Published: 30 June 2024

A bulk queue’s analysis with two-stage heterogeneous services, multiple vacations, closedown with server breakdown, and two types of renovation

Palaniammal S1
Kumar K2
1epartment of Mathematics, Sri Krishna Adithya College of Arts and Science, Coimbatore, Tamilnadu, India
2Department of Science and Humanities, Sri Krishna College of Engineering and Technology, Coimbatore, Tamilnadu, India
Corresponding Author:
Palaniammal S
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Abstract

A queueing system with two stages of heterogeneous services, multiple vacations, and closedown upon server failure is considered in this study for MX/Ga,b data. Two different services must be provided to a group of customers one at a time. At time C, the server finishes its shut-down procedures following the conclusion of each SPS. A server that has a queue length shorter than a takes numerous, different-duration vacations. Upon returning from vacation, if the server detects that there are less than a customers in the queue, they will continue to serve the following batch if the server determines that there are at least a customers waiting for service. During the server’s first phase of operation, the service channels will momentarily fail with a probability of (π1) at any time, and during the server’s second phase of operation, the service channels will fail with a probability of (π2) at any time.

Highlights

  • Closedown completion epoch found
  • Vacation completion epoch found
  • Service in phase I completion epoch found
  • Service in phase II completion epoch found
  • Renovation in phase I completion epoch found
  • Renovation in phase II completion epoch found

1. Introduction

Queues with server vacations are a crucial component of queueing theory and have been thoroughly and fruitfully researched due to their various applications in manufacturing systems, communication systems, textile, food, and chemical processing industries, among other industries. In a traditional vacation queue, a server can entirely halt service or perform additional work while away. Offering several vacation rules gives the system's design and operation control additional flexibility.

B. T. Doshi [1] have discussed a queueing system with vacations (1986). H. S. Lee [2] Steady state probabilities for the server vacation model with group arrivals and under control operation policy (1991). H. Takagi [3] developed a foundation of performance evaluation with queueing analysis (1991). Madan K. C. [4] considered the M/G/1 queue with second optional service (2000). K. C. Madan [5] On a single server queue with two-stage heterogeneous service and deterministic server vacations (2001). Medhi [6] generalized the model by considering that the second optional service is also governed by a general distribution (2002).

Madan et al. [7] considered the classical M/G/1 queueing system in which the server provides the first essential service to all the arriving customers whereas some of them receive second optional service (2003). Arumuganathan and Jeyakumar [8] have studied Bulk queueing models with different parameters. The queueing model with two phases of heterogeneous service under Bernoulli schedule and a general vacation time is considered Madan and Choudhury [9]. After first-stage service the server must provide the second stage service (2005).

The various system performance measures for optimization of the T policy M/G/1 queue with server breakdowns and general startup times was presented by Wang et al. [10] (2007). An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair has studied by Choudhury, Tadj and M. Paul [11]. This model generalized both the classical M/G/1 queue subject to random breakdown and delayed repair as well as an M/G/1 queue with second optional service and server breakdowns (2007).

Wang et al [12] have presented the various system performance measures for optimization of the Tpolicy M/G/1 queue with server breakdowns and general startup times (2007). Jain and Agrawal [13] analyzed the optimal policy for bulk queue with multiple types of server breakdown. In this paper breakdown occurs only when the server is in busy state and each type of breakdown requires a random number of finite stages of repair (2009).

Choudhury et al. [14] deals with an M/G/1 queueing system with two phases of service and Bernoulli vacation schedule for an unreliable server, which consist of a breakdown period and a delay period, under N –policy and a random setup time (2009). Jain, M. and Upadhyaya, S. [15] considered the Optimal repairable MX /G/1 queue with multi-optional services and Bernoulli vacation (2010).

Thangaraj and Vanitha [16] have analyzed a single server queue with Poisson arrivals, two stages of heterogeneous service with different service time distributions subject to random breakdowns and compulsory server vacations with general vacation periods (2010). The steady state behaviour of a batch arrival queue with two phases of heterogeneous service along and Bernoulli schedule vacation under multiple vacation policy is examined Choudhury et al. [17] (2011).

