COST & PROFIT ANALYSIS OF TWO-DIMENSIONAL STATE M/M/2 QUEUING MODEL WITH CORRELATED SERVERS, MULTIPLE VACATION, BALKING AND CATASTROPHES Текст научной статьи по специальности «Медицинские технологии»

COST & PROFIT ANALYSIS OF TWO-DIMENSIONAL STATE M/M/2 QUEUING MODEL WITH CORRELATED SERVERS, MULTIPLE VACATION, BALKING AND

CATASTROPHES

Sharvan Kumar1, Indra2 •

1*, 2 Department of Statistics and O. R, Kurukshetra University, Kurukshetra, Haryana,

India-136119

[email protected] , [email protected] Abstract

The present study obtains the time-dependent solution of a two-dimensional state Markovian queuing model with infinite capacity, correlated servers, multiple vacation, balking and catastrophes. Inter arrival times follow an exponential distribution with parameters A and service times follow Bivariate exponential distribution BVE (p, p, v) where p is the service time parameter and v is the correlation parameter. Both the servers go on vacation with probability one when there are no units in the system and the servers keeps on taking a sequence of vacations of random length each time the system becomes empty, till it finds at least one unit in the system to start each busy period referred as multiple vacation. The unit finds a long queue and decides not to join it; may be considered as balking. All the units are ejected from the system when catastrophes occur and the system becomes temporarily unavailable. The system reactivates when new units arrive. Occurrence of catastrophes follow Poisson distribution with rate £,. Laplace transform approach has been used to find the time-dependent solution. By using differential-difference equations, the recursive expressions for probabilities of exactly i arrivals and j departures by time t are obtained. The probabilities of this model are consistent to the results of "Pegden & Rosenshine". The model estimates the total expected cost, total expected profit and obtained the optimal values by varying time t for cost and profit. These important key measures give a greater understanding of the model behaviour. Numerical analysis and graphical representations have been done by using Maple software.

Keywords: Correlated servers, Multiple vacation, Balking, Catastrophes

1. Introduction

A two-dimensional state model has been used to deal with complicated transient analysis of some queuing problem. This model is used to examine the queuing system for exact number of arrivals and departures by given time t. In case of a one-dimensional state model, it is difficult to determine how many units have entered, left or waiting units in the system, while the two-dimensional state model exactly identifies the numbers of units that have entered, left, or waiting in the system. The idea of two-dimensional state model for the M/M/1 queue was first given by Pegden & Rosenshine [4]. After that, the two-dimensional state model has attracted the attention of a lot of researchers.

A system of queues in series or in parallel should ordinarily be studied taking into account the interdependence of servers, but this leads to very complicated mathematics even in very simple case of systems. So to reduce such complications of analysis the servers are considered to be independent. But this independence of servers cause impact in time bound operations such as vehicle inspection counters, toll booths, large bars and cafeterias etc. where for efficient system functioning the correlation between the servers contributes significantly. Nishida et al. [3] investigated a two-server Markovian queue assuming the correlation between the servers and obtained steady-state results for a limited waiting space capacity of two units. Sharma [6] investigated the transient solution to this problem again using only two units waiting spaces capacity. Sharma and Maheswar [8] developed a computable matrix approach to study a correlated two-server Markovian queue with finite waiting space. They also derived waiting time distribution for steady-state and obtained the transient probabilities through steady-state by using a matrix approach and Laplace transform approach.

Various studies have been conducted to evaluate different performance measures to verify the robustness of the system in which a server takes a break for a random period of time i.e. vacation. When the server returns from a vacation and finds the empty queue, it immediately goes on another vacation i.e. multiple vacation and if it finds at least one waiting unit, then it will commence service according to the prevailing service policy. Different queuing systems with multiple vacation have been extensively investigated and effectively used in several fields including industries, computer & communication systems, telecommunication systems etc. Different types of vacation policies are available in literature such as single vacation, multiple vacation and working vacations. Researches on multiple vacation systems have grown tremendously in the last several years. Cooper [2] was the first to study the vacation model and determined the mean waiting time for a unit arrive at a queue served in cyclic order. Doshi [5] and Wu & Zhang [15] have done outstanding researches on queuing system with vacations and released some excellent surveys. Xu and Zhang [12] considered the Markovian multi-server queue with a single vacation (e, d)-policy. They also formulated the system as a quasi-birth-and-death process and computed the various stationary performance measures. Altman and Yechiali [13] studied the customer's impatience in queues with server vacations. Kalidaas et al. [18] obtained the time-dependent solution of a single server queue with multiple vacation. Ammar [19] analysed M/M/1 queue with impatient units and multiple vacation. Sharma and Indra [24] investigated the dynamic aspects of a two-dimensional state single server Markovian queuing system with multiple vacation and reneging. Gahlawat et al. [25] studied the time-dependent first in first out queuing model with a single intermittently available server and variable-sized bulk arrivals and bulk departures by using the Laplace transform and inverse transform approaches.

