Problems of Information Transmission, Vol. 41, No. 3, 2005, pp. 296–299. Translated from Problemy Peredachi Informatsii, No. 3, 2005, pp. 123–127.
Original Russian Text Copyright
2005 by Lebedev.
COMMUNICATION NETWORK THEORY
Maxima of Waiting Times for the
Random Order Service M|M|1 Queue
A. V. Lebedev
M.V. Lomonosov Moscow State University
Received February 1, 2005
Abstract—A random order service M|M|1 queueing system is considered. A stochastic esti-
mate for the asymptotic distribution of normalized maxima of waiting times and an estimate
for the upper limit almost sure are obtained.
Consider a random order service (ROS) M|M|1 system. In other words, if there are several
customers in queue when the server becomes free, each of them may occupy the server equiprobably
[1, Section 5.12]. From the point of view of dynamics of the number of customers, such a system is
similar to the classical one, but customers get service in a random order, which results in another
waiting time distribution.
Exact asymptotic behavior of the waiting time distribution tail was obtained in . It was found
that this asymptotics is drastically diﬀerent from that observed in the classical M|M|1systemand
has a rather speciﬁc nature. It was also found that, for the waiting times in the random order service
system, both larger and smaller (as compared to the usual) values are typical. This phenomenon
can be explained by the fact that some customers get service “out of turn,” while some others, on
the contrary, have to wait longer. Clearly, deviations towards larger waiting times are unfavorable
for the customers.
Recently, some results were also obtained for heavy tail M |G|1 queues . It is claimed that
such models better describe processes in data processing and data transmission computer networks.
In particular, for the case of a regularly varying tail of the waiting time distribution with index
ν ∈ (1, 2), exact asymptotic behavior of the waiting time distribution tail was obtained, which was
found to be also regularly varying, but with index ν − 1 ∈ (0, 1). Thus, the average waiting time
was found to be inﬁnite, which looks not too realistic.
Below, we ﬁnd a stochastic upper estimate for the asymptotic distribution of normalized maxima
of waiting times in the form of a double exponential law and an estimate for the upper limit almost
Note that precise results in this direction, not only for the M|M|1 but even for a GI|G|1 queue 
with a standard service discipline (FIFO) were known long ago.
2. MAIN RESULTS AND COMMENTS
In what follows, we assume for simplicity that the input ﬂow intensity equals one, and the
average service time is ρ ∈ (0, 1), i.e., coincides with the system load.
03-01-00724, 04-01-00700 1758.2003.1.
2005 Pleiades Publishing, Inc.