4.6.2 Case 2: m Operators, Infinite Number of LinesSuppose now that, while keeping everything else in the emergency call center exactly the same as before, the number of telephone operators is increased to m (> 1). The service time pdf's associated with each operator are identical and negative exponential with parameter ju. When all operators are busy, the next call in line is assigned to the first operator to become free, while when two or more operators are free, the next incoming call is assigned to an operator in some arbitrary way.
The state-transition diagram for this case is shown in Figure 4.7. In terms of the queueing system code, this is a M/M/m system with infinite queue capacity and FCFS service. With respect to our fundamental model,
and, substituting in (4.25),
Expressions for other quantities of interest can now be derived using the steady-state probabilities, Pn.
Limiting case: Infinite number of servers. The limiting extension
of case 2 is when the number of servers m is (countably) infinite. In such a situation no
user of the queueing system will ever have to wait in line. Since in this case we have
This is a remarkable result, stating that the
steady-state probability distribution for the number of users present (and, consequently,
for the number of busy servers as well) in a M/M/ system is Poisson with parameter /. It follows, of course, that 0. Note also
that steady state is inevitably reached in this case, since there are always sufficiently
many servers to assure that the service rate will eventually exceed the rate of arrivals