analysis of a multi-server markovian queue with working...
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P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
ISSN 2250-0588, Impact Factor: 6.452, Volume 08, Special Issue, June 2018, Page 10-24
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Analysis of a Multi-server Markovian Queue with Working Vacations
and Impatience of Customers
P. Vijaya Laxmi
Assistant Professor, Department of Applied Mathematics, Andhra University,
Visakhapatnam, India
T. W. Kassahun
Research Scholar, Department of Applied Mathematics, Andhra University, Visakhapatnam,
India
Abstract
This paper deals with a multi-server Markovian queue with discouraged arrivals and
reneging of customers due to impatience. The matrix geometric method and truncation
procedures are utilized to derive the probability distributions of the queue length and other
system characteristics. A cost function is constructed to determine the optimum value of the
service rate that minimizes the cost function using a quadratic fit search method (QFSM). In
addition, we have included numerical results to show the effects of the system parameters on
the various performance measures of the system.
Keywords— Multiple working vacations, Discouraged arrivals, Matrix geometric method,
Optimization, Queue, Quadratic fit search method.
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
ISSN 2250-0588, Impact Factor: 6.452, Volume 08, Special Issue, June 2018, Page 10-24
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INTRODUCTION
Queueing theory is used to model many real life problems involving congestion in various
fields of applied sciences. Organizations like banks, airlines, telecommunication companies
and police departments make use of queueing models to manage and allocate resources in
order to respond to demands in a timely and cost efficient fashion. When a queue is formed
and the size of the queue gets longer, the interest of the customer to join the queue declines
and hence may balk (decide not to join the queue) or discouraged to join the queue. Abdul
Rasheed et. al. [1] have studied discouraged arrival of Markovian queueing systems whose
service speed is regulated according to the number of customers in the system. A queueing
system where a customer is discouraged from joining the queue is referred to as a
discouraged arrival queueing system. After joining the queue, a customer would stay in the
queue until a certain level of patience after which he will decide to leave the system before
getting service. This phenomenon is known as reneging. In this model, we have considered
both discouragement of arrivals and reneging of impatient customers.
The concept of working vacation has been discussed in a number of literatures on queueing
theory. Unlike the classical vacation where the server stops service during vacation, during
working vacation, the server still provides service with a slower rate. Other maintenance or
secondary activities could be accomplished during working vacation without completely
stopping the service. When a vacation ends, if there are customers in the queue, a service
period begins and the servers serve the queue with the regular busy period service rate,
otherwise on return from a vacation, if there is no customer waiting in the queue, the servers
take another working vacation and continues to do so until it finds at least one customer
waiting in the queue at a vacation termination epoch. Such a vacation policy is known as
multiple working vacation [12]. Servi and Finn [7] have analyzed the classical
single server vacation model in which a server works at a different rate rather than completely
stopping service during the vacation period. Baba [8] considered a queue with
vacations such that the server works with different rates rather than completely stopping
service during a vacation period and derived the steady-state distributions for the number of
customers in the system and for the sojourn time of an arbitrary customer. Jain and Jain [9]
studied a single server working vacation queueing model with multiple types of server
breakdowns.
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
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Matrix geometric method has been used in different literatures [3]–[6] to solve queueing
systems that give rise to a repetitive structure of the state transitions. Some models could be
truncated to approximate the matrix geometric structure through the assumption that after a
certain point onwards, the state transitions repeat themselves.
In this paper, our focus is on a multi-server synchronous multiple working vacation queue
with reneging of customers due to impatience and discouragement of arrivals. The inter-
arrival times of customers are assumed to be independent and exponentially distributed. The
service times during a busy period, working vacation period, vacation times and reneging
times are all assumed to be exponentially distributed. We have used matrix geometric method
and recursive approach to get the steady-state system length distributions using the rate
matrices and the Neuts and Rao’s truncation method [10]. Some performance measures have
been discussed and a cost model is constructed to determine the optimum service rate that
minimizes the cost function.
MODEL DESCRIPTION
We consider an queueing model where the arrivals are discouraged as the queue size
increases. The arrivals follow a Poisson distribution with parameter where
(1)
If a server is available upon arrival of a customer, the customer is served immediately. The
service facility consists of c identical servers and an infinite waiting space. The servers go
for a synchronous working vacation as soon as the system is empty. The service rates during
busy period and working vacation period are both exponentially distributed with parameters
and respectively where . When the vacation ends, if the system is found empty,
the servers go for another working vacation. Otherwise, they will start service with the
regular busy period service rate . The vacation period is also exponentially distributed
with parameter . Due to impatience, customers may renege from the system with some
probability . The reneging times are assumed to follow exponential distribution with
parameter , where
(2)
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
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where in equation (1) and equation (2) is a sufficiently large fixed integer such that the
discouragement rate and the rate of reneging keeps the same once the number of customers in
the system reaches . The inter arrival times, the service times, the vacation times and the
reneging times are all assumed to be identically and independently distributed. In addition,
service is rendered on the basis of first-come-first-served.
