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Transient and Numerical Solution of a Feedback Queueing System with Correlated Departures
1Department of Statistics, Punjabi University, Patiala, India
Abstract
This paper studies a feedback queueing system with correlated departures. Departures take place only at transition marks. Inter-arrival times and inter-transition times follow exponential distributions. Transient-state queue length probabilities and laplace transform of the generating function of transient-state queue length probabilities are obtained. A few special cases of interest are also derived. Various probabilities relating the model are obtained numerically and are compared graphically.
At a glance: Figures
Keywords: queueing, feedback, server, probability, generating function, numerical solution
American Journal of Numerical Analysis, 2014 2 (1),
pp 20-28.
DOI: 10.12691/ajna-2-1-5
Received December 12, 2013; Revised January 10, 2014; Accepted February 07, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.Cite this article:
- Singla, Neelam, and P.C. Garg. "Transient and Numerical Solution of a Feedback Queueing System with Correlated Departures." American Journal of Numerical Analysis 2.1 (2014): 20-28.
- Singla, N. , & Garg, P. (2014). Transient and Numerical Solution of a Feedback Queueing System with Correlated Departures. American Journal of Numerical Analysis, 2(1), 20-28.
- Singla, Neelam, and P.C. Garg. "Transient and Numerical Solution of a Feedback Queueing System with Correlated Departures." American Journal of Numerical Analysis 2, no. 1 (2014): 20-28.
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1. Introduction
It was [5] who introduced the idea, in discrete time, of dependence of probability of an arrival or of no arrival at a time mark is dependent on the probability of an arrival or of no arrival at the previous time mark. Afterwards a good many research problems are solved in discrete time by [1, 3, 7, 9, 10]. Again [6] handled the same concept in continuous time. Later on [12] studied a number of random memory queueing models for the exact number of arrivals and departures by given time. These type of queues in discrete and continuous time are named in literature as correlated queues.
References [4, 8] solved a number of queueing problems with feedback. However the models solved by them do not assume the dependence of probability of the happening of an event (arrival or departure) at a time mark on the previous time mark. But situations do occur in day-to-day life wherein such dependence is observed. In the present paper, a continuous time single server feedback queueing system in which probability of a departure or of no departure at a time mark is dependent on the probability of a departure or of no departure at the previous time mark, is considered.
Feedback concept is used in the sense that units are given another chance of service if required. However the units will have to leave the system definitely after getting second service. Recently [13] obtained the steady state solution of a feedback queueing system with correlated departures.
The practical situation which corresponds to the above situation can be that of a gymnasium room with some kind of exercising machine, wherein people come for exercising. It is supposed that the departures can occur only at transition points. There may arise two types of situations:
(I) Departure has occurred at previous transition mark.
Now there may be two cases according as
(a) departure occurs.
(b) departure does not occur at the present transition mark.
(II)Departure has not occurred at previous transition mark.
Now there may be two cases according as
(a) departure occurs.
(b) departure does not occur at the present transition mark.
The queueing system investigated in this paper is described by the following assumptions:
1. Arrivals are Poisson with parameter 
.
2. All departures take place only at the transition marks 
where 
, 
are identically distributed random variables.
3. If at any instance, the queue length is 
then the departure probabilities at two successive transition marks are governed by the following assumptions:
(a) If departure has occurred at previous transition mark then the probability of occurring a departure at present transition mark is 
and the probability of not occurring a departure at present transition mark is 
.
(b) If departure has not occurred at previous transition mark then the probability of occurring a departure at present transition mark is 
and the probability of not occurring a departure at present transition mark is 
.
4. The inter-transition times 
are exponentially distributed with mean transition time 
.
5. Units are served in their order.
6. The probability of rejoining the system is 
and that of leaving the system is 
for the units getting service first time, so that 
. However the units will have to leave the system definitely after getting second service.
7. The probability that the unit departs the service channel for the first time is assumed to be 
and that for the second time is 
, so that 
.
8. The stochastic processes involved, viz.
(a) arrival of units
(b) departure of units
(c) transition times
are statistically independent of each other.
2. Description
Probability that there are n units in the system at time t and the next unit is to depart for the first time or second time according as 
or 
and 
departure has occurred at the previous transition mark 
.
Probability that there are n units in the system at time t and the next unit is to depart for the first time or second time according as 
or 
.
 |
= Probability that there are 
units in the system at time 
and 
departure has occurred at the previous transition mark
 |
= Probability that there are 
units in the system at time 
.
 |
 |
Initially,
 |
 |
Elementary probability reasoning leads to the following difference differential equations
 | (1) |
 | (2) |
 | (3) |
 | (4) |
 | (5) |
 | (6) |
Taking the Laplace Transformation 
; Re s > 0 of (1)-(6).
 | (7) |
 | (8) |
 | (9) |
 | (10) |
 | (11) |
 | (12) |
3. Mathematical Analysis of the Problem
Description of the generating functions
 |
 |
 |
and
 |
 |
all for 
or 
with 
Laplace Transformation of Probability Generating Function of transient-state queue length probabilities
 | (13) |
 | (14) |
 | (15) |
 | (16) |
 | (17) |
Where 
 |
Let 
where
 |
 |
 |
 |
Obviously 
have at least one zero inside the unit circle.
