CC BY
26
Analysis of M, MAP/PH\, PH2/I non-preemptive priority Queueing model with Delayed working vacations, immediate feedback,impatient customer, differentiate breakdown and phase type repair
G. Ayyappan, N. Arulmozhi •
Department of Mathematics, Puducherry Technological University, Puducherry, India.
Chennai 600005, India. ayyappan@ptuniv.edu.in, arulmozhi.n@pec.edu,
Abstract
The arrival of high priority customers is governed by the Poisson process while that of low priority customers is governed by the Markovian Arrival Process, and the service times are determined by a distinct Phase-type distribution. When the service is finished and the system is empty, the server stays idle for a random period (delay time). If a customer arrives within the delayed period, the server resumes normal service to the customer immediately. Otherwise, at the end of the delayed period, the server will take a working vacation and will instantly provide slow service to customers (high priority customers only). The Matrix analytic method is used to investigate the system. We also discussed the steady-state vector and busy period for our concept. The estimated and visually displayed performance measures of the system
Keywords: Non-preemptive Priority, Working vacation policy, Phase-type repair, Immediate feedback, Differentiate breakdown, Delay time.
AMS Subject Classification (2010): 60K25, 68M30, 90B22.
1. Introduction
For the past two decades, the priority queue hypothesis has been used in communication strategies. Because priority does not come under FCFO, it distinguishes it from a normal queue. It is a special type of queue in which each customer is dealt with priority and served according to its priority. There are two different types of priority service available in a queueing system: preemptive and non-preemptive. Priority customers that arrive early will wait until the service is finished while regular customers are serviced. This belongs under the non-preemptive priority rule. In the event of a preemptive rule, high-priority consumers would frequently interrupt low-priority service.
Ayyappan et al. [21] looked at M/M/1 for retrials, with negative arrival while using non-preemptive priority service. Bhagat and Jain [5] described a multi-server, non-preemptive priority service that is susceptible to failure and maintenance. According to Jeganathan et al. [9], the inventory system and non-preemptive priority service for retrials have been discussed. Additionally, discretionary priority service is utilized, taking into account both disciplines. Ayyappan and Somasundaram [3] analyzed discretionary priority service for retrials used MX1,MX2/G1, G2/1.
Krishnamoorthy and Divya [13] examined queueing models with MAP and PH distributions, as well as working vacations under N-policy.
In many real-world queueing situations, the server can be seen working during its rest period if necessary. Working vacation means that the server offers service at a lower rate throughout the vacation period rather than entirely shutting down. In the past few decades have seen, queueing systems with server working vacation, owing to similarities between telecommunication system, manufacturing system, and computer system. Yang et al. [22] applied the spectral expansion method to deal with a single server queueing model with delayed working vacations and working breakdown: The author showed the steady-state probability vector, LST of sojourn time, and expected sojourn time. After service completion, the server is idle when there are no customers in the system for a certain amount of time (changeover time) (Pikkala et al. [19], Krishna Reddy and Anitha [11]). The server begins offering service if customers access the system during changeover time; if not, the server goes on vacation at the end of the changeover time.
After obtaining service from the server, customers may be satisfied or unsatisfied. Customers who are satisfied with the system will leave, while those who are not satisfied will get feedback right away. A single server model with starting failures, standby server, single vacation, delayed repair, breakdown, immediate feedback, and impatient customers was extensively analyzed by Ayyappan and Thilagavathy [i], who found the expected results for both the system size and orbit size. In their 2008 study, Badamchi Zadeh and Shahkar [4] examined queuing systems that included two phases of heterogeneous service, optional second service, and feedback for each service. In contrast to the current study, when services are parallel, they had sequential services during their studies. Afterward, performance measures for the Poisson arrival queuing system and probability-generating functions are obtained under the assumption of exponential service times. Ayyappan and Thilagavathy [2] explored closedown, breakdown and multiple vacation used MAP/PH/1.
When the system is inactive or when a customer is being served, random failures can happen. The terms "hard failure" and "soft failure" refer to two different kinds of system failure. Hard failure's typically takes a long period and needs the repairman's actual presence. On the other side, soft failure's takes less time because the system may be recovered with a simple reboot. Markovian queueing models with two different forms of server breakdown have already been studied by Jain and Jain (2010) [7], Kalyanaraman (2019) [10], Krishna Kumar(2008)[l2], Li (2013) [16], and many others. Using the matrix geometric technique, stability conditions for a single server infinite capacity Markovian queue were obtained. According to Janani [8], the final value theorem of the Laplace transform is used to convert the transient state probabilities of the model into steady-state probabilities.
When customers abandon the line because they have waited too long for service, they are considered impatient customers. Kumar [i4] investigated a non-Markovian queue with an unreliable server that first provides an essential service and then one of the m optional services. He has described the balking techniques as well as cost analysis for the objective of model optimization. A single server queueing system with associated reneging, feedback, and balking was investigated by Rakesh Kumar and Soodan [20]. We explored the time-dependent behavior of the model using the Runge-Kutta method. Additionally, they discovered the average waiting time and system size. In modeling, the arrival using a Markovian Arrival Process, a particular type of Versatile Markovian Point Process was proposed by Neuts [18]. Lucantoni et al. [17], with considerable VMPP as BMAP notational simplifications since it started in 1990. Due to its ability to simulate a broad spectrum of real-world events, MAP is an effective point process in stochastic modeling. Chakravarthy [6], describes two parameter matrices of m dimensions, let's say D0 and Di. Transitions in the MAP are determined by the generator matrix D = D0 + Di.
