## 4.10.1 Important Property of M/M/m Queueing Systems

For M/M/m queueing systems with infinite queue capacity, we now state a property that often plays an important simplifying role in the analysis of queueing networks. It is sometimes referred to as the equivalence property for M/M/m systems.

Let the arrival process at a M/M/m queueing system with infinite queue capacity have parameter . Then, under steady-state conditions  (i.e., for < m), the departure process from the queueing system is also Poisson with parameter .

The proof of this property is quite involved. However, it is relatively straightforward for the special case M/M/1 (see Problem 4.10).

The implications of the property for the analysis of queueing networks are quite obvious. If some "component" (facility) of a queueing network can be modeled as a M/M/m system with infinite capacity, the "output" of this "component" is also Poisson with parameter . That is, users will leave this specific facility according to a probability distribution identical to the probability distribution for the arrival of users at the facility.

Thus, if the served users of the M/M/m facility are subsequently routed to another facility, the arrival process to this other facility is a Poisson process. In fact, it is clear that if

1.    the queueing network consists of, say, K facilities in series (Figure 4.17), each of which
contains m1, m2, . . . , mk (mi = 1, 2, 3, ...) identical servers with negative exponential
service and rates i, (i =1, 2, . . . , K);

2.    there is infinite queueing space between successive facilities; and

3.    the arrival process for the first facility (facility 1 in Figure 4.17) is Poisson with rate ;

then, under steady-state conditions, the queueing network can be analyzed as K independent M/M/m queues and the results of Section 4.6 are directly

FIGURE 4.17 Poisson input at rate at queueing system 1 will result in a Poisson input at rate for all four queueing systems.

applicable. The condition for steady state in this case is that < mii, for all i(i = 1, 2, . . . , K).