The system can be modelled by a two dimensional continuous time Markov process
, where is the number of customers in the system at time and
is the state of the server at time such that
The state space of the Markov process is
Let be the steady-state probability that the system is in state and there are number of
customers in the system.
The difference equations that govern the Quasi birth-death model are the following:
Let and Then, the system of equations given
above can be expressed in matrix form as
, (3)
where and the infinitesimal generator matrix is given by
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
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0 0
0 1 1
2 2 2
c c c
N N N
A C
B A C
B A C
Q B A C
B A C
,
where ,
where
,
and
with
,
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
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Stability Conditions
If the number of customers waiting in the queue go beyond a certain value (say ), the
majority of the waiting customers fail to get served and so they do not change the state of the
system. Thus, the number of customers who can possibly be served can be restricted
(truncated) to an appropriately chosen number . The main idea of truncation method is to
change the original intractable model to a
tractable one by altering or restricting the capacity of the queue. Neuts and Rao [10] have
proposed a useful method to make the number of waiting customers as a constant after a
sufficiently large number of customers waiting in the queue and solve the resulting system by
matrix geometric method. Domenech-Benlloch et al. [2] also discussed the use of generalized
truncated methods which approximate the retrial queue by some infinite but solvable
system based on the work of Neuts and Rao [10]. We determine the stability conditions using
Neuts and Rao’s truncation method. We assume that for a sufficiently large N, the reneging
rate remains constant at . Thus, we assume that
The truncated model then becomes a level independent whose
infinitesimal generator matrix is given by According to the theorem given by Neuts
[11], it is known that the necessary and sufficient condition for the existence of probability
vector at steady-state is
, (4)
where represents the invariant probability of the matrix . The
equations and where are satisfied by . Subramanian et
al. [13] indicated that since there is no clear choice of N, we may start the iterative process by
taking say N = 1 and increase it until the individual elements of P do not change significantly.
That is, if denotes the truncation point, then
(5)
Where is a predefined error tolerance. Therefore, the truncated system and
is stable if and only if .
STEADY-STATE ANALYSIS OF THE MODEL
From equation (3), we have the following system of equations:
(6)
(7)
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
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(8)
(9)
where .
A. Rate Matrices
For , under the stability conditions and substitution of the matrices
, and assuming an upper triangular rate matrix , we can show that
the equation
(10)
has a minimal nonnegative solution given by
(11)
where
,
with
For under the stability conditions and substitution of the matrices
and assuming an upper triangular rate matrix , we can show
that the equation
, (12)
where
,
where
,
− and −1= − +100 − +1 is satisfied by the rate matrix
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
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, (13)
where
,
,
,
,
, ,
, ,
, .
Here and , are obtained from for .
For , under the stability conditions and substitution of the matrices
and and assuming an upper triangular rate matrix we can show
that the equation
, (14)
where
,
and
is satisfied by the rate matrix
, (15)
where
,
and
.
Here can be computed from the entries of and are obtained from
for .
B. Steady-state Probabilities
Theorem 3.1: Under the assumption that the stability conditions are satisfied, the steady-
state vector is given by
, (16)
and
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
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, 1 (17)
where and
Proof: For , substitution of equation (16) in equation (9) gives
(from
equation 10).
Similarly, for , substitution of equation (17) in equation (8) gives
(from
equation 12).
And for , substitution of equation (17) in equation (8) gives
(from
equation 14).
Furthermore, from the normalization condition, we have that where is a column
vector whose entries are ones. This means that the sum of all probabilities equals one. Thus
we have
Since and
such that and
, using the
theorem we obtain
where
and
, .
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
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PERFORMANCE MEASURES
In this section, we shall derive some important performance measures of the queueing system
under consideration as given below:
Probability that the system is in working vacation ( and probability that the system
is busy ( are given respectively as:
.
Expected system size during busy period ( ), expected system size during vacation
period ( ) and expected system size ( ) are given respectively as:
.
Expected number of customers served ( ) is given as:
Expected waiting time in the system ( ) is given as:
where with .
Average discouragement rate , average reneging rate and average loss rate
are given respectively as:
,
where .
NUMERICAL RESULTS
In this section, we shall present the effect of the system parameters on some performance
measures using tables and graphs.
From Table 1, we can see that with the increase in the rate of arrival ( results in the increase
of the performance measures and . That is, as the rate of arrival increases, the number
of customers joining the system increases, and this is accompanied with a very high loss rate
of the system.
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On the other hand, the same increase in the arrival rate ( does not significantly affect the
performance measures and .
We can see from Table 2 that with the increase in the number of servers generally
increases since the presence of more servers increase the service rate and hence the customers
will be served and the likelihood of the system to go to empty state increases. On the other
hand, the expected waiting time decreases due to the increasing service rate. The loss rate
also decreases since customers get the service with a short waiting time. The queue size
shows an increase as increases from 3 to 5 and then starts to decreases afterwards. This is
an indication of the fact that as initially the increase in the number of servers attracts more
customers and hence an increase in the queue size but after a certain number of servers (an
optimum number), the size starts to decline because the service rate dominates the
corresponding increase in the queue size.