Queues with balking have numerous applications in everyday life. Balking occurs if units avoid joining the queue, when they perceive the queue to be too long. Long queues at cash counters, ticket booths, banks, barber shops, grocery stores, toll plaza etc. Kumar et al. [7] obtained the time-dependent solution of an M/M/1 queue with balking. Chauhan and Sharma [10] derived an expression of the probability distribution for the number of customers in the service station for the M/M/r queuing model with balking and reneging. Zhang and Yue [11] analysed the M/M/1/N queuing system with balking, reneging and server vacation. Sharma and Kumar [17] studied a single-server Markovian feedback queuing system with balking. Bouchentouf and Medjahari [23] presented a single-server feedback queuing system under two differentiated multiple vacation with balking and obtained steady-state probabilities for the model. They also derived some important performance measures, including the average number of customers in the system, the average number of customers in the queue, the average balking rate etc.

Queuing systems with catastrophes are also getting a lot of attention nowadays and may be used to solve a wide range of real-world problems. Catastrophes may occur at any time, resulting in the loss of units and the deactivation of the service centre, because they are totally unpredictable in

nature. Such type of queues with catastrophes plays an important role in computer programs, telecommunication and ticket counters etc. For example, virus or hacker attacking a computer system or program causing the system fail or become idle. Chao [9] obtained steady-state probability of the queue size and a product form solution of a queuing network system with catastrophes. Kumar et al. [14] obtained time-dependent solution for M/M/1 queuing system with catastrophes. Kalidass et al. [16] derived explicit closed form analytical expressions for the time-dependent probabilities of the system size. Dharamraja and Kumar [20] studied Markovian queuing system with heterogeneous servers and catastrophes. Chakravarthy [21] studied delayed catastrophic model in steady state using the matrix analytic method. Sampath and Liu [22] studied an M/M/1 queue with reneging, catastrophes, server failures and repairs using modified Bessel function, Laplace transform and probability generating function approach. Souza and Rodriguez [26] worked on fractional M/M/1 queue model with catastrophes.

With above concepts in mind, we analyse a two-dimensional state M/M/2 queuing model with correlated servers, multiple vacation, balking and catastrophes.

Consider a situation in a company, where two colleagues work independently on the same project i.e. they are not able to share the information of project with each other and not helping each other. Then it will take a long time to complete the project, some information will be lost due to communication gap and the results obtained are not much reliable. But if both of them work together (interdependent servers) i.e. they will share all information of project and help each other, then there are more chances that it will take less time and results obtained will be more reliable. Hence interdependent servers are more reliable than the independent servers. When servers work interdependently then they termed as correlated servers. After project completion, the colleagues may take a break, when they find there is no further work, considered as vacation. During the project, if someone wants to work with these colleagues on different project but due to their busy schedule decides not to join them; it may be considered as balking. If due to disease or any other reason the colleagues are not working this may be considered as catastrophes.

The present paper has been structured as follows. In section 1 introduction and in section 2 the model assumptions, notations and description are given. In section 3 the differential-difference equations to find out the time-dependent solution are given and section 4 describes important performance measures. Section 5 investigates the total expected cost function and total expected profit function for the given queuing system. In section 6, we present the numerical results in the form of tables and section 7 contains the tables and graphs to illustrate the impact of various factors on performance measures. The last section contains discussion on the findings and suggestions for further work.

2. Model Assumptions, Notations and Description

• Arrivals follow Poisson distribution with parameter A.

• There are two servers and the service times follow Bivariate exponential distribution BVE* (p, p, v) where p is the service time parameter and v is the correlation parameter.

• The vacation time of the server follows an exponential distribution with parameter w.

• On arrival a unit either decides to join the queue with probability 5 or not to join the queue with probability 1-5.

• Occurrence of catastrophes follows Poisson distribution with parameter

• Various stochastic processes involved in the system are statistically independent of each other.

^introduced by Marshall and Olkin [1]

Initially, the system starts with zero units and the server is on vacation, i.e.

P0AV (0) = 1 ; P0AB (0) = 0 (1)

Î1 ; for i = j j

¿,, =1 ; >= 0 for j < i (2)

|0 ; for i * j 4-

The Two-Dimensional State Model

Pi, jy (t)= The probability that there are exactly i arrivals and j departures by time t and the server is on vacation.

P j B (t) = The probability that there are exactly i arrivals and j departures by time t and the server

is busy in relation to the queue. Pi j (t) = The probability that there are exactly i arrivals and j departures by time t.

