Научная статья на тему 'PHASE TYPE QUEUEING MODEL OF SERVER VACATION, REPAIR AND DEGRADING SERVICE WITH BREAKDOWN, STARTING FAILURE AND CLOSE-DOWN'

PHASE TYPE QUEUEING MODEL OF SERVER VACATION, REPAIR AND DEGRADING SERVICE WITH BREAKDOWN, STARTING FAILURE AND CLOSE-DOWN Текст научной статьи по специальности «Медицинские технологии»

CC BY
246
50
i Надоели баннеры? Вы всегда можете отключить рекламу.
Ключевые слова
Phase type Distribution / Markovian Arrival Process / Degrading Service / Server Vacation / Breakdown / Repair / Starting failure / Close-down / Matrix-analytic method

Аннотация научной статьи по медицинским технологиям, автор научной работы — G. Ayyappan, S. Meena

We consider a single server phase type queueing model with server vacation, repair, breakdown, degrading service, starting failure and closedown. When the arrival rate of the customer follows the Markovian Arrival Process (MAP) and the service rate of the server follows the phase-type distribution. If no one is in the system when the server is back from the vacation, then the server will wait until the customer arrives. If the customer arrives at the moment with no starting failure, then he provides service, otherwise the server immediately goes to the repair process. Here, the service rate declining until degradation fixed. After completion of K services the degradation is addressed. During the period of service, the server may get a breakdown at any moment, and then the server immediately goes for a repair process. After completing the service, he switches to the close-down process, and then he goes on vacation. Using the Matrix-Analytic method, The stationary probability vector representing the total number of customers in the system is examined. The analysis of the busy period, the mean waiting time, and cost analysis are discussed. A few significant performance measures are attained. Finally, some numerical examples are given.

i Надоели баннеры? Вы всегда можете отключить рекламу.
iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.
i Надоели баннеры? Вы всегда можете отключить рекламу.

Текст научной работы на тему «PHASE TYPE QUEUEING MODEL OF SERVER VACATION, REPAIR AND DEGRADING SERVICE WITH BREAKDOWN, STARTING FAILURE AND CLOSE-DOWN»

PHASE TYPE QUEUEING MODEL OF SERVER VACATION, REPAIR AND DEGRADING SERVICE WITH BREAKDOWN, STARTING FAILURE AND

CLOSE-DOWN

G. Ayyappan, S. Meena •

Department of Mathematics, Puducherry Technological University, Puducherry, India. ayyyappan@ptuniv.edu.in, meenasundar2296@gmail.com

Abstract

We consider a single server phase type queueing model with server vacation, repair, breakdown, degrading service, starting failure and closedown. When the arrival rate of the customer follows the Markovian Arrival Process (MAP) and the service rate of the server follows the phase-type distribution. If no one is in the system when the server is back from the vacation, then the server will wait until the customer arrives. If the customer arrives at the moment with no starting failure, then he provides service, otherwise the server immediately goes to the repair process. Here, the service rate declining until degradation fixed. After completion of K services the degradation is addressed. During the period of service, the server may get a breakdown at any moment, and then the server immediately goes for a repair process. After completing the service, he switches to the close-down process, and then he goes on vacation. Using the Matrix-Analytic method, The stationary probability vector representing the total number of customers in the system is examined. The analysis of the busy period, the mean waiting time, and cost analysis are discussed. A few significant performance measures are attained. Finally, some numerical examples are given.

Keywords: Phase type Distribution, Markovian Arrival Process, Degrading Service, Server Vacation, Breakdown, Repair, Starting failure, Close-down, Matrix-analytic method.

AMS Subject Classification (2010): 60K25, 68M30, 90B22.

1. Introduction

The Markovian arrival process is one of the modelling techniques for studying point processes that is most flexible. In order to define arrival processes that are not fundamentally renewal processes, Neuts [13] proposed the concept of a versatile Markovian point process (VMPP). Neuts [14] first introduced and investigated the underlying Markov structure of the MAP, which fits perfectly into the framework of matrix-analytic methods and is one of its most notable properties. Qi-Ming He [16] investigated the foundations of matrix analytical methodologies in order to comprehend the idea of service and arrival process.

Chakravarthy [5] made a significant contribution to MAP. Markovian Arrival Process represents by (D0, D1) and the service times with representation (a, T) that follow phase type distribution and whose matrices of order m and n, respectively. He described several types of arrivals and services. The irreducible stochastic matrix D = D0 + D1 defines the generator D. If the irreducible generator D describes the Markov process, then n is the steady state probability

vector, and it is defined as nD = 0 and ne = 1. Based on the Markovian arrival process, the constant A = nD1e represents the basic customer arrival rate per unit time.

MAP/PH/1-type queueing models with degradation and phase type vacation have been analysed by Alka et al. [6]. Degradation can be included in a service system in a number of ways. The service rate will decrease unless the degradation is addressed. In other words, the service rate will decrease as more services are provided. For vacation queueing models, we refer to Doshi's survey paper [7] and Tian and Zhang's book [20]. Li and Tian [12] investigated the M/M/1 model with working vacation and proposed an interruption in vacation, where the server returns without completing the ongoing vacation due to certain conditions. Krishna Kumar et al. [10] have analysed the several server model with server vacations under the Bernoulli schedule. Sreenivasan et al. [18] have examined the MAP/P H/1 queueing model with N-Policy, vacation interruption and working vacations.

One of the main queueing theory subfields has recently been queueing models with server breakdown. Wang et al. [21] have investigated the batch arrival queueing model with multiple vacations and the server struck with breakdown. Ayyappan and Nirmala [2] have explored the non-Markovian queueing model and the server provides service to the customers based on general bulk service rule with multiple vacations, breakdown and two-phase repair . Ayyappan and Deepa [1] have studied the batch arrival and bulk service queueing model with multiple vacations and optional repair. A single server queueing model with MAP arrival and phase type service, vacation, instantaneous feedback and breakdown has been looked into by Ayyappan and Thilagavathy [3]. In this model, they obtained stability condition and busy period analysis. Senthil Vadivu et al. [17] have performed a cost function of the bulk service queueing model of a single server with finite capacity and close-down times by using embedded Markov chain and supplementary variable techniques.