2. Model Formulation
Within this part, our focus is on a system for queueing with a single server, utilizing non-preemptive priority. Customers categorized as high priority (HP) arrive through a Poisson process with rate denoted by A2, while low priority (LP) customers arrive via a Markovian Arrival Process represented by (D0, Di) of order m. The matrix D0 means no arrival LP customer, while the matrix Di depicts LP customer arrival. HP customers have a limited capacity of K size, while LP customers have unlimited capacity. The fundamental arrival rate, denoted as Ai, is equivalent to n1 Die, where n1 represents the stationary probability vector. A customer categorized as HP is assumed to have a service time that follows a phase-type distribution with the notation (7, U) of order n, while an LP customer's service time is assumed to follow a phase-type distribution with the notation (y1, U') of order n'.
Upon completion of the service, if no customer is in the system, then the server will remain inactive for a random duration. That time is referred to as the delayed period. The delayed period follows an exponential distribution with parameter w. when a customer arrives during the delayed period, prompt resumption of regular service is initiated by the server. However, if the delayed period ends and any customer does not arrive, the server will proceed on a working vacation. The vacation period is generated by an exponentially distributed parameter n HP customers who arrive during this period will be served at a lower service rate and it is followed by phase-type distribution with representation (7,9U), where 0 < d < 1. As such, the mean service rate in normal mode is = [y( — U)—1e]—1, and the vacation mode of service rate is d^.
After completion of service for HP customer during working vacation, if there exists no HP customer awaiting service, then the server will doemant in vacation mode, irrespective of the presence of LP customers in the system. After the expiration of the vacation clock during a WV, the server shall revert to its normal working mode. At the end of vacation period, LP customers shall be considered for service during no HP customer present in the system. The expected service rate of an LP customer is denoted by ^2 = [7'( — U')-1e]-1.
The server is affected by soft failure(short time) during idle period and hard failure (long time) during normal busy period (both HP and LP customers). The rates of soft and hard failure are exponentially distributed with parameters and ^2. When a soft and hard failure happens, the server repair process starts immediately. The customer who is receiving service at that point must join the front of the waiting queue. If there are any customers in line when the repair is finished, the server will start servicing them. Or else, the server remains idle and repair times follows a phase-type distribution (a, T) of order l for soft failure, where T0 + Te = 0 and (a.1, T') of order l' for hard failure, where T 0 + T'e = 0. The repair rate is indicated as T1 = [a(-T)-1e]-1 and t2 = [a' (-T' )—1 e]—1 respectively.
The arriving LP Customers may balk the system with probability b during working vacation or join the system with probability (1 — b). After receiving normal service (both HP and Lp customers), the satisfied customer leave the system with probability p1 and if the customer is not satisfied with probability q1 then they will get feedback immediately.
3. The QBD process infinitesimal generation matrix
Notations
We will need the following notations:
• 0 -Kronecker product of two matrices of various dimensions resulting in a block matrix.
• ® - Kronecker sum of two matrices of various dimensions resulting in a block matrix.
• Im stand for identity matrix of m x m order.
• e - a column vector of the suitable order. Each of its entries is one.
• e0-e3m+2Knm+Kl' m+(K+1)lm •
• e1-eKmn+(K+1)n' m+(K+i)l'm+(K+i)lm+Kmn+nf
• N1(t): the total number of LP customers in the system at epoch t.
• N2(t): the total number of HP customers in the system at epoch t.
• J(t) represents the server's status at epoch t.
As a result, the server is in one of the following states at any given time t:
J(t)
0, idle during normal mode,
1, if the server is offering service to HP customers during normal mode,
2, if the server is offering service to LP customers during normal mode,
3, hard failure (during normal busy mode),
4, delay time,
5, soft failure (during idle),
6, busy(HP) in working vacation mode,
7, idle in working vacation mode.
R(t) stands for the repair process considered by phases. K(t) stands for phases of the service.
A(t)- The Markovian arrival process is considered in phases.
Let Y-{Y(t) : t > 0}, where Y(t) = {N1 (t),N2(t), J(t), R(t), K(t), A(t)} is a CTMC with state space
0(0) U <K0.
i=1
(1)
where
0(0) ={(0,0,0,a) : 1 < a < m} u {(0,r,1,k1,a) : 1 < r < K, 1 < k1 < n, 1 < a < m}
u {(0,r,3,k4,a) : 1 < r < K, 1 < k4 < l', 1 < a < m} u {(0,0,4,a) : 1 < a < m} u {(0,0,5,k3,a) : 0 < r < K, 1 < k3 < l, 1 < a < m}
u {(0,1,6,fci,a) : 1 < r < K, 1 < < n, 1 < a < m} u {(0,0,7,a) : 1 < a < m}, and for i > 1,
0(0 ={(i,r, 1,k1,a) : 1 < r < K, 1 < k1 < n, 1 < a < m}
u {(i, r, 2,k2, a) u {(i, r, 3,k4, a) u {(i, r, 5,k3, a) u {(i, r, 6,k1, a)
0 < r < K, 1 < k2 < n', 1 < a < m}
0 < r < K, 1 < k4 < l', 1 < a < m}
0 < r < K, 1 < k3 < l, 1 < a < m}
1 < r < K, 1 < k1 < n, 1 < a < m} u {(i,0,7,a) : 1 < a < m}.
3.1. The Infinitesimal Generator Matrix
The quasi-birth-death process has the generator matrix Q given by
Q
B00 B01 0 0 0 0
B10 A1 A0 0 0 0
0 A2 A1 A0 0 0
0 0 A2 A1 A0 0
(2)
300
"Boo B00 0 0 Ri5 B00 0 0
0 B0202 r23 B00 B24 B00 0 0 0
0 R32 B00 B33 B00 0 0 0 0
0 r42 B00 0 B44 B00 0 0 B47 B00
R5i B00 B52 B00 0 0 B55 B00 0 0
0 62 B00 0 0 0 B66 B00 B67 B00
-b00 0 0 0 0 W76 B00 B77
where
B^ = Do — (A2 + fai)Im, Bq2 = el ® a ® A2Im, BJ5 = e[(K + 1) ® a ® faI„
Boo
L1 L2 0 .
