. An M/M/1 queue with control-limit policy and exponential start-up time (Baker, 1973).

Here the control policy is to turn off the system and withdraw the server when the system becomes empty and to turn on the system when the system size reaches n(>0). When the system is turned on, it cannot immediately serve customers but requires some time to start up. It is assumed that the time required for start-up is exponential with mean 1/y. The interarrival and service-time distributions are exponential with means 1/À and 1/u, respectively. Suppose that pis denotes the steady-

state probability that there are i customers in the system and the server state is s (where s = 0 implies an idle state of the server in which the start-up has not begun or has not been completed) and s = 1 denotes the busy state of the server (in which service is being performed). Suppose that P; = Pi.o + Pi,1 is the steady-state probability that there are i in the system.

Denote

Show further that P = x(1 -p'+'), on where p / 0, and @ = (1 - 0)/[0 + n(1 - 0)]. Further show that the mean number in the system E ( N) equals E(N) = EiP = 00 an(n-1)
(p - 200 +0)
+
2 (1 - p)(1-0)
1=0 See Borthakur et al. (1987) for a more general model, and Böhm and Mohanty (1990) for transient solution (through discrete-time analogue and use of combinatorial arguments) of M/ M/1 queue under control limit policy and zero start-up time. See also Lee (1990).

p ==(