Yang et al. [22] have discussed the Markovian model of the retrial queue with multi-server and starting failure. They analyzed their model with the aid of the matrix geometric method. With respect to the stability condition, the cost analysis is built to calculate the ideal number of servers, the ideal average service rate, and the ideal average repair rate. Karpagam et al. [9] have been analysed the batch arrival and bulk service queueing system with starting failure and additional service. They obtained system performance measures and the stability condition. Ayyappan and Gowthami [4] has analysed a Phase type model with impatient customers, Setup time, vacation, feedback, Breakdown and Repair. In this article, they compute the average waiting time.

2. Description of the Model

Assume that there is a single server in a queueing model, and that customers arrive at the system according to the MAP with representation (Do, D1), where Do and D1 are m-dimensional square matrices. Let D = Do + D1 be the generator matrix, where Do governs for no arrival at the system and D1 governs for an arrival at the system. The stationary vector of D is denoted by n, so we have nD = 0 and ne = 1. The arrival rate A is given by A = nD\e. The system is performed on an FCFS basis. With the notation (a, T), that is of order n, the length of the server's service is thought to be a PH-distribution, where T0 + Te = 0 so that T0 = — Te. The average service rate £ is given by £ = [a( — T)— 1e]—1. The service rate decreases after each service is completed. Let £ be the first service rate and £, be the ith service rate such that £ = £\ > £2 > £3 > ■ ■ ■ > £K, where = 0,£ and 0 < 0, < 1 for all i = 1,2,3,...,K. After K services are completed, the original rate of £ is immediately applied to the degraded service rate. Because Q\ = 1, After the degradation has been corrected, the service rate for the first customer is always £. The server that customers use to access services could breakdown at any time and needs to be repaired.

The repair procedure is based on the PH-distribution with representation (ft, S) of order n2 and S0 + Se = 0 so that S0 = -Se. If no one is present in the system when the server's service is completed, the close-down process begins, and then the server goes on vacation. The vacation period is thought to be a PH-distribution with the notation (7, V) of order n1, where V0 + Ve = 0 so that V0 = — Ve. After completion of the vacation period if no customer present in the system, then the server is idle; otherwise the server starts the service. If a customer arrives while the server is idle, it may experience a starting failure with probability p or no starting failure with probability q, resulting in p+q=1. In the event of a server breakdown, the customer who is currently providing the service from the server will remain in a frozen state until the server gets rid of the repair process. After completion of the repair process, the server will serve a fresh service for the current frozen customer. The breakdown and close-down time follows an exponential distribution with the parameters ff and 5 respectively. The average repair rate and vacation rate are given by Z and n respectively.

Figure 1: Schematic Representation of the model

3. The QBD Process of Matrix Generation

We have described our model's notation for the basis of generating the QBD process in this section as follows.

Matrix Generation Notations

• 0 - Kronecker product represents the product of any two different order matrices, can refer to the works in Steeb et al. [19].

• ® - The Kronecker Sum represents the sum of any two of the different orders of matrices.

• Ik - An identity matrix of order k.

• e[(m) - An m-dimensional row vector with 1 in the ith position and 0 elsewhere.

• e-Each entry in a column vector of appropriate dimension is 1.

• The customer's arrival rate is denoted by A and is defined by A = nD\em

• The server's service rate is denoted by £ and is defined by

I =[a(—T)—1 en ]—1

• The server's vacation rate is denoted by n and is defined by n = M-V)-1 enx ]-1

• The server's repair rate is denoted by Z and is defined by

Z =[^(-S)-1 e„2 ]-1

/01 0 ......\

0 02

Define 0 = (01,02,..., 0K)f and A(0)

\0 ...... 0k)

• Let N(t) be the number of customers in the system at epoch t

• Let V(t) be the server's status at epoch t

'0, if the server is on vacation,

1, if the server is in idle,

V(t) = < 2, if the server is in busy,

3, if the server is in repair process,

4, if the server is in closedown process

I(t) is the type of service at time t

/ (t) represents the vacation process as framed by phases.

(t) represents the repair process as framed by phases. S(t) represents the service process as framed by phases. M(t) represents the arrival process as framed by phases.

Let { N(t),V(t),I(t),/1(i),/2(t),S(t),M(t):t > 0} denote the Continuous Time Markov Chain (CTMC) with state level independent Quasi-Birth and Death process, the state space of which is as follows:

n = 1(0) U 1(q) ,

where

1 (0) = {(0,0,j1,k) : 1 < j1 < n1,1 < k < m} U {(0,1,k) : 1 < k < m} U {(0,4,k) : 1 < k < m} for q > 1,

1(q) = {q,0,j1,k) : 1 < j1 < n1,1 < k < m}U {(q,2,1,j,k) : 1 < 1 < K,1 < j < n,1 < k < m} U{(q,3,1, j2,k) : 1 < 1 < K,1 < j2 < n2,1 < k < m} U {(q,4,k) : 1 < k < m}.

The QBD process's infinitesimal matrix generation is given by

Q

B00 B01 0 0 0 0

B10 Ai A0 0 0 0

0 A2 Ai A0 0 0

0 0 A2 Ai A0 0

The entries in the block matrices of Q are defined as follows,

V ® D0 V0 ® Im 0 B00 =0 D0 0

Y ® ^Im 0 D0 - ^Im

301

A

0 —

A1

A2

"In1 ® D1 0 0 0

0 e[(K) ® a ® qD1 e 1 (K) ® p ® pD1 0 , B10 —

_ 0 0 0 D1

in1 ® D1 0 0 0"

0 Ik ® In ® D1 0 0

0 0 Ik ® In 2 ® D1 0

- 0 0 0 D1

"V © D0 e1 (K) ® qV0a ® Im e1 (K) ® pV0p ® Im

0 (A(0) ® T) © D0 VIKnm Ik ® (en ® P) ® Vim

0 Ik ® S0a <g> Im (Ik ® S) © D0

Y ® 5 Im 0 0

00 0 0"

0 A22 0 0

00 00 0 0 00

A22

0 0

0

_9kT°a ® Im

01T0 a ® Im 0

0 02 T0a ® Im

0 ...

