Analysis of a Multi-server Markovian Queue with Working...

P. Vijaya Laxmi and T.W. Kassahun, International Journal of Research in Engineering, IT and Social Sciences,

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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.


ISSN 2250-0588, Impact Factor: 6.452, Volume 08, Special Issue, June 2018, Page 10-24



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.





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)





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





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

,





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)





(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





, (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





, 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

, .


ISSN 2250-0588, Impact Factor: 6.452, Volume 08, Special Issue, June 2018, -24



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.





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






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:


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





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.





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.





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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