3. The Differential-Difference Equations for the Queuing Model under Study

d_ dt d

¿P.UV (t ) = -m^V (t )+ M + V)P„ 1B (t )(l - ¿,0 )+ <i-2, B (t )(l - ¿,0 - 4.1 )+ ^ (1 - P,,vV (t )) i>0 (3)

Pub (t) = (t)+2juPmm b (t)(1-4,0 )+P,-2b (t )(1-^0-41)+wPmy (t) i>0 (4)

1, i +1, i, B

dfcPi'j'V (t )= -(W + w + jV (t )+ -1, j,v (t ) i>j>0(5)

d

P, j b (t )-

+ P-2,B (t)+ WPj V (t)

dt P, j,B (t ) = -(W + 2U +v + & J B (t ) + WP-1, j B (t )(1 - 4-1, j )+ 2uP, j-1B (t )(1 - ¿j, 0 ) (6)

The preceding equations (3) to (6) are solved by taking the Laplace transforms together with initial condition

^^W^Ti) <7)

P-(s)= ( Z' iY *> (8)

s (s + A/3 + ,)(s + A/3 + w + ,)

P'iV (s )=i ZL^ Pj ^ (s )+i + 7 + (s) i>0 (9)

^ s + Ap+i) ^ s +Ap+i)

Pi,0,B (s) =

w(Aß)1 (% + s)

s(s + Aß + %)(s + Aß + w + %){s + Aß + p + v + %)(s + Aß + 2p + v +

1-1 (# + s)

i-1

+ w(Aß)1 Y—--- , w ,

s (s + Aß + %\s + Aß + w + %)m+1 (s + Aß + 2p + v + %)'

j-m

Pi+1,i,B (s) =

2p

\ (

in 7 Pi+1,1-1,b(s) +

s + Aß+ p + v + %)

(p + v) wAß

v

s + Aß + p + v + %

P-1,i-1,B (s)

Pi +1,1 -2,B (s ) +

(s + Aß + + Aß + w + %)(s + Aß + p + v + %)

Pi, JV (s) =

p+v

Aß

s + Aß+% I s + Aß+w+%

Pi,Ms) =

f

Aß

s+Aß+ 2p+v+%

l'-1,j,B

(s)

f

+

P j-1B (s) + 2p

Aß

\1-Ji \ II v

s + Aß+w+%) I s + Aß+%

s+Aß+ 2p+v+%)

HB(s)

f

+

s+Aß+ 2p+v+%

i>1 (10)

¿>0 (11)

Pj, j-2,B (s) ¿>7>°(12)

P,j-ZB(s)

+

w

p+v Y

s+Aß+%[ s+Aß+ 2p+v+s+Aß+ w+%

g^Tb^h-PjJsH

i>j+1, j>0 (13)

W i - "II

YY\PijV (s)+Pj (s tl-S.j )] = -

It is seen that

W i

^ ... (5)+Pi,j,B(s) i=0 j=0

and hence

H kjV (t) + PjB (t )(l-SUi )]= 1

i = 0 j=0

a verification.

(14)

(15)

4. Performance Measures

(i) The Laplace transform of Pi (t) of the probability that exactly i units arrive by time t; when initially there are no unit in the system is given by

Pi(s )=i [p ' J.V (s ) + P i .j.B (s )(l - )] = ± P i, j (s ) =

j = 0 j = 0

(A.Bt)

And its inverse Laplace transform is: Pj.(t) =---

(Aß )i

(s + Aß )

(16)

The arrivals follow a Poisson distribution as the probability of the total number of arrivals is not affected by vacation time of the server.

(ii) Pj (t) is the probability that exactly j units have been served by time t. In terms of Pj (t)

■ j

have

we

p. (t )=Y Pj (')

(17)

1-J

V

Sharvan Kumar, Indra RT&A, No 4 (76)

COST & PROFIT ANALYSIS OF TWO-DIMENSIONAL STATE Volume 18, December 2023

(iii) The Laplace transform of mean number of arrival is: i iPi (s) = (18)

xp

IP'(S ) =

i=0

TO

And its inverse Laplace transform is: i iPi (t) = Xpt (19)

i=0

(iv)The mean number of units in the queue is calculated as follows

TO TO

Ql (t) = Z NPV (t) + X (N - 1)PB (t) (20)

N=0 N=1

Where N = i-j.

5. Cost Function and Profit Function

For the given queuing system, the following notations have been used to represent various costs to

find out the total expected cost and total expected profit per unit time:

Let

Ch: Cost per unit time for unit in the queue.

Cb: Cost per unit time for a busy server.

Cm: Cost of service per unit time.

Cv: Cost per unit time when the server is on vacation.