0 ...

0 0 0 0 0 0 ® T0 ® Im 0

00 00

0

0 0 0

D0 - 5Im

0 0

0K-1 T0a ® Im 0

4. Analysis of Stability Condition We examined our model under the assumption that the system is stable.

4.1. Condition for Stableness

Let us specify the matrix A as A = A0 + A1 + A2. It clearly demonstrates that the order of the square matrix A is n\m + Knm + Kn2m + m and this matrix is an irreducible infinitesimal generator matrix. Let f indicate the steady-state probability vector of A and it satisfying fA = 0 and fe = 1. The vector f is partitioned by f = (fo, f 1, f2, f3)=( fo, fii, f 12, f 13,..., f 1^1, fiK, f 21, f22, f23,..., f2K-1, f2K, f3), where f0 is of dimension n1m, f 1 is of dimension Knm, f2 is of dimension Kn2m, f3 is of dimension m. Our model's stability should satisfy the necessary and sufficient condition f Aoe < f A2e when the Markov Process is investigated using the Quasi-Birth-and-Death structure. The probability vector f is calculated by solving the following equations

(V ® D) fo + (y 0 5Im)f3 = 0,

(qV0a 0 Im) fo + (01T ® D - oInm) fii + (0iT°a 0 Im) fiK + (S°a 0 Im)f2i = 0, (0j-iT°a 0 Im) fij-i + (0jT ® D - aInm) fij + (S0a 0 Im) f2j = 0 for 2 < j < K, (pV0p 0 Im ) fo + (en 0 P 0 O-Im) fii + (S ® D) f 21 = 0, (en 0 P 0 oIm) fij + (S ® D) f2j = 0 for 2 < j < K, (D - 5In)f3 = 0.

subject to normalizing condition

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

<Pljenm + ^^ ty2j£n2m + Ç3 em — 1. j—1 j—1

G. Ayyappan, S. Meena RT&A, No 1 (72)

PHASE TYPE QUEUEING MODEL OF SERVER VACATION... Volume 18, March 2023

The stability condition f Aoe < is obtained after some algebraic manipulation, which turns out to be

K K K

fo (eni ® Diem) + £ fij(e„ ® Diem) + £ f2j (e«2 ® Diem) + f3D^m < £ fij (fyT0 ® em) j=i j=i j=i

4.2. Analysis of Steady-State Probability Vector

Consider the steady-state probability vector x of Q and it is divided into x = (x0, xi, x2,...). x0 has a dimension 2m + nim while xi, x2,... have a dimension nim + Knm + Kn2m + m. Then x satisfied the condition xQ = 0 and xe = i.

Furthermore, if the system is stable with the vector x, the following equation provides the remaining sub vectors except for the boundary states.

xq = xiRq-i, q > 2

where the rate matrix R indicates the minimal non-negative solution of the matrix quadratic equation as R2A2 + RAi + A0 = 0, as referred by Neuts [i5] and satisfies the relation RA2e = A0e.

The sub vectors of x0 and xi were calculated by solving the subsequent equations.

x0 B00 + xi Bio = 0 xo Boi + xi( Ai + RA2) = 0

The normalizing condition is subject to

x0e2m+nim + xi(I - R)- e«i m+Knm+K«2 m+m — i

As a result, the rate matrix R could be mathematically calculated using crucial procedures in the Latouche algorithm for logarithmic reduction of R [ii].

5. Busy period Analysis

The time between customers entering into an empty system and the system becoming empty again after the first interval can be used to measure a busy period. This is the first passage in the transition from level i to 0. Thus, it is the first time returns to level 0, followed by at least one visit to a state at any other level is known as the busy cycle. We give an overview of the fundamental period before moving on to the busy period. The QBD process takes into account the first transition time, q > 2, from level q to level q-i. It is necessary to examine each of the cases q = 0, i that correspond to the boundary states individually. It should be noted that for each level j with q > 2, there are (nim + Lnm + Ln2m + m) states that correspond. Similarly, when the states are organised in lexicographic order, the state(q, j) at level j signifies that jth state at the level q is mentioned. The variable Gj (v, x) represents the conditional probability that the QBD process, which begins in the state (q, j) at time t=0 and visits the level q-i but not before time x, can make changes v transition to the left and enter the state (q, j'). Let us first define the joint transform

f m

Gj (z, s) = £ zv / e-sxdGjj' (v, x); |z| < i, Re(s) > 0

v=i -'0

and the matrix is represented as G(z, s) = Gj (z, s) [i4]then the previously defined matrix G (z, s) satisfied the equation

G(z,s) = z(sI - Ai)-iA2 + (si - Ai)-iAqG2(z,s).

• The matrix G = Gj = G (i, 0), excluding the boundary states, would be used for the first passage time. If we are already familiar with the matrix R, we can use the results to discover the matrix G

G = -(Ai + RA2 )-i A2.

Or else, the idea of a logarithmic reduction algorithm method [ii] could be used to determine the values of the G matrix.

Notations

• Gjy,0) (v, x) shows that at time t = 0, the conditional probability has been discussed for the first time during the passage from level i to level 0.

• GjjQ,0) (v, x) shows that the conditional probability has been discussed for the return time to level 0.

• Hiq shows the average first passage time between levels q and q-i, assuming the process is in the state (q, j) at time t=0.