L3 Li L2 . 0 L3 Li .
, B23 = Ik ® en ® a' ® faIm, B24 = eiK ® qU0 ® I„
0 0 0... L1 L2 0 0 0 ... L3 L1 + L2_
where Li = (U + pU%) ® D0 - (A2 + fa)Inm, L2 = A2Inm, L3 = qU°j ® I„
■l4 l5 0 ... 0 0
0 l4 L5 ... 0 0
0 0 L4 ... 0 0
B32 = Ik ® T 7 ® Im, Bi
00
L4 L5
0 L4 + L5.
0 0 0. 0 0 0.
where L4 = T' ® D0 — A2 Ii'm, L5 = A2I//m.
B42 = eiK ® a ® A2Im, B44 = D0 — (A2 + w)Im, B47 = wIm, B5i = ei (K + i) ® T0 ® In
R52 B00
0 0 0 ... 0 0
T0 7 ® Im 0 0 ... 0 0
0 T0 7 ® I m 0 ... 0 0
0 0 T0 7 ® Im ... 0 0 ,
0 0 0 ... T0 7 ® Im 0
- 0 0 0 ... 0 T0 7 ® Im.
L6 L7 0 0 0 " Ls L9 0 .. 0 0
0 L6 L7 0 0 Li0 Ls L9... 0 0
0 0 L6 0 0 0 Li0 Ls . . . 0 0
66 , B00 =
0 0 0 L6 L7 0 0 0 .. Ls L9
0 0 0 0 Lo + Ly. 0 0 0... Li0 Ls + L9
R55 B00
where L6 = T ® D0 — A2Iim, L7 = A2Iim, Ls = 9U ® (D0 + bDi) — (n + A2)Inm, L9 = A2Inm Li0 = dU' ® Im.
Boo = ik ® en ® n7 ® Im, Bg7 = eiK ® 9U° ® Im, b70 = nIm, B76 = eiK ® 7 ® A2Im B77 = (D0 + bDi) — (n + A2) Im.
301
0 B0112 0 0 0 0
B21 0 0 0 0 0
0 0 B33 B01 0 0 0
0 B42 B01 0 0 0 0
0 0 0 B54 B01 0 0
0 0 0 0 B65 B01 0
0 0 0 0 00 B76 B01
where
b12 = e[(K + 1) 0 y 0 di, b21 = ik 0 In 0 di,
R33 B01
'0 Iv 0 D1 0 0
0 0 Iv0 D1 0 0 0 0 Iv 0 D1
b51 = ik+1 0 Il 0 d1, b65
I// 0 D1 0 0 Ip 0 Dl
, B42 = e1(K + 1) 0 Y 0 D1,
Ik 0 In 0 (1 - b)D1, B76 = (1 - b)D1.
10
0 0 0 0 0 0 0
0 B22 0 B24 B10 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
where
b22
0
qU'07 0 im 0 0
00 00 qU'07 0 Im 0
0 qU'0y 0 Im
B24 = e1(K + 1) 0 qU'0 0 I„
qU'0 Im 0
0 qU'0y 0 Im.
where
B1111
11
B11 B1112 B11 0 0
0 b22 B23 0 0
B31 B31 B33 B11 0 0
B114111 B114112 0 B1414 0
B115111 0 0 0 B55 Bl1
0 B1612 0 0 65 BU
0 0 0 0
B56 B11 B66 B11-
"¿1 ¿2 0 . .0 0 qU0y 0 Im 0 0 .. .0 0
Li ¿1 L2.. .0 0 0 0 0 .. .0 0
0 Li L1 . . .0 0 , B12 = 0 0 0 .. .0 0
0 0 0 .. .0 0
0 0 0 . . L1 L2
0 0 0.. . L1 ¿1 + ¿2. 0 0 0 .. .0 0
R13 _ B11 =
'0 e„ 0 02«' 0 im
0 0 e„ 0 02a' 0 im 0
0 e„ 0 02«' 0 im
e„ 0 02a' 0 Im 0
0 e„ 0 02a' 0 im.
L11 L12 0 . .0 0
0 L11 L12.. .0 0
00 L11 . . .0 0
00 0 . . L11 L12
00 0.. .0 L11 + L12
, b21 = ik+1 0 en 0 02a' 0 im,
where Ln = (U' + pU'071) ® D0 - (A2 + 02)W L12 = A2i„
11 B11
0
T '0 7 0 im 0 0
0 0
T'07 0 im 0
T'0 7 0 i„
0 0
t'07 0 im 0
T'0 7 0 imJ
' 0 im 0 0 . .0 0 -L4 L5 0 . .0 0
0 0 0.. .0 0 0 L4 L5.. .0 0
0 0 0 . .0 0 R11 = , B11 = 0 0 L4 . .0 0
0 0 0.. .0 0 ,
0 0 0.. . L4 L5
0 0 0 . .0 0 0 0 0 . .0 L4 + L5.
0
T0 7 0 im 0 0
0 0
T0 7 0 im 0
0 0 0
T0 7 0 im
0 0 0
0 0 0
"T°7 0 im 0 0 .. .0 0"
0 0 0 .. .0 0
0 0 0 .. .0 0
0 0 0 .. .0 0
0 0 0 .. .0 0
Ik 0 e„ 0 ^7 0 i m,
T0 7 0 im 0
r44 , B11
00 00
0 0 0 0
0
T0 7 0 im.
