ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 1, pp. 14–23.
Pleiades Publishing, Inc., 2016.
Original Russian Text
V.N. Tarasov, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 1, pp. 16–26.
Analysis of Queues with Hyperexponential
V. N. Tarasov
Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia
Received November 17, 2014; in ﬁnal form, November 10, 2015
Abstract—We study H
/M/1, and M/H
/1 queueing systems with hyperexponen-
tial arrival distributions for the purpose of ﬁnding a solution for the mean waiting time in the
queue. To this end we use the spectral decomposition method for solving the Lindley integral
equation. For practical application of the obtained results, we use the method of moments.
Since the hyperexponential distribution law has three unknown parameters, it allows to ap-
proximate arbitrary distributions with respect to the ﬁrst three moments. The choice of this
distribution law is due to its simplicity and the fact that in the class of distributions with coef-
ﬁcients of variation greater than 1, such as log-normal, Weibull, etc., only the hyperexponential
distribution makes it possible to obtain an analytical solution.
In modeling traﬃc in modern telecommunication networks, exponential distributions are widely
used, which include the Weibull, log-normal, hyperexponential, and other distributions whose co-
eﬃcients of variation are greater than 1 (c
> 1) for certain values of parameters. The coeﬃcient
of variation greater than one 1 indicates that the probability of large values of the random variable
is much greater than for the classical exponential distribution. As is known, e.g., from , for the
G/G/1 queueing system (QS) the mean waiting time is given by
2(1 − ρ)/λ
where ρ is the system load factor (0 <ρ=
< 1), λ is the arrival ﬂow rate, μ is the service rate,
are variances of the arrival interval length and service time, respectively, I and I
the mean and the second initial moment of an idle period. Hence, the ﬁrst term on the right-hand
side of (1) is quadratic in the coeﬃcients of variation of the arrival and service intervals. The second
term on the right-hand side of (1) remains unknown, and it is quite possible that it may depend on
moments of arrival intervals and service times of orders higher than two. Therefore, in the analysis
of the G/G/1 QS one should take into account not only the ﬁrst two moments of random arrival
and service intervals but also moments of higher orders.
Now we pass to the deﬁnition of a hyperexponential distribution law. A distribution with the
density function f(t)=
,wheret ≥ 0and
and is denoted by H
(see ). It is proved in  that the coeﬃcient of variation of a random
variable distributed according to this law is greater than one. A similar deﬁnition of H
respect to a probability distribution function is given in .