• Hi identifies the column vector containing the entries Hiq.

• H2q shows the average number of customers expected to be served during the first passage time from level q to q-i, assuming that the state's first passage time has already begun (q, j).

• H2 identifies the column vector containing the entries H2q.

• Hii,0) shows the average first passage time between level i and level 0.

• H2i,0) shows the expected number of services finished during the first passage time from level i to level 0.

• Hi0,0) shows the initial return time to level 0.

• H20,0) shows the expected number of services finished between the first return time and level 0.

The following equations, which are given by G(i,0)(z,s) and G(0,0)(z,s), are for the boundary levels i and 0 respectively.

G(i0)(z,s) = z(si - AiBi0 + (si - AiAoG(z,s)G(i,0)(z,s), G(q,q)(z,s) = (si - BooBoiG(i,Q)(z,s).

The matrices are used to calculate the following instances because G, G(0,0)(i,0) and G(i,0)(i,0) are all stochastic in nature. We can compute the instants as follows:

Hi = - ^ G (z, s) e = -[Ai + Ao( I + G)]-ie,

z=i,s=0

as

H2 = d G (z, s) e = -[Ai + Ao ( I + G)]-1 A2 e,

^ z=1,s=0

Hi1,0) = - ¿G (1,0)(z, s) ^ j = -[Ai + Ao G]-1 (AoHi + e),

z=1,s=0

H21,0) = JZGG (1,0)(z, s) , „e = -[A1 + A0 G]-1(A0H2 + Bwe),

z=1,s=0

H1

1°,0) = - d G (0,0)(z, s) , ne = -B-1 [B01H11,0)+ e],

ds

h20,0) = ¿G (0,0)(z, s) ^ j = -B0-01[B01H21,0)].

z=1,s=0

" -H00 [B01H2

z=1,s=0

6. System Performance Measures

• The average system size

Esystem = qxqe

q=i

• Probability of the server is busy

to Knm

Pbusy ^^^^ xq2ijk q=11=1 j=1 k=1

• Probability of the server is in idle

m

Pidle = ^ X01k k=1

• Probability of the server is on vacation

to ni m Pvac = ^ ^ ^ xq0jik

q=0 j1=1 k=1

• Probability of the server is breakdown

to K n2 m

Pbd = ^ ^ ^ ^ xq3lj2k q=11=1 j2=1 k=1

• Probability of the server is on closedown

to m Pcd = ^ ^ xq4k

q=0 k=1

• The average system size during vacation

to ni m Evac = ^ ^ ^ qxq0jikenim

q=1 j1=1 k=1

• The average system size of the server is busy

to Knm Ebusy = ^ ^ ^ ^ qxq2ljkeKnm

q=1 l=1 j=1 k=1

• The average system size during breakdown

to K n2 m

Ebd = ^ ^ ^ ^ qxq3lj2keKn2m q=11=1 j2=1 k=1

• The average system size when the server is close-down

to m

Ecd = qxq4kem

q=1 k=1

7. Waiting Time Distribution

The first passage time analysis is used in this section to analyse the distribution of a customer's waiting time when they enter the queueing line. Let W(t) be the waiting time distribution function, which takes into account new customers joining the queue. If the server is idle when a customer arrives, they will get service immediately; otherwise, if the server is busy or on vacation, they will have to wait in a queue to receive service from the server.

Let's look at the absorption time in the state space of a Markov chain, which is given by

n = (*) u {0,1,2,... }

where

and for q > 1,

0 = {(0,0,ji) : 1 < ji < ni}U{(0,4)}

q = {(q,0,j1) : 1 < j1 < n1 }U{(q,2,l,j) : 1 < l < K,1 < j < n} U {(q,3,l,j2) : 1 < l < K,1 < j2 < n2} U {(q,4)}

The state space (*)obtained by considering the states that have the server in the idle state at the instant of arrival is as below

(*) = {(0,1)} Let this Markov process's transition matrix Q be

Q

0 0 0 0 0 0

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

J0 L0 0 0 0 0

J1 L10 L 0 0 0

0 0 L2 L 0 0

0 0 0 L2 L 0

where

Jo

V0 o

' V

o o

Y ® S

Lo

V 0 " Y ® S —S

e1(K) ® qV0a A(0) ® T — alKn Ik ® S0 a 0

, Ji

e1(K) ® pV0p 0'

IK ® (en ® 0

Ik ® S 0 0S

L2

0 0 0 0' 0 L22 0 0

00 00

L22

0 0

0

0kT0 a

01 T0a 0

0

02 T0 a

0K-

0 0 0

0 0 , L10 — 0 0 0 ® T0 0

0 0 0

0 0

0

_i T0 a 0

With the aim of determining the arriving tagged customer's waiting time distribution W(t), where(t > 0). To start, we search for the system size probability vector at the arrival epoch in a steady state and it is indicated by Z(0) = (Zo(0),Z1 (0),Z2(0),...). The vector Z0(0) may be further partitioned as follows Z0(0) = (Z00, Z04). The system size probability vector at the arrival epoch in the steady state is as follows because the arrival process abides by the Markovian property:

Z,

00

x00

In1 ®

Diem A

Z

04

x04

Di em A

Zq(0)

Di em

In1 +Kn+Kn2+1 ^ —A—

for q > 1

where A denotes the fundamental arrival rate of Markovian Arrival Process.