0 . .0 0
L7 . .0 0
L6 . .0 0
0 . . L6 L7
0 . .0 L6 + L7
B
r55 R11
'l8 L9 0
¿10 L8 L9 0 L10 ¿8
B62 = e1(K + 1) ® 77' ® I„
r56 R11
00 00 00
¿8 ¿9 ¿10 ¿8 + ¿9.
B65 = e[(K) ® a ® A2Im, B66
e1K ® 0U0 ® I„
(D0 + bD1) - (7 + A2)I„
312
where
B11 = Ik ® In ® D1,
B22
11 R12 0 0 0 0 0
0 R22 R12 0 0 0 0
0 0 R33 R12 0 0 0
0 0 0 R44 B12 0 0
0 0 0 0 R55 R12 0
0 0 0 0 0 R66 S12-l
ik+1 ® In' ® D1,
r33 _ R12 =
ik+1 ® Iv ® D1,
B42 = Ik+1 ® I/ ® D1, B55 = Ik ® in ® (1 - b)D1, B^ = (1 - b)D1,
where
21 R21
0
qU'0y ® Im 0 0
B21
0 0
qU'0y ® Im 0
0 0 0 0 0 0
R21 R21 0 R23 R21 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0
qU'0y ® Im
qU'0y ® Im 0
qU'0y ® Im.
r23 R21
■qU'V ® Im 0 0
0 0 0
0 0 0
0 0 0
0 0 0
00 00 00 00
00
4. Analysis of Stability Condition We examined our model under the assumption that the system is stable.
4.1. Condition for Stability
Let A = A0 + A1 + A2 be the square matrix of order Kmn + (K + 1)n'm + (K + 1)/'m + (K + 1)/m + Kmn + m and it is an infinitesimal generator matrix is an irreducible. Let x indicate the steady-state probability vector of A satisfying xA = 0 and xe = 1. The vector x is partitioned by
X = (X0, Xh X2, X3, X4, X5)=(X00, X01/.../ X0K-1/X0K, X\h Xu^.^ XiK-i/Xik , X20, Xn,..., X2K-1/X2K, X30, X31,..., X3K-1, X3K, X40, X41,..., X4K-1, X4K, X50, X51,..., X5K-1, X5K), where X0 is of dimension Kmn, X1 is of dimension (K + 1)n'm, x2 is of dimension (K + 1)1'm, x3 is of dimension (K + 1)/m, x4 is of dimension Kmn and x5 is of dimension m. The probability vector x is calculated by solving the following equations:
X00[(U + pU%) ® im - (A2 + n)inm] + X01 (qU0Y ® im) + X11 ® im)
+ X21(T'0Y ® im) + X31 (T0Y ® Im) + X40(en ® nY ® Im) = 0. X0j -1( A2 ^nm ) + X0j [(U + pU0y) ® im - (A2 + n)Inm] + X0j+1 (qU0Y ® im) + X1j+1(qU'0Y ® im) + X2j+1 (T'0Y ® Im)+ X3j+1 (T0Y ® Im )+ X4j (*n ® nY ® im) = 0, for 1 < j < K - 1. X0K-1 (A2inm) + X0K[(U + pU0Y) ® Im - ninm] + X4K(en ® nY ® Im) = 0. X00(qU0y' ® Im) + X10[(U' + pU'0y') ® Im - (A2 + fa)In'm] + X20(T'V ® Im) + X30(T0Y ® Im)
+ X50 (nY ® Im) = 0.
X1j-1 (A2In'm) + X1j[(U' + pU'0Y') ® Im - (A2 + fa)In'm] + X5j(nY' ® Im) = 0, for 1 < j < K - 1.
X1L-1 (A2In'm) + X1L[(U' + pU'0Y') ® Im] = 0.
X10[((«n ® ^2a!) + qU'0y') ® Im] + X20(T' ® Im - A2i/'m) = 0.
X0j-1 [en ® faa' ® im] + X1j[«n ® ® im] + X2j-1 (A2i/'m) + X2j(T' ® im - A2i/'m) = 0,
for 1 < j < K - 1.
X0K [en ® faa' ® im] + X1K [en ® ^2a' ® im] + X2K-1 (A2i/'m) + X2K(T' ® im) = 0. X30(T ® im - A2i/m) = 0,
X3j-1 (A2i/m) + X3j(T ® im - A2i/m) = 0, f Or 1 < j < K - 1. X3K-1 (A2i/m) + X3K(T ® im) = 0.
X40[QU ® im - (n + A2)inm] + X41 (QU'® im) + X50(a ® A2im) = 0.
X4j-1 (A2inm) + X4j [QU ® im - (n + A2)inm] + X4j+1 (QU' ® im) + X5j(a ® A2im) = 0,
for 1 < j < K - 1.
X4K-1 (A2 inm ) + X4K[QU ® im - ninm] + X5K(a ® A2im) = 0. X4K-1 (e1K ® QU0 ® im) - X5K(n + A2)im = 0.