Now
 |
for 
 |
Hence 
on 
Since all the conditions of Rouche’s theorem are satisfied, so 
has at least one zero inside the unit circle. Let it be 
. Numerator must also vanish for this zero since 
, and 
are analytical functions of 
. From this we will get one equation in six unknowns 
and 
. Eliminating common factor from equations (10) and (12), we get second equation in above six unknowns for 
. Other four equations required for evaluating six unknowns are equations (8) & (11) and equations (7) & (9) for 
.
Hence the generating functions 
, and 
are completely known. 
and 
can be obtained by using the following formulae
at 
for 
at 
for 
at 
for 
at 
for 
at 
for 
In either case 
, 
and 
can be found by inverting the Laplace transforms 
, 
and 
using [2].
Further 
, as desired
And
 |
Also 
& 
.
4. Special Cases
Case (i). When the units are not given the option of rejoining the system after first service i.e. a queueing system without feedback when departures at two successive transition marks are correlated with inter-arrival times and inter-transition times following exponential distribution. Putting 
, 
; 
, 
, 
, 
, 
, 
; 
in (13), (15, and (17) and on simplification , we get
 | (18) |
 | (19) |
 | (20) |
Case (ii). When departures at two successive transition marks are independent i.e. 
, then (13) to (17) give
 | (21) |
 | (22) |
 | (23) |
 | (24) |
 | (25) |
5. Numerical Solution and Graphical Representation
Following the work of [11], the numerical results are generated from equations (1)–(6) by Runga Kutta Method of order four for solving ordinary differential equations using MATLAB programming. In Figure 1 to Figure 8, various probabilities are plotted versus time for the case when 
.
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Larger image(png format)Figure 2 shows relative change in probabilities 
, 
, 
and 
. All these probabilities first increase then decrease to some extent and finally attain some steady values for higher values of 
. Figure 3 and Figure 4 show respectively the probability plots of 
, 
, 
, 
and 
, 
, 
, 
versus time. All these probabilities firstly increase and finally settle down to some steady values.
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Larger image(png format)Comparison among four probabilities when no departure has occurred at the previous transition mark and the unit at the head of the queue is to join the server for the first time i.e. among 
, 
, 
and 
is done through Figure 5. It is apparent that probability 
decreases as 
increases. It is also seen that all the four probabilities increase rapidly in the starting moments, then 
decreases to some extent and all show slight variability before finally approaching to some steady values. Figure 6 shows similar type of relationship among probabilities 
, 
, 
and 
as shown by Figure 5 among 
, 
, 
and 
with a difference that decrease in 
is not that much pronounced as is there in case of 
(Figure 5). It is also observed that as 
increases, probabilities increase with less rapidity in their initial times as compared to the probabilities with lower 
values.
Figure 7 illustrates the relationship among four probabilities when a departure has occurred at the previous transition mark and the unit at the head of the queue is to join the server for the first time i.e. among 
, 
, 
and 
. It shows that probability decreases as 
(number of units in the system) increases. It is also seen that 
increases rapidly in the starting moments, then decreases to some extent for higher values of 
(average transition times) and show slight variability before finally approaching to a steady value. Other three probabilities increase rapidly in the initial moments, then increase slowly and finally attain some constant values for higher values of 
. Figure 8 shows similar type of relationship among probabilities 
, 
, 
and 
when plotted against time as is shown by the Figure 7 among 
, 
, 
and 
.
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Larger image(png format)To study the effect of various parameters on different probabilities of the model, data of various probabilities is generated for different values of 
keeping other parameters constant. The set of values that 
took is {0.2, 0.3, 0.5, 0.8}. The other parameters were fixed at 
. The probability 
is plotted against time for different values of 
in Figure 9. From the figure it is concluded that the probability 
decreases with increasing 
. So more the 
i.e. more customers are arriving per unit average transition time less is the probability of zero units in the system.
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Larger image(png format)Behavior of the probabilities 
and 
, with respect to time 
(average transition times) and changing 
is apparent from Figure 10 to Figure 11.
Effect of change in probability 
(probability of leaving the system after getting first service) is studied by generating data of various probabilities for different values of probability 
keeping other parameters constant. The set of values that 
took is {0.45, 0.6, 0.75}. The other parameters were fixed at 
. In Figure 12, the probability 
is plotted against time for different values of 
. From the figure it is clear that there is no significant change in the value of probability 
as 
increases. So no significant effect of varying probability 
is observed when there is no unit in the system.
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Larger image(png format)Figure 13 shows probability plot of 
with respect to 
(average transition times) for a set of 
values. It is clear from the graph that in all the three cases probability 
increases rapidly in the starting moments then decreases and finally becomes almost steady for higher values of 
. It is observed that probability 
takes higher values for high value of 
. This can be explained as 
(i.e. zero state) is more probable when 
value is high. Probability 
is plotted vs time in Figure 14. Here it is observed that probability 
takes smaller values for high value of 
as 
(i.e. one state) is less probable when 
takes a high value.
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Larger image(png format)Figure 15 & Figure 16 show the probability plots of 
and 
. These also show similar kind of relationship as shown by probabilities 
and 
in Figure 13 & Figure 14 respectively and also can be explained by same argument.
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Larger image(png format)From the above discussion and on observing all the above graphs, it is interpreted that all the probabilities finally tend to attain some steady value for high values of time 
(average transition times). It is also checked numerically that at any moment sum of all probabilities related to the queueing system is one.
References
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| In article | |||
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| In article | |||
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| In article | |||
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| In article | |||
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