Define Z(t) = (Z* (t), Z0(t), Z1(t),...),

where

Zq(t), q > 1 - vector of order 1 x (n1 + Kn + Kn2 + 1)

L

x

q

Z0(t) = (Zoo, Z04) - vector of order 1 x (n1 + 1)

and their components give the probability that at epoch t, the CTMC generator matrix is Q, will be in the appropriate level q state. Since the tagged customer's probability of being in the absorbing state at epoch t is specified by Z* (t), we get W(t) = Z* (t), where t > 0. The differential equation Z'(t) = Z(t)Q for t > 0 becomes

Z* (t) = Zo (t) Jo + Zi(t) Ji, Z'o(t) = Zo (t)Lo + Zi(t)Lio,

Z'q (t) = Zq (t)L + Zq+1 (t) L2, q > 1

where ' specifies the derivative concerning t. Let's use the method suggested by Neuts et al. [15] to compute the LST for W(t). The row vector j(s) specifies the Laplace-Stieltjes Transform (LST) of the first passage time to level 1 by starting the process at state q and using Zq (0), q > 1 as the initial probability vector. Neuts et al. [15] state that we get,

to

j(s) = £ Zq(0)[(sI - L)-1 L2]q-i (1)

q=1

With the restriction that the process begins at level q = 0, 1, let the LST of the time to

become absorbed into the state (*) be specified by 0(q, s). Similar to Neuts et al. [15], we have

(0, s) = [sI - Lo]-1 Jo, (2)

0(1,s) = [sI - L]-1 Lio0(o,s) + [sI - L]-1 Ji. (3)

This allows us to quickly note that the LST for the distribution of sojourn time is as follows.

W (s) = Zo (0)0(0, s) + j(s)0(1, s) (4)

Expected Waiting Time

The mean waiting time is given as

E(W) = -W'(0) = -Zo(0)0(0,0) - J(0)eni+Kn+Kn2+i - J(o)0(1,0) (5)

The initial term of the previous equation gives the expected time to reach the absorbing state (*), assuming that the system is at level 0. The final two components of the previous equation provide the expected time for accessing the absorbing state (*) if the system is resting at level q > 1. By differentiating (2) and (3) and making s=0,

<&' (0,0) = -[-Lo ]-2 Jo (6)

<&'(1,0) = -[-L]-2Lio<&(0,0) + [-L]-1 Lio&'(0,0) - [-L]-2Ji (7)

Using (6) and the vector Z(0) = (Z0(0), Z1 (0),...), it is simple to calculate the first term of (5). From (1), we get

to

j(0)= £ Zq(0)Mq-i (8)

q=1

where M = [-L]-1 L2. As M is a stochastic matrix, we get

J(0)en1 +Kn+Kn2+i = 1 - Zo(0)en1+i (9)

Equations (7) and (8), as well as the vector Z(0) = (Z0(0), Z1 (0),...), allow us to quickly calculate the last term of (5).

We obtain by differentiating (1) and setting s=0,

to q-i

J(0) = (-1) £ Zi+q(0) £ Mj[-L]-1 Mq-'. (10)

q=1 j=0

Due to the stochastic nature of matrix M,

«> q—1

(-l)w'(0)e„1+Kn+Kn2+1 = E zi+q(0) E Mj[—ll—1 (11)

q=1 j=0

Let's assess the value of ( — 1)w'(0)en1 +k«+k«2+1 using the technique described in Neuts et al. [15] and Kao et al. [8]. We start by building a matrix M2 that is generalised inverse of I-M and stochastic, with I — M + M2 being non-singular and M2 being stochastic. The matrix M2 can be viewed as M2 = en1 +Kn+Kn2+1 m0, where m0 is the invariant probability vector of M. Additionally, using the property MM2 = M2M = M2, we have

q—1

E Mj (I — M + M2) = I — Mq + qM2 for q > 1. (12)

j=0

By using (12) in (11), we obtain

" D1 e

Diem

In1+Kn+Kn2+1 & ^

- œ(0)

(13)

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

M2

(-1)w'(0)en1+Kn+Kn2+1 = jx1( 1 - R) 1 In1 +Kn+Kn2+1 ® -A

+ X1 R( 1 - R)-2

x [ 1 - M + M2]-1 [-L]-1en1+Kn+Kn2+1. Since we have calculated all the terms in (5), we can easily calculate the average waiting time.

8. Cost Analysis

Our model's cost analysis has been created below by assuming the cost elements (per unit time) correspond to distinct measures of the system.

= CHEsystem + CbMSyPbMSy + Qd/gPfd/g + CvacPvac + + CcdPcd

K

+ E C1i Oi z+^C2 + £ C3 + SC4 i=1

where

TC - Total cost per unit time CH - Each customer's holding cost in the system Cfcusy - Cost acquired by the system during server being busy Cid/e - Cost acquired due to server being idle Cvac - Cost acquired during server's vacation period Cbd - Cost acquired by the server during breakdown time Ccd - Cost acquired by the server during close-down process C1f - Cost acquired by the server for offering ith type service, i = 1,2,..., K C2 - Cost acquired when the server caused by breakdowns C3 - Cost acquired in carrying out the repair process C4 - Cost acquired in carrying out the close-down process

9. Numerical

In this section, we are using numerical and graphical representations to analyze model behavior. The mean value of the subsequent five different MAP representations is 1, which is the same for all the various arrival processes. In published studies, these five sets of arrival values have been used as input data (see Chakravarthy [5]).

Arrival in Erlang(ERL-A):

Do

-2 2 ' 0 -2

Di

0 0 2 0

Arrival in Exponential(EXP-A):

D0

[-1] , Di = [1]

• Arrival in Hyper-exponential(HEX-A):

D0

-1.90 0 0 -0.19

Di

1.710 0.171

0.190 0.019

• Arrival in MAP-Negative Correlation(MAPNC-A):

D0

-1.25 1.25

0

0

-1.25 0

D

1

0

0

0

0.0125 0 1.2375

0 0 -2.5 J [2.4750 0 0.0250

• Arrival in MAP-Positive Correlation(MAPPC-A):

D0

-1.25 1.25 0 0 -1.25 0

0

0

2.5

D1

0 0 0 1.2375 0 0.0125 0.0250 0 2.4750

Let's think about the service, repair, and vacation processes as three phase type distributions. In the literature, these sets of service, vacation, and repair values have been used as input data [5].

