (Haghighi-Montazer et al ., 1986). See also Abou-El-Ata and Hariri (1992) and Mohanty et al. (1993) for some generalizations.