Subject to normalizing condition
K K K K K
^ X0renm + ^ X1ren'm + ^ X2re/'m + ^ X3r«/m + ^ X4r«nm + X50«m = 1.
r=1 r=0 r=0 r=0 r=1
The stability condition xA0e < xA2e is obtained after some algebraic simplification, which turns out to be
K K K K
^ X0r(en ® D1 em) + ^ X1r (en' ® D1 em) + ^ X2r (e/' ® D^m) + ^ X3r(e/ ® D1 em)
r=1 r=0 r=0 r=0
KK
+ ^ X4r(en ® (1 - b)D1em) + X50(1 - b)D1 em < ^ X1r(qU'0 ® em).
r=1 r=0
4.2. The Stationary Probability Vector
Let y be the stationary probability vector of the infinitesimal generator Q of the process {Y(t): t > 0}. The subdivision of y by level as, y = (y0, y1,y2,...), where y0 is of dimension (3m + 2Knm + K/'m + (K + 1)/m) for i = 0 and y1, y2,... are of dimension Kmn + (K + 1)n'm + (K + 1)/'m + (K +
1)1m + Kmn + m for i > 1. As y is a stationary probability vector satisfies the relation yQ = 0 and ye = 1. Furthermore, while the stability criterion is satisfied, the equation gives the various levels.
y = y1 Rj-1, j > 2 (3)
where R is the smallest non-negative solution of the quadratic equation
R2 A2 + RA1 + A0 = 0
and satisfies the relation RA2e = A0e and the vector y0, y1 are obtained with the help of succeeding equations:
1 + y1 B10 = 0, (4)
y0 B01 + y1 [ A + RA2]= 0, (5)
subject to normalizing condition
y0e0 + y1 [i - R]-1 e1 = 1. (6)
As a result, we can compute matrix R using Logarithmic reduction algorithm in Latouche and Ramaswami[15] and the vector y by using the special structure of something like the coefficient matrices.
5. Busy Period Analysis
• In a single-server queueing demonstration, the word busy period is characterized as the length of time between the entry of a customer into the void system and the first time from that point that the system size reaches zero. As, the first passage epoch to level zero, starting from level one. It is the first return time of level zero, taken after by a least one visit to a few other levels, which is the analog of the busy cycle.
• We have to present an outline of the fundamental period to analyze the busy period. when the QBD process is taken into thought the first passage time from level i to i - 1, where i > 2.
• It is worth pointing out that for each level i, i > 2, there are (3m + 3nm + 1m) states. The state (i, j) of level i signifies the jth state of level i when the states are sorted alphabetically.
• Let Gjj' (u, y) represent the conditional probability that the QBD process, starting at time t = 0 in the state (i, j) and keep track of the time until the first visit to the level (i - 1) but not later than time y. We can modify after exactly m transitions to the left and enter the state (i, j'),
t = 0.
Let the joint transform matrix
__TO f to
Gj(z,s) = £ zM e-sydGj;v(u,y) ; |z| < 1, Re(s) > 0, (7)
u=1 -'0
and put the matrix G(z, s) = Gjj' (z, s). Specifically, computed the matrix G(z, s) satisfy the equation,
G(z,s) = z(Si - A1 )-1 A2 + (Si - A1 )-1 A0G2(z,s). (8)
The matrix G = Gjj' = G(1,0) is concerned with negating the boundary states during the first passage times. knowing the rate matrix R allows us to use the below result to find the matrix G
G = -(A1 + RA2 )-1 A2. (9)
The matrix G can be found with the assistance of the Logarithmic reduction algorithm [15]. We find the matrix with the succeeding equation
G(1,0)(z,s) = z(si - A1 )-1 B10 + (si - A1 )-1 AoG(z,s)G(1,0)(z,s) G(0,0)(z,s) = (si - Boo)-1 Bo1G(1,0)(z,s).
(10) (11)
Thus, the moments that obey are calculated using the matrices G, G(0,0)(1,0) and G(1,0)(1,0) are stochastic at z = 1 and s = 0. We can find the moments as follows:
f = - — G(z, s)e = -[A1 + A0 (i + G)]-1 e, ds
f2 = dz G(z, s)e = -[ A + A0 (i + G)]-1 A2e,
f(1,0) = - g G(1,0) (z, s)e = -[ A1 + A)G]-1( A0f1 + e),
f2(1,0) = dz G(1,0)(z, s)e = -[ A + A0G]-1 (A0 f + B10e), fr = G0,0 (z, s)e = -B0-01[B01 + e],
f(0,0) = I G(0,0) (z, s)e = -B0-1 [B01 f(1,0)].
dr (1,0),
(12)
(13)
(14)
(15)
(16) (17)
6. System Performance Measures
• Expected number of LP customers in the system Elp = 1 ¿y/e.
• Probability that the server is idle
P/die = KT=1 y000«.
• Probability that the server busy with HP customers
D __V^K v^n V^m
P Hbusy = £i=0 £r=1 £k1=1 £a=1 yir1fc1 a
• Probability that the server is on hard failure
PHF = Ef=1 Efc4=1 £m=1 y0r3k4a + Ef=0 Efc4=1 £m=1 yir3k4a
• Probability that the server is Delay time to go for vacation
PDT = KT=1 y004a
• Probability that the server is busy during working vacation
D __V^K v^n V^m
P BWV = £i=0 £r=0 £k1=1 £a=1 yir6fc1 a
7. Numerical Implementation
To compute numerical outcomes, we have employed distinct MAP representations for the arrival process in a manner that ensures their mean values are 1, as recommended by Chakravarthy [6].
Erlang of order 2 (ERL-A):
D0
Exponential (EXP-A): Hyper exponential (HYP-A)
D0 =
-22 , D1 = 0 0
0-2 2 0
D0 =[-1], D1 = [1].