• Service in Erlang(ERL-S):

a = (1,0), T = Repair in Erlang(ERL-R):

P =(1,0), S

-2 2 0 -2

-2 2 0 -2

Vacation in Erlang(ERL-V):

Y =(1,0), V

-2 2 0 -2

Service in Exponential(EXP-S):

a = [-1], T =[1] Repair in Exponential(EXP-R):

P =[-1], S = [1] Vacation in Exponential(EXP-V):

Y =[-1], V =[1]

Service in Hyper-exponential(HEX-S):

a = (0.8,0.2), T

-2.8 0 0 -0.28

• Repair in Hyper-exponential(HEX-R):

P =(0.8,0.2), S

-2.8 0 0 -0.28

Vacation in Hyper-exponential(HEX-V):

Y =(0.8,0.2), V

-2.8 0 0 -0.28

9.1. Illustration 1

We investigated the consequence of the repair rate (Z) on the average system size(Esystem). We fix A = 2, £ = 6, n = 10, & = 1, S = 5, K = 10, d* = [1,0.97,0.93,0.9,0.87,0.83,0.8,0.75,0.7,0.6], p = 0.6, q = 0.4.

Table 1: Repair rate (Z) vs Esystem - ERL-S

ERL-S

Z ERL-A EXP-A HEX-A MAPNC-A MAPPC-A

4 1.240013 1.428884 2.424636 1.337204 10.61745

5 1.100282 1.263127 2.098389 1.18343 8.801677

6 1.015791 1.16274 1.90369 1.090236 7.692035

7 0.959243 1.095599 1.775197 1.027872 6.946641

8 0.918743 1.047606 1.684428 0.98328 6.412992

9 0.888298 1.011625 1.617078 0.949845 6.012954

10 0.864568 0.983665 1.565214 0.923863 5.702437

11 0.845543 0.961321 1.524099 0.903103 5.454723

Table 2: Repair rate (Z) vs Esystem - EXP-S

EXP-S

Z ERL-A EXP-A HEX-A MAPNC-A MAPPC-A

4 1.300513 1.477625 2.385188 1.39259 8.815389

5 1.14504 1.300952 2.072725 1.226593 7.266135

6 1.05166 1.194414 1.885579 1.126538 6.322526

7 0.989618 1.1235 1.761902 1.059955 5.692052

8 0.945487 1.073041 1.674513 1.012588 5.243369

9 0.912517 1.035369 1.609694 0.977232 4.909026

10 0.886957 1.006202 1.559813 0.949868 4.650976

11 0.866562 0.982972 1.520303 0.928081 4.446203

Table 3: Repair rate (Z) vs Esystem - HEX-S

HEX-S

Z ERL-A EXP-A HEX-A MAPNC-A MAPPC-A

4 1.558351 1.660665 2.248675 1.59979 4.230083

5 1.325099 1.423599 1.946641 1.369357 3.457511

6 1.188053 1.283658 1.765584 1.233688 3.008707

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

7 1.099269 1.19259 1.646429 1.145591 2.721063

8 1.037696 1.12918 1.562789 1.084356 2.523394

9 0.992794 1.082778 1.501231 1.039609 2.380326

10 0.958759 1.047505 1.454251 1.005634 2.272559

11 0.932162 1.019875 1.417352 0.979044 2.188777

With the help of tables 1, 2 and 3, we can determine that increasing the repair rate reduces the average system size in various arrangement of services and arrivals of ERL-A, EXP-A, HEX-A, MAPNC-A and MAPPC-A. The positive correlation arrival decreases rapidly compared to all other arrivals.

9.2. Illustration 2

We investigated the consequence of the vacation rate (n) on the average waiting time E(W). We fix A = 2, g = 6, a = 1, £ = 4, 5 = 5, K = 5, Of = [1,0.9,0.8,0.7,0.6], p = 0.4, q = 0.6.