If I is the total expected amount of income generated by delivering a service per unit time then

(i) Total expected cost per unit at time t is given by

TC (t) = CH * Ql (t) + Cb * Pb (t) + CV * Pv (t) + j* CM (21)

(ii) Total expected income per unit at time t is given by

TEj (t) = I *u* (1-Pv (t)) = I * u* Pb (t) (22)

(iii) Total expected profit per unit at time t is given by

TEP (t ) = TEI (t) - TC(t) (23)

6. Numerical Results

6.1. Numerical Validity Check

(i) For the state when the server is on vacation

i

PV (t) = Z PjV(t) (24)

j=o

(ii) For the state when the server is busy in relation to the queue

Pb (t) = Z Pi.JB (t) (25)

j=o

(iii) The probability Pi (t) that exactly i units arrive by time t is

ip,, (' )=ip,j ,v (' )+z PjB (>)

j=0 j=0 j=o

(26)

(iv) A numerical validity check of inversion of Pi, j (s) is based on the relationship

e-Apt (iPt ) ®

Pr {i arrivals in (0, t)} =-V P ' Pt j (t) = Pt (t) (27)

i! j=o

The probabilities of this model shown in last column of table 1 given below are consistent to the last column of "Pegden & Rosenshine" [4] by keeping constant values of w=1, £=0, 6=1 and V=0.25 shown in table

Table-1: Numerical validity check of inversion of Pi, j (s )

A P t i e * (At) i! i 7=0 i-i 7=0 i Y^Pijv 7=0

1 2 3 1 0.149361 0.129196 0.020165 0.149361

1 2 3 3 0.224041 0.158076 0.065965 0.224041

1 2 3 5 0.100818 0.057803 0.043016 0.100818

2 2 3 1 0.014873 0.012865 0.002008 0.014873

2 2 3 3 0.089235 0.062961 0.026274 0.089235

2 2 3 5 0.160623 0.092090 0.068533 0.160623

1 2 4 1 0.073263 0.065390 0.007873 0.073263

1 2 4 3 0.195367 0.148001 0.047366 0.195367

1 2 4 5 0.156294 0.100998 0.055296 0.156294

2 2 4 1 0.002683 0.002395 0.000288 0.002683

2 2 4 3 0.028626 0.021686 0.006940 0.028626

2 2 4 5 0.091604 0.059195 0.032409 0.091604

2 4 4 5 0.091604 0.073396 0.018208 0.091604

1 2 4 4 0.195367 0.136810 0.058557 0.195367

1 2 3 6 0.050409 0.025824 0.024585 0.050409

7. Sensitivity Analysis

This part focuses on the impact of the arrival rate (A), service rate (p), vacation rate (w), correlation parameter (v), balking probability (1-6) and catastrophes rate (4) on the probability when the server is on vacation (Pv(t)), probability when the server is busy (PB(t)), expected queue length (QL(t)), total expected cost (TC(t)), total expected income (TEi(t)) and total expected profit (TEp(t)) at time t. To determine the numerical results for the sensitivity of the queuing system one parameter varied while keeping all the other parameters fixed and taking cost per unit time for unit in the queue=10, cost per unit time for a busy server=8, cost per unit time when the server is on vacation=5, cost of service per unit time=4, total expected amount of income=100 and number of units in the system=8.

7.1. Impact of Arrival Rate (A)

We examine the behaviour of the queuing system using measures of effectiveness along with cost and profit analysis by varying arrival parameter with time, while keeping all other parameters fixed; ^=5, w=3, v=0.25, 4=0.0001 and ¡3=1. In table 2, we observe that as the value of A increases with time t, PB(t), QL(t), TC(t), TEi(t) and TEp(t) increases but Pv(t) decreases.

Table-2: Measures of Effectiveness versus A

t A Pv(t) PB(t) Q(t) TC(t) TEi(t) TEP(t)

1 1.00 0.8550827 0.1449161 0.4618549 30.0532913 72.45805 42.4047587

2 0.8451239 0.1546386 0.4824289 30.2870173 77.31930 47.0322827

3 0.8427343 0.1534636 0.4779723 30.2211033 76.73180 46.5106967

4 0.8414541 0.1511897 0.4597185 30.0139731 75.59485 45.5808769

5 0.8403975 0.1485374 0.4210111 29.6003977 74.26870 44.6683023

1 1.10 0.8427120 0.1572855 0.5062882 30.5347260 78.64275 48.1080240

2 0.8321542 0.1673763 0.5277639 30.7774204 83.68815 52.9107296

3 0.8279867 0.1661032 0.5192708 30.6614671 83.05160 52.3901329

4 0.8278007 0.1634085 0.4891622 30.3378935 81.70425 51.3663565

5 0.8264318 0.1599687 0.4312130 29.7240386 79.98435 50.2603114

1 1.20 0.8306335 0.1693615 0.5505522 31.0135815 84.68075 53.6671685

2 0.8194239 0.1797142 0.5725572 31.2604051 89.85710 58.5966949

3 0.8165607 0.1777708 0.5579035 31.0840049 88.88540 57.8013951

4 0.8147421 0.1744598 0.5117878 30.5872669 87.22990 56.6426331

5 0.8130834 0.1702170 0.4412499 29.8396520 85.10850 55.2688480

Figure 1 shows the variation of cost with time by varying arrival rate while keeping the other parameters fixed.

