## 4.8.4 G/G/m Queueing Systems

About the only general results that have been obtained to date for the G/G/m case are in the form of quite loose upper and lower bounds on average steady-state queueing characteristics. [BRUM 71]. These bounds are often computed by, first, comparing a G/G/m system with a G/G/1 system that has the same "service potential" as the G/G/m system (i.e., the single server in G/G/1 works m times as fast as each of the servers in G/G/m) and, then, by using the earlier results on G/G/1.
The best generally applicable bounds on the average waiting time in queue which have been published to date for G/G/m systems give the inequalities

where , , and E[S] are the service rate, variance of service time, and s second moment of service time, respectively, for each of the m servers. is the average waiting time for a G/G/1 system with a service time described by a random variable S* = S/m (i.e., with service m times faster than that of each of the m servers in the G/G/m system) and with an arrival process identical to that for the G/G/m system.
For , one should obviously use either an exact expression, if one is available, or, as is more likely, a lower bound on by using (4.92) or, if applicable, (4.94). For example, for the M/G/m queueing system, one should use the exact expression (4.81) for with 1/m and    /m, for  the expected value and variance of the service times, respectively.
Finally, a result analogous to the heavy-traffic approximation  for G/G/1 systems has also been derived recently [K0LI, 74] for G/G/m systems. This result states:

For m 1 in a G/G/m system, the waiting time in queue under steady-state conditions assumes a distribution that is  approximately negative exponential with mean value

Note once more that expected waiting time is dominated by a (1 - ) term, as approaches 1 ( = /m for multiserver systems).