Table 4: Vacation rate (n) vs E(W) - ERL-S

ERL-S

n ERL-A EXP-A HEX-A MAPNC-A MAPPC-A

10 0.513310803 0.652469261 1.316678245 0.597597919 6.305631788

11 0.502141873 0.641535425 1.304722472 0.586775589 6.294830579

12 0.493006347 0.632560036 1.294845632 0.577894144 6.285940644

13 0.485401455 0.625064256 1.286552181 0.570478428 6.27849855

14 0.478976132 0.618712658 1.279491771 0.564195772 6.272178893

15 0.4734782 0.613263522 1.273409875 0.55880656 6.266746629

16 0.46872206 0.608538394 1.268117234 0.554133955 6.262027841

17 0.46456821 0.604402741 1.263470256 0.550044688 6.257891127

18 0.460909846 0.600753272 1.259358071 0.546436465 6.254235397

19 0.457663866 0.597509402 1.255693735 0.543229496 6.250981623

Table 5: Vacation rate 67) vs E(W) - EXP-S

EXP-S

n ERL-A EXP-A HEX-A MAPNC-A MAPPC-A

10 0.557188985 0.69195187 1.314692759 0.63940605 5.40504825

11 0.545405475 0.680428504 1.302062559 0.627990745 5.393620067

12 0.535811233 0.671007926 1.291662266 0.618660721 5.384251971

13 0.527857453 0.663169128 1.282954639 0.610898654 5.376437568

14 0.521162512 0.656548669 1.27556107 0.604343883 5.369822843

15 0.515453367 0.650885673 1.269207338 0.598737649 5.364153128

16 0.510529823 0.645988247 1.263690113 0.593889693 5.359240685

17 0.506241951 0.641712219 1.258855539 0.589657117 5.354944174

18 0.502475357 0.637947256 1.254585115 0.585930588 5.351155205

19 0.49914132 0.634607531 1.250786132 0.58262508 5.347789301

Table 6: Vacation rate (ц) vs E(W) - HEX-S

HEX-S

n ERL-A EXP-A HEX-A MAPNC-A MAPPC-A

10 0.765770257 0.873794399 1.367546979 0.829456953 2.90799282

11 0.749707796 0.858258947 1.350700199 0.814080592 2.891938923

12 0.736875156 0.84579312 1.337039635 0.801751095 2.879049871

13 0.726421715 0.835595592 1.325762738 0.791670904 2.868502654

14 0.717764748 0.827116374 1.316311418 0.783293156 2.859731284

15 0.710493246 0.819966703 1.30828648 0.77623166 2.852335034

16 0.704309955 0.813864754 1.301395404 0.770206698 2.846023015

17 0.698995159 0.808601698 1.29541931 0.765011173 2.840579451

18 0.694383327 0.804019799 1.290191394 0.760488778 2.835841216

19 0.690347624 0.799997883 1.285582472 0.756519514 2.831682906

With the help of tables 4, 5 and 6, we can determine that increasing the vacation rate reduces the average waiting time in various arrangement of services and arrivals of ERL-A, EXP-A, HEX-A, MAPNC-A and MAPPC-A.

9.3. Illustration 3

We examined the consequence of the vacation rate(n) on the Total cost(TC) of the system. We fix Я = 2, I = 6, Z = 4, a = 1, Ô = 5, K = 5, вt = [1,0.9,0.8,0.7,0.6], p = 0.6, q = 0.4, CH = 10,

Cvac = 2 Cidle = 1 Cbusy = 4 Cbd = 2 Ccd = ^ C11 = ^ C12 = 2.9, C13 = 2.7, C14 = 2.5,

Ci5 = 2.2, C2 = 1, C3 = 2, C4 = 2.

Table 7: Vacation rate (ц) vs TC - ERL-S

ERL-S

n ERL-A EXP-A HEX-A MAPNC-A MAPPC-A

10 99.68315738 102.0191734 114.5649299 100.8662306 220.3575308

11 99.58654131 101.9327141 114.4864353 100.7816754 220.2740852

12 99.50790684 101.8624352 114.422411 100.7129863 220.2062508

13 99.44275241 101.8042608 114.369262 100.6561557 220.150093

14 99.38794402 101.7553616 114.3244797 100.6084052 220.1028821

15 99.34123751 101.7137163 114.2862631 100.5677517 220.0626686

16 99.30098652 101.677845 114.253288 100.5327444 220.0280254

17 99.26595761 101.6466406 114.2245603 100.5022988 219.9978848

18 99.23520954 101.6192592 114.1993197 100.4755886 219.971433

19 99.20801222 101.595047 114.1769756 100.4519741 219.9480397

Table 8: Vacation rate (y) vs TC - EXP-S

EXP-S

n ERL-A EXP-A HEX-A MAPNC-A MAPPC-A

10 100.2339903 102.3592967 113.45068 101.3175747 196.2037767

11 100.1280443 102.2634003 113.3608307 101.2234308 196.1105677

12 100.0424323 102.1859477 113.28795 101.1474223 196.0352852

13 99.97195226 102.122204 113.2277544 101.0848836 195.973326

14 99.91300867 102.0689027 113.1772692 101.0325998 195.9215153

15 99.86304362 102.0237237 113.134369 100.9882889 195.8775984

16 99.8201921 101.9849774 113.0974976 100.9502904 195.8399333

17 99.78306479 101.9514061 113.0654919 100.9173687 195.8072978

18 99.75060709 101.9220556 113.0374655 100.8885871 195.7787647

19 99.72200511 101.8961899 113.0127327 100.8632232 195.7536187

Table 9: Vacation rate (y) vs TC - HEX-S

HEX-S

n ERL-A EXP-A HEX-A MAPNC-A MAPPC-A

10 102.3258123 103.4719568 110.0500597 102.7836354 135.054583

11 102.1616111 103.3188195 109.8998592 102.6325837 134.9021191

12 102.0324917 103.1981907 109.7806956 102.5136915 134.7822937

13 101.9288319 103.1011839 109.684308 102.4181401 134.6861809

14 101.8441297 103.0217915 109.6050482 102.3399749 134.6077393

15 101.7738585 102.9558273 109.5389409 102.2750516 134.5427562

16 101.7147844 102.9002977 109.4831181 102.2204095 134.4882179

17 101.6645447 102.8530131 109.4354652 102.1738855 134.4419202

18 101.621379 102.8123398 109.3943937 102.1338671 134.402219

19 101.5839523 102.7770377 109.3586901 102.0991312 134.3678669

With the help of tables 7, 8 and 9, we can determine that increasing the vacation rate reduces the total cost of the system in various arrangement of services and arrivals of ERL-A, EXP-A, HEX-A, MAPNC-A and MAPPC-A.

9.4. Illustration 4

We investigated the consequence of the breakdown rate (a) on the average system size(Esystem). We fix A = 2, £ = 6, n = 10, Z = 4, S = 5, K = 10, d* = [1,0.97,0.93,0.9,0.87,0.83,0.8,0.75,0.7,0.6], p = 0.6, q = 0.4.

With the help of figures 2, 3, 4, 5 and 6, we analyze the breakdown rate versus the average system size with the combination of arrival and service time groupings. The breakdown rate increases then the corresponding average system size is also increases rapidly in Erlang services and, increases gradually in Exponential services and slowly in Hyper-exponential services but in case of MAP positive correlation arrival increases rapidly than compared to all other arrivals.

Figure 2: Breakdown rate(c) vs Esystem - ERL-A Figure 3: Breakdown rate(^) vs Esystem - EXP-A

Figure 4: Breakdown rate(^) vs Esystem - HEX-A Figure 5: Breakdown rate(^) vs Esystem

- MAPNC-A

Figure 6: Breakdown raie(^) vs Esystem - MAPPC-A

9.5. Illustration 5

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

We have examined both the vacation rate(n) and repair rate(Z) against the average system size(Esystem). We fix A = 2, £ = 6, a = 1, S = 5, K = 10, d* = [1,0.97,0.93,0.9,0.87,0.83,0.8, 0.75,0.7,0.6], p = 0.6, q = 0.4.