Table 3 shows the effect of the busy period service rate on the performance measures. With
the increase in the service rate , the probability of empty state increases, where as the
queue size , loss rate and waiting time decrease. This is logical since higher service
rate implies faster service thus shorter waiting time, small queue size (since customers get
served quickly) and low loss rate. On the other hand, the probability of the system to go to
empty state increases since the customers would be served at a higher service rate and the
system becomes empty.
Table 1 Effect of on performance measures
30 0.1271 2.4742 0.0875 5.1744 1.7345
35 0.1257 2.5971 0.0801 5.1485 2.6180
40 0.1237 2.7150 0.0747 5.1293 3.7036
45 0.1157 2.9254 0.0741 5.1229 5.5788
50 0.1183 2.9497 0.0677 5.1050 6.5089
55 0.1149 3.1916 0.0639 5.0922 10.1943
60 0.1111 3.1916 0.0639 5.0922 10.1943
65 0.1072 3.3162 0.0628 5.0885 12.3750
70 0.1030 3.4434 0.0621 5.0861 14.7820
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Table 2 Effect of on performance measures
3 0.2181 1.6478 0.0639 5.0763 4.2984
4 0.2141 1.7702 0.0635 5.0707 2.1642
5 0.2190 1.7937 0.0618 5.0649 0.9836
6 0.2276 1.7524 0.0592 5.0598 0.3990
7 0.2369 1.6817 0.0563 5.0557 0.1443
8 0.2456 1.6059 0.0536 5.0525 0.0467
9 0.2534 1.5367 0.0512 5.0500 0.0137
10 0.2601 1.4774 0.0493 5.0481 0.0036
COST FUNCTION AND OPTIMIZATION
The total expected cost function per unit time is constructed based on the following cost
factors.
Holding cost per unit time per customer in the system,
Cost per customer served per unit time,
Cost per customer lost per unit time,
Fixed cost per server.
Total expected cost function per unit time is given as:
Table 3 Effect of on performance measures
3 0.1356 2.8156 0.1063 2.0651 3.5843
4 0.1347 2.8102 0.1059 2.1428 3.5315
5 0.1336 2.8038 0.1054 2.2351 3.4792
6 0.1326 2.7961 0.1050 2.3414 3.4277
7 0.1317 2.7870 0.1044 2.4577 3.3787
8 0.1311 2.7769 0.1039 2.5769 3.3346
9 0.1307 2.7665 0.1034 2.6909 3.2980
10 0.1306 2.7565 0.1029 2.7924 3.2702
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Our objective is to evaluate the optimum value of which is that gives a minimum total
expected cost per unit time using an optimization method called quadratic fit search method
(QFSM). We do so by keeping all other parameters constant. The steps of quadratic fit search
algorithm are described as follows:
Step 1: Initialization: Choose a starting 3-point pattern along with a stopping
tolerance and initialize the iteration counter .
Step 2: Stopping: If , stop and report approximate optimum solution .
Step 3: Quadratic Fit: Compute a quadratic fit optimum such that
.
Then if , go to step 5 and if , go to step 6.
Step 4: Coincide: New coincides essentially with current . If is farther from than
from , perturb left
, and proceed to step 5. Otherwise, adjust right
and proceed to step 6.
Step 5: Left: If is superior to (less for minimization function, greater for
maximization function), then update , otherwise replace , . Either
way, advance and return to step 2.
Step 6: Right: If is superior to (less for minimization function, greater for
maximization function), then update otherwise replace , . Either
way, advance and return to step 2.
It is evident from Table 4 that the QFSM provides the minimum total expected cost
which is attained after 9 iterations at the optimum busy period service rate
. Furthermore, Fig. 1 shows the surface plot for the expected cost function per unit
time as a function of and for different values of . Fig. 2 shows for a fixed vacation
service rate , the effect of the change in for the expected cost function per unit
time. In figures 1 and 2, the values assumed for the cost factors are = 6, = 5, = 8 and
= 30.
P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,
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Fig. 1. Exp. Cost per unit time as a function of and
Table 4 Search for optimum service rate
l m h q F( q)
1 4.50000 4.75000 5.00000 4.77412 207.038
2 4.75000 4.77412 5.00000 4.77362 207.038
3 4.75000 4.77362 4.77412 4.77357 207.038
4 4.75000 4.77357 4.77362 4.77357 207.038
5 4.75000 4.77357 4.77357 4.77358 207.038
6 4.75000 4.77357 4.77358 4.77357 207.038
7 4.77357 4.77357 4.77358 8.88000 207.540
8 4.77357 4.77357 8.88000 4.77356 207.038
9 4.77357 4.77356 4.77357 4.77356 207.038
Fig. 2. Effect of busy service rate on the cost function.
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CONCLUSION
We considered a multi-server Markovian queue with discouraged Poisson arrivals, reneging
of impatient customers and multiple working vacations of servers. We applied the matrix-
geometric method and the truncation method to obtain the steady state probabilities which are
further employed to derive some performance measures. The impact of some parameters on
performance measures is shown numerically. We also constructed a cost function to
investigate the optimum service rate that minimizes the cost function using the quadratic fit
search method.
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