-1.90 0 0 -0.19
D1
1.710 0.190 0.171 0.019
MAP-Negative Correlation (MAP-NC-A):
D0
-1.00243 0
1.00243 1.00243
0 0 -225.797
MAP-Positive Correlation (MAP-PC-A):
D1
D0
-1.00243 1.00243 0 0 -1.00243 0 0 0 225.797
, D1
000 0.01002 0 0.99241 223.539 0 2.258
000 0.99241 0 0.01002 2.258 0 223.539
-2 2 02
-2 2 02
Let us consider PH-distributions for the service and repair process as follows: Erlang of order 2 (ERL-S):
Y = y' =[1,0], U = U' Erlang of order 2 (ERL-R):
a = a' = [1,0], T = T'
Exponential (EXP-S): Exponential (EXP-R):
Hyper exponential (HYP-S):
Y = y' =[0.8,0.2], U = U' Hyper exponential (HYP-R):
a = a' =[0.8,0.2], T = T'
Y = y' =[1], U = U' =[-1]. a = a = [1], T = T = [-1].
-2.8 0 0 0.28
-2.8 0 0 0.28
7.1. Illustration 1
We have examined the consequence of the hard failure rate fa against the Expected number of LP customers in the system(ELP) in the following tables 1-3. Fix = 20, = 15, K = 5, A1 = 1, A2 = 1.5, n = 8, w = 0.5, fa = 0.5, r1 = 2, t2 = 6, Q = 0.6, b = 0.7, p1 = 0.3, q1 = 0.7 such that the system is stable.
• As the hard failure rate (fa) increases, the variety of arrangements of arrival and service times than the corresponding Elp also increases.
• Observe the arrival times, Elp increases highly in MAP - PC - A and increases much slower in ERL — A than all other arrival times.
7.2. Illustration 2
We investigated the impact of the vacation rate (n) against the probability of the server being idle (Pid/e) in the following tables 4-6. Fix = 20, = 15, K = 5, A1 = 1, A2 = 1.5, w = 0.5, fa = 0.5, fa = 1, T1 = 2, t2 = 6, Q = 0.6, b = 0.7, p1 = 0.3, q1 = 0.7 such that the system is stable.
• As the vacation rate (n) increases, the variety of arrangements of arrival and service times than the corresponding Pid/e also increases.
• While comparing to EXP - S and HYP - S, Pid/e increases more rapidly for ERL - S. Similarly, Pid/e increases slowly for HYP - S.
7.3. Illustration 3
We analyze the effect of the repair rate (t2) on the probability of the server being busy for HP customer (PHbuSy) in the following tables 7-9. Fix = 16, ^2 = 15, K = 5, A1 = 1, A2 = 1.5, n = 8, w = 0.5, fa = 0.5, fa = 1, T1 = 2, Q = 0.6, b = 0.7, p = 0.3, q = 0.7 such that the system is stable.
• While maximizing the repair rate (T2), PHbwsy minimizes for various possible arrangements of arrival and service times.
• When correlating the distinct arrival times, PHbuSy decreases more quickly in the case of MAP - PC - A whereas slowly in ERL - A. Similarly, considering the service times, PHbwsy decreases gradually in ERL - S and highly in HYP - S.
7.4. Illustration 4
To determine the existence of the service rate of HP customer (^2) versus the expected system size for LP customer (ELP) in Figures 1-5. Fix = 15, K = 5, A1 = 1, A2 = 1.5, n = 8, w = 0.5, fa = 0.5, fa = 1, T1 = 2, t2 = 6, Q = 0.6, b = 0.7, p1 = 0.3, q1 = 0.7 such that the system remains stable.
A quick observation from Figures 1-5, Elp decreases while increasing the service rate of HP customers for all combinations of arrival and service time groupings. Due to the availability of the HP service rate in the system, the customers will get service successfully which leads to Elp decreases. However, ELP decreases slowly for ERL - A with the combination of ERL - S whereas slowly in HYP - S. Likewise, Elp decreases highly for HYP - A in HYP - S whereas slowly in ERL - S.
7.5. Illustration 5
To see the features of both the HP service rate (^1) and repair rate of hard failure (t2) on the expected number of LP customers in the system (Elp ) in the Figures 6 -10. Fix ^2 = 15, K = 5, A1 = 1, A2 = 1.5, n = 8, w = 0.5, fa = 0.5, fa = 1, T1 = 2, Q = 0.6, b = 0.7, p = 0.3, q = 0.7 such that stability condition is satisfied.
Observation in Figures 6 -10, we increase the values of both the HP service rate and repair rate of hard failure, then Elp decreases with various arrival groupings. Due to the HP customer increase in the service rate, Elp decreases likewise increase the repair rate of hard failure decrease in the Elp. Let's look at the arrival times, Elp decreases slowly for ERL - A and decreases fastly for MAP PC A.
