Mathematics /4. Applied mathematics
Nsangou M. M.
People’s
Friendship University of Russia, Moscow, Russia
ANALYSIS OF THE QUEUE
MODEL WITH THE BUSY PERIOD, ORDINARY AND WORKING VACATIONS
In this paper, we consider a simplified
vacation model, investigating a system with a fixed size of the costumers
queue. When the server becomes empty, it either goes on an ordinary vacation or
takes a working vacation. In the working vacation, a costumer is served at a
lower rate, and at the instants of the service completion, the vacation is
interrupted and the server resumes to a regular busy period or continues the
vacation according to Bernoulli schedule.
Keywords: server vacations, working vacations, Bernoulli
vacation schedule.
Introduction
The
vacation queue models have been investigated extensively in view of their
application in computer systems, production managing, communication networks
particularly the IP access networks [3]. In a classical vacation queue, the
server completely stops serving customers and may do some additional works or
maintain servers during a vacation [2]; while on a working vacation, the server
continues to work at a lower rate. The goal of the paper is to model some
server ability characteristics during a busy cycle, fixing the size of the
costumers queue as for some really systems. The recurrent formulas of the
stationary probabilities are obtained, the state or blocking probabilities and
queue length managing parameters are evaluated.
1. Model
description
In this
section, we make the following assumptions to describe the model with both
ordinary and working vacations. Costumers reach the system according to a
Poisson process with compound intensity
and there is one server
in the system. Service times are assumed to be exponentially distributed with
mean
; the rate is
. As soon as the server finishes a service, it can begin a vacation of
random variable, and takes an ordinary vacation or a working vacation with
probability
, where
. The ordinary and working vacations lengths are exponentially
distributed with parameters
and
respectively. During an ordinary
vacation the server will stops serving even if there are new arrivals in the
system. In the working vacation, costumers are served at a lower rate
; furthermore, at the instants of service completion, the vacation is interrupted
and the server resumes to a regular busy period with probability
(if there are costumers in the queue) or continues the vacation with probability
. The decisions of choosing vacations as well as interruptions are
mutually independent and all aforementioned variables are independent of each
other; the service order is First Come First Served.
2. Model
analysis
Let be the number of costumers in the system at time
and let define,
, the states
of the server, respectively for working, ordinary vacations and busy period at
time t. Let consider, that all the steady state probabilities
existence conditions are realized, then
can be analyzed as a QBD (Queuing
Birth and Dearth) process with states space
.
Under
the steady state conditions, and according to the exponentially distributed
service delay, is a
standard continuous-time Markov chain. The followed system
can be illustrated by the states transition diagram in figure 1, with
a queue size is fixed and equal to
.
Fig. 1. State
transition diagram of the QBD process
Let, be the steady state probability and
; from the Markov chains theory, it follows that
has a unique
equilibrium distribution which satisfies the following family of equations.
;
;
And the partial balance equations system,
;
.
To find the state probabilities, let define and
, we will
check
,
as
so that,
and
,
. Making
required transformations, and solving the systems of above equations, we can enounce
the next algorithm for
, which calculate the distribution
,
.
Step 1. Finding of the solution of the recurrent equation:
.
Step 2. Calculating of
the probabilities constants:
,
,
and
, where,
;
and
.
Step 3. Calculating of the parameter:
, where,
intermediate constants
,
,
.
Step 4. Calculating of all steady state probabilities, as
follows
, where,
;
and
.
3. Distribution
of the server ability characteristics
The
most important QoS parameters are the different types of probabilities, the
queue length and average sojourn time of a costumer in the system. Let define states
probabilities as follows: ;
;
, the probabilities that the server is in “regular vacation”, “busy period” and “working vacation”
respectively and can be expressed as:
Another important length characteristic is the
blocking probability defined as:
For the
average queue length analysis and the sojourn time of a costumer in the system,
we use the probability generating function (pgf) of the queue and, after some transformations,
we obtained,
;
,
.
The
average queue length can be founded as:
;
,
Concluding this part, let remark that,
.
è
is the average queue length for the ordinary Ì/Ì/1/R model.
The sojourn time is founded using the Littla
formula as follows:
.
Furthermore, the highest load time can be
measured trough the variation and derivate parameters like:
;
- Queue length variation.
4. Numerical
analysis
In this section, we take the initial values from [2] and [5] under the
assumptions that. So let take
,
,
,
,
, maximum of
and
.
Fig. 2. The average sojourn
time in function of intensities and
The fig. 2 demonstrates that the model we built,
satisfies the vacation type one as the costumer sojourn time property stochastically
was proofed. As we can observe, the growth of the customers or the vacation
service loads improve the sojourn time.
Conclusion
The work in this paper can be used to model many practical problems. For example, the wireless mobile
devices accessing the wireless networks with different
data rates can also be analyzed. Numerically, the decomposition property of the
queue length and sojourn time distributions was proofed and the algorithm of
steady state probabilities calculation was founded; using these results, we
defined the states and blocking probabilities, the queue length and his
variation during a time period.
References
1. Hongbo Zhang and
Dinghua Shi, The M/M/1 Queue with Bernoulli-Schedule-Controlled Vacation and
Vacation Interruption. International Journal of Information and Management
Sciences 20 (2009), 579-587.
2. Naishuo Tian ,
Xinqiu Zhao and Kaiyu Wang, The M/M/1 Queue with Single Working Vacation. International
Journal of Information and Management Sciences Volume 19, Number 4, pp.
621-634, 2008.
3. Servi, L. and Finn,
S., M/M/1 queues with working vacations (M/M/1/WV), Perform. Eval., Vol.50,
pp.41-52, 2002.
4. Neuts, M.,
Matrix-Geometric Solutions in Stochastic models, Johns Hopkins University
Press, Baltimore, 1983.
5.
C. Chi, R.Hao, D.Wang, Z.Cao “IMS Presence Server: Traffic Analysis
& Performance Modelling.