With the help of figures 7 to 11, we analyze the both vacation rate and repair rate versus the average system size with the combination of arrival and service time groupings. Both the vacation rate and repair rate increases then the corresponding average system size is decreases rapidly in MAP positive correlation compared to all other arrivals.

Figure 7: Ek /Hk /1 - Vacation rate(n) and Repair rate(Z) vs Esystem

Hk/Hk/1

Figure 8: M/Hk/1 - Vacation rate(n) and Repair rate(Z) vs Esystem

MAPNC/Hk/1

Figure 9: H/Hk /1 - Vacation rate(n) and Repair rate(Z) vs Esystem

MAPPC/Hk/1

Figure 10: MAPNC/Hk/1 - Vacation rate(n) and Repair rate(Z) vs Esystem

Figure 11: MAPPC/Hk/1 - Vacation rate(n) and Repair rate(Z) vs Esystem

Ek/Hk/1

M/Hk/1

10. Conclusion

In our paper, customers arrive in a Markovian Arrival Process and the service process follows a phase-type distribution with degrading service, server breakdown, vacation process in phase type distribution, repair process in phase type distribution, starting failure and close-down. We also perform the busy period analysis, waiting time distribution and cost analysis in our work. Using numerical values of arrival and service times, we tabulated the repair rate versus expected system size and the vacation rate versus the expected waiting time numerically. We compared the breakdown rate to the expected system size, as well as the vacation and repair rates to the expected system size, as shown by the graphical demonstrations.

References

[1] Ayyappan, G. and Deepa, T. (2018). Analysis of batch arrival bulk service queue with multiple vacation closedown essential and optional repair, Applications and Applied Mathematics, 13(2):578-598.

[2] Ayyappan, G. and Nirmala, M. (2018). An M[X]/G(a, b)/1 queue with breakdown and delay time to two phase repair under multiple vacation, Applications and Applied Mathematics, 13(2):639-663.

[3] Ayyappan, G. and Thilagavathy, K. (2020). Analysis of MAP/PH/1 Queueing Model with Breakdown, Instantaneous Feedback and Server Vacation, Applications and Applied Mathematics, 15(2):673-707.

[4] Ayyappan, G. and Gowthami, R. (2021). A MAP/PH/1 Queue with Setup time, Bernoulli vacation, Reneging, Balking, Bernoulli feedback, Breakdown and repair, Reliability: Theory and Applications, 16(2):191-221.

[5] Chakravarthy, S. R. (2010). Markovian arrival process, Wiley Encyclopaedia of Operations Research and Management Science.

[6] Choudhary, A., Chakravarthy, S. R. and Sharma, D. C. (2021). Analysis of MAP/PH/1 Queueing System with Degrading Service Rate and Phase Type Vacation, Mathematics, 9, 2387.

[7] Doshi, B. T. (1986). Queueing System with vacations-A Survey, Queueing Systems, 1:29-66.

[8] Kao, E. P. C. and Narayanan, K. S. (1991). Analysis of an M/M/N queue with server's vacations, European Journal of Operational Research, 54(2):256-266.

[9] Karpagam, S., Ayyappan, G. and Somasundaram, B. (2020). A Bulk Queueing System with Rework in Manufacturing Industry with Starting Failure and Single Vacation, International Journal of Applied and Computational Mathematics, 6(6):1-22.

[10] Krishna Kumar, B., Rukumani, R. and Thangaraj, V. (2008). Analysis of MAP/P H(1), P H(2)/2 queue with Bernoulli vacations, Journal of Applied Mathematics and Stochastic Analysis, Article ID: 396871,1-20.

[11] Latouche, G. and Ramaswami, V. Introduction of Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics, Philadelphia, 1999.

[12] Li, J. and Tian, N. (2007). The M/M/1 queue with working vacations and vacation interruptions, The M/M/2 queue with working vacations and vacation interruptions, 16:121-127.

[13] Neuts M. F. (1979). A Versatile Markovian point process Journal of Applied Probability, 16:764-779.

[14] Neuts M. F. Matri geometric Solutions in Stochastic Models: an algorithmic approach. John Hopkins Series in Mathematical Sciences, John Hopkins University Press, Baltimore, Md, USA, 1981.

[15] Neuts, M. F. and Lucantoni, D. M. (1979). A Markovian Queue with N Servers Subject toBreakdowns and Repairs, Management Science, 25(9):849-861.

[16] Qi-Ming He. Fundamentals of Matrix - Analytic Methods, Springer, New York, 2004.

[17] Senthil Vadivu, A. and Arumuganathan, R. (2015). Cost Analysis of MAP/G(a,b)/1/N Queue with Multiple Vacations and Closedown Times, Quality Technology and Quantitative Management, 12(4):605-626.

[18] Sreenivasan, C., Chakravarthy, S.R. and Krishnamoorthy, A. (2013). A MAP/PH/1 queue with working vacations, vacation interruptions and N-Policy, Applied Mathematical Mode//ing,37:3879-3893.

[19] Steeb, W. H. and Hardy, Y. Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra; World Scientific Publishing: Singapore, 2011.

[20] Tian, N. and Zhang, Z. G. Vacation Queueing Models: Theory and Applications; Springer Publishers: New York, NY, USA, 2006.

[21] Wang, K. H. Chan, M. C. and Ke, J. C. (2007). Maximum entropy analysis of the M[x]/M/1 queueing system with multiple vacations and server breakdowns, Computers and industrial Engineering, 52(2):192-202.

[22] Yang, D. Y. Ke, J. C. and Wu, C. H. (2014). The multi-server retrial system with Bernoulli feedback and starting failures, Internationa/ Journal of Computer Mathematics, 92(5):954-969.

i Надоели баннеры? Вы всегда можете отключить рекламу.