Problems of Information Transmission, Vol. 39, No. 3, 2003, pp. 309–316. Translated from Problemy Peredachi Informatsii, No. 3, 2003, pp. 87–94.
Original Russian Text Copyright
2003 by Lebedev.
COMMUNICATION NETWORK THEORY
The Gated Inﬁnite-Server Queue with
Unbounded Service Times and Heavy Traﬃc
A. V. Lebedev
M.V. Lomonosov Moscow State University
Received June 7, 2002
Abstract—A queueing system with inﬁnitely many servers is considered where access of cus-
tomers to service is controlled by a gate. The gate is open only if all servers are free. Customers
are served in stages. We study the asymptotic behavior of the number of customers served in
a stage and of the stage duration in the heavy traﬃc regime in the case where the service time
distribution belongs to the attraction domain of a double exponential law.
A gated inﬁnite-server queue is a queueing system with inﬁnitely many servers where access of
customers to service is controlled by a gate. We assume that the gate is open only in the case of
all servers free. Arrivals enter an inﬁnite queue in a Poisson stream of a constant intensity λ,and
the service is performed in stages.
At the beginning of a stage, when the gate is open, all customers from the queue instantaneously
access the servers and then are served in parallel and independently until all servers are totally
free. The stage duration is equal to the maximum of service times of these customers. At the point
when all servers become free, the gate opens again for new customers and the next stage. If the
queue is empty, the system waits for the ﬁrst arrival, which opens a new stage.
Among applications of this model are data transmission stations, where “servers” are indepen-
dent communication channels. Another application is task-oriented simulation, where tasks are
processed in parallel. Parallel simulation requires synchronization points to ensure the correctness
of computations. A gating mechanism can provide the synchronization at each computation stage.
Gated inﬁnite-server queues were considered in [1–3]. In , the possibility for the system to go
on vacation if the queue is empty was also considered. In this case, a new stage starts no with one
customer but with a random number of customers arrived during the vacation.
Previously, the main attention was paid to the case of bounded (especially, uniformly distributed)
service times, with values in the interval [0,T
]. Clearly, in this case, as λ →∞, the stage du-
ration tends to T
, and the number of customers served during the stage is approximately Pois-
sonian with mean λT
, which, in turn, admits Gaussian approximation. The case of unbounded
(in particular, exponential) service times was studied less, and mostly by numerical methods.
In the present paper, we study the case of unbounded service times with distribution belonging
to the attraction domain of a double exponential law [4, 5]. Results on the asymptotic behavior of
the number of customers served in a stage and of the stage duration as λ →∞are obtained.
Supported in part by the Russian Foundation for Basic Research, project nos. 00-01-00131 and 03-01-
2003 MAIK “Nauka/Interperiodica”