Problems of Information Transmission, Vol. 40, No. 3, 2004, pp. 237–242. Translated from Problemy Peredachi Informatsii, No. 3, 2004, pp. 62–68.
Original Russian Text Copyright
2004 by Lebedev.
COMMUNICATION NETWORK THEORY
Gated Inﬁnite-Server Queue with Heavy Traﬃc
and Power Tail
A. V. Lebedev
M.V. Lomonosov Moscow State University
Received March 10, 2004
Abstract—An inﬁnite-server queueing system is considered where access of customers to ser-
vice is controlled by a gate. The gate is open only if all servers are free. Otherwise, customers
are put on a queue. Asymptotic behavior of the system in heavy traﬃc is studied under the
assumption that the service time distribution has a power tail.
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. Thus the gated service policy is deﬁned. 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 the gate opens, and all customers from the queue instantaneously
access the servers. Then they 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.
Analysis of such queues was initiated in [1,2] and then continued in [3, 4]. 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 not with one customer but with a random number of customers arrived during the vacation.
Among possible applications, models of data transmission stations with independent communi-
cation channels, models of parallel computing with a large number of processors, and models of
distributed computing were discussed in [1,2].
Note that the system is very easy to operate: there is no need for permanent registering of
arriving and leaving customers, free and occupied servers, etc. Allocation of customers to servers
(which are currently free) is made once only, at the beginning of each stage. Another advantage of
a gated queue can appear in the situation where customers in the queue and servers are somehow
separated, and establishing a connection requires certain expenses. For example, it can be not
proﬁtable (or impossible) to always keep on a data transmission channel; short connections from
time to time can be preferable.
Of course, an inﬁnite-server queue is only an approximation for the case where the actual number
of servers is large. On the other hand, such queues are worth studying for estimation of various
characteristics of the service grade (which, for any ﬁnite-server queueing system can only be worse).
In the analysis of gated inﬁnite-server queues, the main attention was previously paid to the case
of bounded (especially, uniformly distributed) service times. The case of unbounded (in particular,
Supported in part by the Russian Foundation for Basic Research, project nos. 03-01-00724 and 04-01-
2004 MAIK “Nauka/Interperiodica”