Table 1: Hard Faz'Zure rate (02) vs Elp - ERL-S
02 ERL - A EXP - A HYP - A NC - A PC - A
1 0.163936 0.206320 0.293281 0.330092 20.674732
1.2 0.172609 0.217533 0.312405 0.342647 21.257457
1.4 0.181624 0.229181 0.332498 0.355637 21.857954
1.6 0.190997 0.241285 0.353617 0.369083 22.477042
1.8 0.200747 0.253867 0.375822 0.383009 23.115590
2 0.210893 0.266951 0.399180 0.397438 23.774519
2.2 0.221454 0.280563 0.423761 0.412394 24.454807
2.4 0.232454 0.294730 0.449638 0.427907 25.157494
2.6 0.243915 0.309480 0.476893 0.444003 25.883685
Table 2: Hard Faz'/wre rate (02) vs Elp - EXP-S
02 ERL - A EXP - A HYP - A NC - A PC - A
1 0.174277 0.217929 0.311487 0.340494 20.533617
1.2 0.182994 0.229005 0.330339 0.352724 21.059251
1.4 0.191976 0.240406 0.349917 0.365273 21.596139
1.6 0.201232 0.252143 0.370253 0.378153 22.144696
1.8 0.210771 0.264230 0.391379 0.391377 22.705353
2 0.220605 0.276679 0.413329 0.404958 23.278556
2.2 0.230744 0.289502 0.436138 0.418908 23.864775
2.4 0.241200 0.302715 0.459844 0.433243 24.464498
2.6 0.251986 0.316331 0.484485 0.447976 25.078233
Table 3: Hard Faz'/wre rate (02) vs Elp - HYP-S
02 ERL - A EXP - A HYP - A NC - A PC - A
1 0.233555 0.282009 0.402324 0.416602 19.623165
1.2 0.240760 0.290404 0.415758 0.425489 19.866887
1.4 0.247923 0.298753 0.429141 0.434323 20.109536
1.6 0.255059 0.307070 0.442498 0.443121 20.351499
1.8 0.262179 0.315369 0.455850 0.451895 20.593109
2 0.269292 0.323660 0.469217 0.460658 20.834659
2.2 0.276407 0.331955 0.482615 0.469419 21.076406
2.4 0.283532 0.340261 0.496061 0.478189 21.318580
2.6 0.290674 0.348587 0.509566 0.486975 21.561381
Table 4: Vacation rate (n) vs P^ - ERL-S
n ERL - A EXP - A HYP - A NC - A PC - A
5 0.082706 0.088338 0.098335 0.092979 0.094606
6 0.084704 0.090362 0.100285 0.094971 0.096400
7 0.086239 0.091918 0.101790 0.096500 0.097789
8 0.087467 0.093162 0.102999 0.097723 0.098910
9 0.088478 0.094188 0.103999 0.098732 0.099840
10 0.089331 0.095053 0.104845 0.099583 0.100629
11 0.090063 0.095797 0.105573 0.100315 0.101311
12 0.090700 0.096444 0.106210 0.100952 0.101907
13 0.091261 0.097015 0.106772 0.101514 0.102435
Table 5: Vacation rate (n) vs Pid;e - EXP-S
n ERL - A EXP - A HYP - A NC - A PC - A
5 0.084405 0.090174 0.100258 0.095251 0.096557
6 0.086462 0.092258 0.102267 0.097311 0.098407
7 0.088037 0.093853 0.103811 0.098887 0.099835
8 0.089292 0.095124 0.105045 0.100142 0.100980
9 0.090321 0.096167 0.106061 0.101172 0.101925
10 0.091185 0.097042 0.106916 0.102036 0.102722
11 0.091922 0.097789 0.107648 0.102775 0.103407
12 0.092561 0.098437 0.108283 0.103415 0.104002
13 0.093121 0.099006 0.108842 0.103977 0.104527
Table 6: Vacation rate (n) vs Pjd;e - HYP-S
n ERL - A EXP - A HYP - A NC - A PC - A
5 0.090101 0.096426 0.106953 0.102570 0.103410
6 0.092254 0.098602 0.109047 0.104734 0.105339
7 0.093876 0.100237 0.110625 0.106360 0.106796
8 0.095147 0.101518 0.111863 0.107631 0.107942
9 0.096173 0.102552 0.112863 0.108656 0.108870
10 0.097022 0.103406 0.113691 0.109503 0.109639
11 0.097737 0.104125 0.114389 0.110217 0.110290
12 0.098349 0.104741 0.114987 0.110827 0.110848
13 0.098880 0.105275 0.115507 0.111356 0.111333
Table 7: Repa/r rate (T2) vs PHbuSy - ERL-S
T2 ERL - A EXP - A HYP - A NC - A PC - A
5 0.105099 0.105226 0.105427 0.105573 0.105653
6 0.105061 0.105172 0.105341 0.105479 0.105548
7 0.105034 0.105135 0.105284 0.105412 0.105472
8 0.105014 0.105107 0.105243 0.105362 0.105416
9 0.104998 0.105086 0.105213 0.105323 0.105373
10 0.104986 0.105070 0.105190 0.105292 0.105339
11 0.104976 0.105056 0.105171 0.105267 0.105311
12 0.104968 0.105045 0.105156 0.105246 0.105288
13 0.104961 0.105036 0.105143 0.105228 0.105268
Table 8: Repa/r rate (T2) vs PHbuSy - EXP-S
T2 ERL - A EXP - A HYP - A NC - A PC - A
5 0.103623 0.103745 0.103941 0.104090 0.104179
6 0.103590 0.103698 0.103864 0.104004 0.104081
7 0.103565 0.103663 0.103811 0.103941 0.104009
8 0.103545 0.103637 0.103772 0.103892 0.103954
9 0.103530 0.103616 0.103742 0.103854 0.103911
10 0.103518 0.103600 0.103719 0.103824 0.103876
11 0.103509 0.103587 0.103701 0.103798 0.103848
12 0.103500 0.103576 0.103685 0.103778 0.103824
13 0.103493 0.103567 0.103673 0.103760 0.103805
Table 9: Repa/r rate (T2) vs PHbuSy - HYP-S
T2 ERL - A EXP - A HYP - A NC - A PC - A
5 0.095098 0.095149 0.095235 0.095317 0.095365
6 0.095158 0.095216 0.095310 0.095423 0.095481
7 0.095182 0.095241 0.095332 0.095455 0.095514
8 0.095191 0.095247 0.095335 0.095458 0.095515
9 0.095192 0.095247 0.095329 0.095449 0.095503
10 0.095190 0.095243 0.095321 0.095436 0.095487
11 0.095187 0.095238 0.095313 0.095422 0.095470
12 0.095183 0.095232 0.095304 0.095408 0.095453
13 0.095180 0.095227 0.095296 0.095394 0.095438
(a) ERL-A
m
Figure 1: High priority service rate vs. Elp (c) HYP-A
16 18 20 22 24 F1
Figure 3: High priority service rate vs. Elp
(e) MAP-PC-A
Figure 5: High priority service rate vs. ELP
(b) EXP-A
16 18 20 22 24
m
Figure 2: High priority service rate vs. Elp
(d) MAP-NC-A
Figure 4: High priority service rate vs. ELP
M,Ek/Ek,Ek/1
Figure 6: HP service (^1) and Repair(HF) (T2) rates vs. Elp - ERL-S
M,M/E. ,E./1
' k' k
j 0.3
m
12
22
Figure 7: HP service (^1) and Repair(HF) (T2) raies vs. Elp - ERL-S
Figure 9: HP service (^1) and Repair(HF) (T2) raies vs. Elp - ERL-S
M,Hk/Ek>Ek/1
26
Figure 8: HP service (^1) and Repair(HF) (T2) raies vs. Elp - ERL-S
Figure 10: HP service (^1) and Repair(HF) (T2) rates vs. Elp - ERL-S
8. Conclusion
This paper contributes by employing the Matrix analytic method to compute the stationary distribution of the number of customers in the M, MAP/PH1, PH2/I queueing system with delayed working vacations under non-preemptive priority. We discussed some system performance measures using steady-state probabilities and also calculated busy period analysis. We used numerical examples to show how different system parameters affect performance measures.
