M/M/∞ Queue with ON-OFF Service Speeds

Abstract

In this paper we analyze of a special kind of M/M/∞ queueing system. We assume that the rate of the servers alternates between two speeds according to an ON-OFF process whose ON periods are exponentially distributed and with general OFF periods. We look at the particular case where the OFF service rate is zero while the distribution of the OFF periods is heavy tailed, and for this case we derive the tail of the number of customers in the system. This model be applied to transportation systems where the M/M/∞ represents a delay line, such as a highway, whose crossing time may be unexpectedly interrupted for a long time by some accident.

M/M/1 queue in two alternating environments and its heavy traffic approximation

“…Moreover, recalling that η 1 + η 2 > 0, Eq. (10) shows that the existence of the equilibrium distribution is guaranteed if and only if one of the following cases holds:…”

“…Since P (ξ 3 ) = 0, to ensure the convergence of the probability generating functions (8), we impose that their numerators tend to zero as z → ξ 3 . Hence, by virtue of (10), one has (see also Eqs. ( 26) and ( 27) of [39]):…”

“…Moreover, Boxma and Kurkova [5] studied an M/M/1 queue for which the speed of the server alternates between two constant values, according to different time distributions. A similar problem for the M/G/1 queue was studied by the same authors in [6], whereas the case of the M/M/∞ queue is treated by D'Auria [10].…”