Figure 2 shows the variation of profit with time by varying arrival rate while keeping the other parameters fixed.

7.2. Impact of Service Rate (p)

The behaviour of the queuing system measures of effectiveness along with cost and profit analysis by varying p with time t, while keeping all other parameters fixed; A=1, w=3, v=0.25, 4=0.0001 and 6=1. In table 3, we observe that as the value of p increases with time t, PB(t), QL(t), TC(t), TEi(t) and TEp(t) increases but PV(t) decreases.

Table-3: Measures of Effectiveness versus p

t Pv(t) PB(t) QL(t) TC(t) TEi(t) TEp(t)

1 2.00 0.7369325 0.2630663 0.6177736 19.9669289 52.613260 32.6463311

2 0.6762899 0.3234726 0.7090209 21.0594393 64.694520 43.6350807

3 0.6699756 0.3262223 0.7084844 21.0445004 65.244460 44.1999596

4 0.6667285 0.3239153 0.6797388 20.7273529 64.783060 44.0557071

5 0.6649823 0.3189526 0.6203355 20.0798873 63.790520 43.7106327

1 2.50 0.7655438 0.2344550 0.5762758 21.4661170 58.613750 37.1476330

2 0.7243328 0.2754297 0.6356142 22.1812436 68.857425 46.6761814

3 0.7211500 0.2750479 0.6303259 22.1093922 68.761975 46.6525828

4 0.7177601 0.2728837 0.6050702 21.8275721 68.220925 46.3933529

5 0.7134529 0.2684820 0.5531662 21.2567825 67.120500 45.8637175

1 3.00 0.7897314 0.2102673 0.5431578 23.0623734 63.080190 40.0178166

2 0.7609462 0.2388163 0.5847948 23.5632094 71.644890 48.0816806

3 0.7587053 0.2374926 0.5788371 23.4818383 71.247780 47.7659417

4 0.7561697 0.2354741 0.5561210 23.2258513 70.642230 47.4163787

5 0.7501976 0.2317373 0.5088841 22.6937274 69.521190 46.8274626

Time Time

Figure 3: Variation of corn with lime t by varying service Tate fi Figure 4: Variation of profit with time tby varying seirice rale fi

Figure 3 shows the variation of cost with time by varying service rate while keeping the other parameters fixed.

Figure 4 shows the variation of profit with time by varying service rate while keeping the other parameters fixed.

7.3. Impact of Vacation Rate (w)

We observe that the behaviour of the queuing system measures of effectiveness along with cost and profit by varying w with time t, while keeping all other parameters fixed; A=1, ^=5, v=0.25, 4=0.0001 and jS=1. In table 4, we observe that as the value of w increases with time t, PB(t), QL(t), TC(t), TEi(t) and TEp(t) increases but Pv(t) decreases.

Table-4: Measures of Effectiveness versus w

t w Pv(t) PB(t) QL(t) TC(t) TEi(t) TEp(t)

1 3.00 0.8550827 0.1449161 0.4618549 30.0532913 72.45805 42.4047587

2 0.8451239 0.1546386 0.4824289 30.2870173 77.31930 47.0322827

3 0.8427343 0.1534636 0.4779723 30.2211033 76.73180 46.5106967

4 0.8414541 0.1511897 0.4597185 30.0139731 75.59485 45.5808769

5 0.8403975 0.1495374 0.4210111 29.6083977 74.76870 45.1603023

1 4.00 0.8462857 0.1537131 0.4015245 29.4763783 76.85655 47.3801717

2 0.8411568 0.1586057 0.4083021 29.5576506 79.30285 49.7451994

3 0.8389988 0.1571991 0.4040524 29.4931108 78.59955 49.1064392

4 0.8378927 0.1547511 0.3886032 29.3135043 77.37555 48.0620457

5 0.8321931 0.1517418 0.3558749 28.9336489 75.87090 46.9372511

1 5.00 0.8415912 0.1584076 0.3613974 29.0891908 79.20380 50.1146092

2 0.8386494 0.1611131 0.3640154 29.1223058 80.55655 51.4342442

3 0.8365297 0.1596682 0.3602046 29.0620401 79.83410 50.7720599

4 0.8345421 0.1575017 0.3464072 28.8967961 78.75085 49.8540539

5 0.8300883 0.1548467 0.3171915 28.5611301 77.42335 48.8622199

Figure 5 shows the variation of cost with time by varying vacation rate while keeping the other parameters fixed.

Figure 6 shows the variation of profit with time by varying vacation rate while keeping the other parameters fixed.

7.4. Impact of Correlation Parameter (v)

We see that the behaviour of the queuing system using measures of effectiveness along with cost and profit analysis by varying v with time t, while keeping all other parameters fixed; A=1, ^=5, w=3, 4=0.0001 and ¡3=1. In table 5, we observe that as the value of v increases with time t, PB(t), QL(t), TC(t), TEi(t) and TEp(t) increases but Pv(t) decreases.