References
Ayyappan, G. and Thilagavathy, K. (2021). Analysis of MAP/PH/1 queueing model with immediate feedback, starting failures, single vacation, standby server, delayed repair, breakdown and impatient customers, International Journal of Mathematics in Operational Research, 19(3):269-301.
Ayyappan, G. and Thilagavathy, K. (2021). Analysis of MAP/PH/1 Queueing model with Setup, Closedown, Multiple Vacations, Standby Server, Breakdown, Repair and Reneging, Reliability: Theory and Applications, 15(2):104-143.
Ayyappan, G. and Somasundaram, B. (2019). Analysis of two-stage MX1,MX2/Gi, G2/1 retrial G-queue with discretionary priority services, working breakdown, Bernoulli vacation, preferred and impatient units, Applications and Applied Mathematics: An International Journal, 14(2):640-671.
Badamchi Zadeh, A. and Shahkar, G. H. (2008). A Two Phases Queue System with Bernoulli Feedback and Bernoulli Schedule Server Vacation, International Journal of Information and Management Sciences, 19:329-338.
Bhagat, A. and Jain, M. (2020). Retrial queue with multiple repairs, multiple services, and non-preemptive priority, OPSEARCH, 57(3):1-28.
Chakravarthy, S.R. (2011). Markovian arrival processes, Wiley Encyclopedia of Operation Research and Management Science.
Jain, M. and Jain, A (2010). Working vacation queueing model with multiple types of server breakdowns, Applied Mathematical Modelling, 34:1-13.
Janani, B. (2022).Transient Analysis of Differentiated Breakdown Model, Applied Mathematics and Computation, 417: p.126779.
Jeganathan, K., Anbazhagan, N. and Kathiresan, J. (2013). A Retrial Inventory System with Non-Preemptive Priority Service, International Journal of Information and Management Sciences, 24(1):57-77.
Kalyanaraman, R. and Sundaramoorthy, A. (2019). A Markovian working vacation queue with server state-dependent arrival rate and with partial breakdown, International Journal of Recent Technology and Engineering, 7:664-668.
Krishna Reddy, G. V., and Anitha, R. (1998). Markovian bulk service queue with delayed vacations, Computers and Operations Research, 25(12):1159-1166.
Krishna Kumar, B., Rukmani, R., Thanikachalam, A., and Kanakasabapathi, V. (2008). Performance analysis of retrial queue with server subject to two types of breakdowns and repairs, Operations Research, 18(2):521-559.
Krishnamoorthy, A. and Divya, V.(2020). (M, MAP)/(PH, PH)/1 Queue with Non-preemptive Priority and Working Vacation Under N-Policy, Journal of the Indian Society for Probability and Statistics, 21(1): 69-122
Kumar, B. (2018). An unreliable bulk queueing model with optional services, Bernoulli vacation schedule and balking, International Journal of Mathematics in Operational Research, 12(3):293-316.
Latouche, G., and Ramaswami, V. (1993). A Logarithmic Reduction Algorithm for Quasi-Birth-Death Processes, Journal of Applied Probability, 30:650-674.
Li, L., Wang, J., and Zhang, F. (2013). Equilibrium customer strategies in Markovian queues with partial breakdowns, Computers and Industrial Engineering, 66:751-757. Lucantoni, D.M., Meier-Hellstern, K.S., and Neuts, M.F. (1990). A single server queue with server vacations and a class of non-renewal arrival processes, Advances in Applied Probability, 22:676-705.
Neuts, M. F. (1979). A Versatile Markovian Point Process, Journal of Applied Probability, 16:764-779.
Pikkala, V. L. and Vepada, S. (2014). Analysis of general input state-dependent working vacation queue with changeover time, ISRN Computational Mathematics, 8, Article ID 248704. https://doi.org/10.1155/2014/248704.
[20] Rakesh Kumar and Soodan, B.S. (2019). Transient Numerical Analysis of a Queueing Model with Correlated Reneging, Balking, and Feedbac,. Reliability: Theory and Applications, 14(4):46-54.
[21] Subramanian, A.M.G., Ayyappan, G. and Sekar, G. (2009). M/M/1 Retrial queueing system with negative arrival under non-preemptive priority service, Journal of Fundamental Sciences, 5(2):129-145.
[22] Yang, D.-Y., Chen, Y.-H. and Wu, C.-H. (2021). Sojourn times in a Markovian queue with working breakdowns and delayed working vacations. Computers and Industrial Engineering 156(107239):1-13.
