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

form4.96.gif (8932 bytes)

where mu.gif
(189 bytes), sig2s.gif (1026
bytes), and E[Ssquare.gif (861
bytes)] are the service rate, variance of service time, and s second moment of service time, respectively, for each of the m servers. w1q.gif (1065 bytes) 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 w1q.gif (1065 bytes), one should obviously use either an exact expression, if one is available, or, as is more likely, a lower bound on w1q.gif (1065 bytes) 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 w1q.gif (1065 bytes) with 1/mmu.gif (189 bytes) and    sig2s.gif (1026
bytes)/msquare.gif (861
bytes), 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 lamda.gif
(291 bytes)mmu.gif (189 bytes) rarrow.gif (63
bytes) 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

form4.97.gif (4214 bytes)

        Note once more that expected waiting time is dominated by a (1 - rho.gif (189 bytes))neg1.gif (875 bytes) term, as rho.gif (189 bytes) approaches 1 (rho.gif (189 bytes) = lamda.gif (291 bytes)/mmu.gif (189 bytes) for multiserver systems).