Table-5: Measures of Effectiveness versus v

t v Pv(t) PB(t) Q(t) TC(t) TEi(t) TEp(t)

1 0.25 0.8550827 0.1449161 0.4618549 30.0532913 72.45805 42.4047587

2 0.8451239 0.1546386 0.4824289 30.2870173 77.31930 47.0322827

3 0.8427343 0.1534636 0.4779723 30.2211033 76.73180 46.5106967

4 0.8414541 0.1511897 0.4597185 30.0139731 75.59485 45.5808769

5 0.8403975 0.1475374 0.4210111 29.5923977 73.76870 44.1763023

1 0.50 0.8612902 0.1387085 0.4555353 29.9714720 69.35425 39.3827780

2 0.8522410 0.1475215 0.4756205 30.1975820 73.76075 43.5631680

3 0.8497855 0.1464124 0.4712878 30.1331047 73.20620 43.0730953

4 0.8472639 0.1433799 0.4532707 29.9160657 71.68995 41.7738843

5 0.8456738 0.1392611 0.4150683 29.4931408 69.63055 40.1374092

1 0.75 0.8670184 0.1329803 0.4497140 29.8960744 66.49015 36.5940756

2 0.8587324 0.1410301 0.4694068 30.1159708 70.51505 40.3990792

3 0.8562189 0.1399790 0.4651801 30.0527275 69.98950 39.9367725

4 0.8544773 0.1361665 0.4473789 29.8355075 68.08325 38.2477425

5 0.8524003 0.1315346 0.4096382 29.4106603 65.76730 36.3566397

Figure 7 shows the variation of cost with time by varying correlation parameter while keeping the other parameters fixed.

Figure 8 shows the variation of profit with time by varying correlation parameter while keeping the other parameters fixed.

7.5. Impact of Joining Probability (/)

We see that the behaviour of the queuing system measures of effectiveness along with cost and profit analysis by varying / with time t. While keeping all other parameters fixed; A=1, p=5, w=3, v=0.25 and 4=0.0001. In table 6, we observe that as the value of // increases with time t, PB(t), QL(t), TC(t), TEi(t) and TEp(t) increases but Pv(t) decreases.

Table-6: Measures of Effectiveness versus ft

t ft Pv(t) PB(t) QL(t) TC(t) TEi(t) TEp(t)

1 0.65 0.8876633 0.0990264 0.2868039 28.0985667 49.51320 21.4146333

2 0.8840590 0.1068084 0.3068008 28.3427702 53.40420 25.0614298

3 0.8880948 0.1069225 0.3132331 28.4281850 53.46125 25.0330650

4 0.8895493 0.1065419 0.3154762 28.4548437 53.27095 24.8161063

5 0.8871414 0.1053500 0.3133391 28.4118980 52.67500 24.2631020

1 0.75 0.8767085 0.1125818 0.3349279 28.6334759 56.29090 27.6574241

2 0.8720983 0.1210185 0.3563601 28.8922405 60.50925 31.6170095

3 0.8751685 0.1210047 0.3615710 28.9595901 60.50235 31.5427599

4 0.8746624 0.1200356 0.3610913 28.9445098 60.01780 31.0732902

5 0.8732121 0.1183004 0.3533446 28.8459097 59.15020 30.3042903

1 0.85 0.8671069 0.1257705 0.3844829 29.1865275 62.88525 33.6987225

2 0.8608738 0.1347799 0.4064078 29.4466862 67.38995 37.9432638

3 0.8623335 0.1344946 0.4090881 29.4785053 67.24730 37.7687947

4 0.8615314 0.1334039 0.4037567 29.4124552 66.70195 37.2894948

5 0.8598611 0.1310590 0.3868133 29.2159105 65.52950 36.3135895

Figure 9 shows the variation of cost with time by varying joining probability while keeping the other parameters fixed.

Figure 10 shows the variation of profit with time by varying joining probability while keeping the other parameters fixed.

7.6. Impact of Catastrophes Rate (4)

We see that the behaviour of the queuing system measures of effectiveness along with cost and profit analysis by varying 4 with time t, while keeping all other parameters fixed; A=1, ^=5, w=3, v=0.25 and jS=1. In table 7, we observe that as the value of 4 increases with time t, PB(t), QL(t), TC(t), TEi(t) and TEp(t) increases but Pv(t) decreases.

Table-7: Measures of Effectiveness versus E

t E Pv(t) PB(t) QL(t) TC(t) TEi(t) TEp(t)

1 0.0001 0.8550827 0.1449161 0.4618549 30.0532913 72.45805 42.4047587

2 0.8451239 0.1546386 0.4824289 30.2870173 77.31930 47.0322827

3 0.8427343 0.1534636 0.4779723 30.2211033 76.73180 46.5106967

4 0.8414541 0.1491897 0.4597185 29.9979731 74.59485 44.5968769

5 0.8403975 0.1445374 0.4210111 29.5683977 72.26870 42.7003023

1 0.0002 0.8550885 0.1449103 0.4618404 30.0531289 72.45515 42.4020211

2 0.8451309 0.1546316 0.4824119 30.2868263 77.31580 47.0289737

3 0.8427420 0.1534569 0.4779565 30.2209302 76.72845 46.5075198

4 0.8414661 0.1491850 0.4597091 29.9979015 74.59250 44.5945985

5 0.8404256 0.1445380 0.4210193 29.5686250 72.26900 42.7003750

1 0.0003 0.8550943 0.1449045 0.4618259 30.0529665 72.45225 42.3992835

2 0.8451380 0.1546246 0.4823948 30.2866348 77.31230 47.0256652

3 0.8427496 0.1534503 0.4779406 30.2207564 76.72515 46.5043936

4 0.8414782 0.1491802 0.4596998 29.9978306 74.59010 44.5922694

5 0.8404537 0.1445385 0.4210276 29.5688525 72.26925 42.7003975

Figure 11 shows the variation of cost with time by varying catastrophes rate while keeping the other parameters fixed.

Figure 12 shows the variation of profit with time by varying catastrophes rate while keeping the other parameters fixed.

8. Discussion

Figure 1 & figure 2 show the variation of cost & profit respectively with time t by varying A(=1.00, 1.10, 1.20). The value of both cost & profit increases with increase in t upto t(=2.00) then decreases slightly. Hence we get the optimal value at t=5 when A=1.00 and t=2 when A=1.20 for minimum cost and maximum profit respectively.

Figure 3 show the variation of cost with time t by varying p(=2.00, 2.50, 3.00). The value of cost increases with increase in t upto t(=2.00) then decreases slightly. The variation in profit with time t represented in figure 4 by varying p(=2.00, 2.50, 3.00). The profit increases with increase in time up to (i) t=3 when p=2.00 (ii) t=2 when p=2.50, 3.00 respectively then decreases slightly. Hence we get the optimal value at t=1 when p=2.00 and t=2 when p=3.00 for minimum cost and maximum profit respectively.

Figure 5 & figure 6 show the variation of cost & profit respectively with time t by varying ^>(=3.00, 4.00, 5.00). The value of both cost & profit increases with increase in t upto t(=2.00) then decreases slightly. Hence we get the optimal value at t=5 when ^>=5.00 and t=2 when ^>=5.00 for minimum cost and maximum profit respectively.

Figure 7 & figure 8 show the variation of cost & profit respectively with time t by varying v(=0.25, 0.50, 0.75). The value of both cost & profit increases with increase in t upto t(=2.00) then decreases slightly. Hence we get the optimal value at t=5 when v=0.75 and t=2 when v=0.25 for minimum cost and maximum profit respectively.

Figure 9 show the variation of cost with time t by varying /(=0.65, 0.75, 0.85). The value of cost increases with increase in t upto (i) t=4 when /3=0.65 (ii) t=3.00 when /3=0.75, 0.85 respectively then decreases slightly. The variation in profit with time t represented in figure 10 by varying /(=0.65, 0.75, 0.85). The profit increases with increase in time up to t(=2.00) then decreases slightly. Hence we get the optimal value at t=1 when /3=0.65 and t=2 when /3=0.85 for minimum cost and maximum profit respectively.

Figure 11 & figure 12 show the variation of cost & profit respectively with time t by varying £(=0.0001, 0.0002, 0.0003). The value of both cost & profit increases with increase in t upto t(=2.00) then decreases slightly. Hence we get the optimal value at t=5 when £=0.0001 and t=2 when £=0.0001 for minimum cost and maximum profit respectively. Finally, the variation in rate of catastrophes shows the minor effect on cost and profit.

9. Conclusions and Future Work

The time-dependent solution, for the two-dimensional state M/M/2 queuing model with correlated servers, multiple vacation, balking and catastrophes, has been obtained. The model estimates the total expected cost and total expected profit, the best optimal value is at t=1 when service rate(=2.00) and t=2 when arrival rate(=1.20) for minimum cost and maximum profit respectively. These key measures give a greater understanding of model behaviour. Finally, the numerical analysis clearly demonstrates the meaningful impact of the correlated servers and multiple vacation on the system performances. This model finds its applications in communication networks, computer networks, supermarkets, hospital administrations, financial sector and many others.

As part of future study, this model may be examined further for Non-Markovian queues, bulk queues, tandem queues etc.

10. Acknowledgement

First and foremost, I am deeply indebted to my supervisor, Prof. (Dr.) Indra Rani for her unwavering guidance and expertise. Her insightful feedback and constant encouragement played a pivotal role in shaping this research. I am grateful to the Department of Statistics and Operational Research, Kurukshetra University, Kurukshetra for providing the resources and research facilities necessary for this study. Additionally, I would like to acknowledge the Kurukshetra University for providing the University research scholarship which made this research possible.

References

[1] Marshall, A. W. and Olkin, I. (1967). A multivariate exponential distribution. Journal American Statistical Association, 62(317):30-40.

[2] Cooper, R. B. (1970). Queues served in cyclic order waiting times. Bell System Technical Journal, 49(3):399-413.

[3] Nishida, T., Watanabe, R. and Tahara, A. (1974). Poisson queue with correlated two servers. Technology Reports of Osaka University, 24:1193.

[4] Pegden, C. D. and Rosenshine, M. (1982). Some new results for the M/M/1 queue. Management Science, 28(7):821-828.

[5] Doshi, B.T. (1986). Queuing systems with vacations- a survey. Queuing Systems, 1(1):29-66.

[6] Sharma, O.P. (1990). Markovian queues. Elis Horwood, ltd, England.

[7] Kumar, B. K., Parthasarathy, P. R. and Sharafali, M. (1993). Transient solution of an M/M/1 queue with balking. Queuing Systems, 13(4):441-448.

[8] Sharma, O. P., and Maheswar, M. V. R. (1994). Analysis of Ml Ml 21 N queue with correlated servers. Optimization, 30(4):379-385.

[9] Chao, X. (1995). A queuing network system with catastrophes and product form solution. Operations Research Letters, 18(2):75-79.

[10] Chauhan, M. S., and Sharma, G. C. (1996). Profit analysis of M/M/R queueing model with balking and reneging. Monte Carlo Methods and Applications, 2(2):139-144.

[11] Zhang, Y., Yue, D. and Yue, W. (2005). Analysis of an M/M/1/N queue with balking, reneging and server vacations. In Proceedings of the 5th International Symposium on OR and its Applications, 63(8):37-47.

[12] Xu, X. and Zhang, Z.G. (2006). Analysis of multi-server queue with a single vacation (e, d)-policy. Performance Evaluation, 63(8):825-838.

[13] Altman, E. and Yechiali, U. (2006). Analysis of customers' impatience in queues with server vacations. Queueing systems, 52(4):261-279.

[14] Kumar, B., Krishnamoorthy, A., Madheswari, S. and Basha, S. (2007). Transient analysis of a single server queue with catastrophes, failures and repairs. Queueing Systems, 56(3):133-141.

[15] Wu, J.C. and Zhang, Z. G. (2010). Recent developments in vacation queuing models: a short survey. International Journal of Operations Research, 7(4):3-8.

[16] Kalidass, K., Gopinath, S., Gnanaraj, J. and Ramanath, K. (2012). Time dependent analysis of an M/M/1/N queue with catastrophes and a repairable server. Opsearch, 49(1):39-61.

[17] Sharma, S. K. and Kumar, R. (2012). A markovian feedback queue with retention of reneged customers and balking. Advanced Modeling and Optimization, 14(3):681-688.

[18] Kalidaas, K., Gnanaraj, J., Gopinath, S. and Kasturi, R. (2014). Transient analysis of an M/M/1 queue with a repairable server and multiple vacations. International Journal of Mathematics in Operational Research, 6(2):193-216.

[19] Ammar, S. I. (2015). Transient analysis of an M/M/1 queue with impatient behaviour and multiple vacations. Applied Mathematics and Computation, 260:97-105.

[20] Dharamraja, S. and Kumar, R. (2015). Transient solution of a markovian queuing system with heterogeneous servers and catastrophes. Opsearch, 52(4):810-826.

[21] Chakravarthy, S. R. (2017). A catastrophic queuing system with delayed action. Applied Mathematical System modelling, 46:631-649.

[22] Suranga Sampath, M. I. G. and Liu, J. (2018). Transient analysis of an M/M/1 queue with reneging, catastrophes, server failures and repairs. Bulletin of the Iranian Mathematical Society, 44(3):585-603.

[23] Bouchentouf, A. A. and Medjahri, L. (2019). Performance and economic evaluation of differentiated multiple vacation queueing system with feedback and balked customers. Applications and Applied Mathematics: An International Journal, 14(1):46-62.

[24] Sharma, R. and Indra (2020). Dynamic aspect of two-dimensional single server markovian queueing model with multiple vacations and reneging. In Journal of Physics: Conference Series, 1531(1):012060.

[25] Gahlawat, V. K., Rahul, K., Mor, R. S., & Malik, M. (2021) A two state time-dependent bulk queue model with intermittently available server. Reliability: Theory & Applications, 16(2(64)):114-122.

[26] De Oliveira Souza, M. and Rodriguez, P. M. (2021). On a fractional queueing model with catastrophes. Applied Mathematics and Computation, 410:126468.