M. Balasubramanian and R.Arumuganathan [18] Steady state analysis of a bulk arrival general bulk service queueing system with modified M- vacation policy and variant arrival rate (2011). Jeyakumar and Senthilnathan [19] have discussed a study on the behaviour of the server breakdown without interruption in a MX/G(a , b)/1 queueing system with multiple vacations and closedown time (2012). Sourav Pradhan and Prasenjit Karan [20] Performance analysis of an infinite-buffer batch- size-dependent bulk service queue with server breakdown and multiple vacation (2022).

In the literature of the queueing system, few authors only have discussed about the repair or renovation due to breakdown of the service station. Practically, in many cases the renovation of the service station due to breakdown may be required. Such breakdowns have a specific effect on the system, particularly on the queue length, busy period of the server and waiting time of the customers.

For the first time, to our knowledge here the generally distributed variable batch size service and bulk arrival queueing system is analyzed in two - stage heterogeneous queueing system. It is important to note that in the literature of two - stage heterogeneous with bulk queueing models, only bulk arrival is considered. Paper on bulk service two - stage heterogeneous does not exist in the literature which is the motivation for the development of this paper. Our paper differs from the existing ones in the following way: Two-stage heterogeneous concept is newly considered for a variable batch size service queuing model which has more practical importance. Probability generating function of queue length distribution at an arbitrary time epoch in steady state is obtained by using supplementary variable technique by Lee’s method.

The paper is structured as follows: 1. Introduction; 2. The queueing model description and its corresponding steady state equations are presented; 3. The distribution of queue sizes is covered; 4. PGF of the queue size at different epochs; 5. The queueing system's anticipated queue length is discovered; 6. Conclusion.

2. Model description and system equations

Crude oil refinery is one real-world use for the single server design. Petroleum contains a variety of chemical compounds. The process of purging impurities and dividing petroleum into products that may be used is called refining. There are two major phases in the refining process: phase I is where water is separated and sulfur compounds are removed, and phase II is where the fractionation process takes place.

Crude oil is a stable combination of oil and brine. To extract salt water, a significant volume of crude oil is pumped between two highly charged electrodes in Phase I. The water droplets evaporate and are disposed of. When oil has been separated from water, copper oxide is employed to treat it. Reactor I produces copper sulphide precipitate by the reaction of copper oxide and sulfur-containing petroleum. To eliminate this precipitate, phase II requires filtration.

The bulk of the crude oil in the furnace must be heated to 400 degrees Celsius during Phase II. Every component has vanished, with the exception of the asphalt residue cake. The vapor (server II) is passing through the fractionation column. The tower-like fractionation column is tall and cylindrical. Within are multiple horizontal trays made of stainless steel. Each tray has an open chimney that is covered with a loose cap. When it comes to fractionation, the lower tray has high boiling points while the top tray has low boiling values. Fractionation yields uncondensed gas and petroleum products such road tar, diesel oil, heavy oil, naphtha, and kerosene. The renovation process starts immediately in the event that server I or server.

Before departing, the operator must carry out the following tasks: cleaning and checking the tools and etc. When the operator returns to the refining of crude petroleum process, if the quantity of crude oil is less than a batch quantity, he continues to work on other projects until he discovers a sufficient quantity.

The above mentioned process can be described as a MX/G(a , b)/1 queueing system with two stages of heterogeneous service and server breakdown, where arrivals take place according to a Poisson process with arrival rate ƛ. The server is turned on to offer each unit with two phases of heterogeneous service in succession of size min ξ , b customers, where ba, once it detects at least a customers waiting for service, let’s say ξ: Two service phases: the service phase I (FPS) and the service phase II (SPS) (Busy periods). It is expected that the service discipline is FCFS. The assumption is that the service times will follow general laws and a PDF. The service channels will fail for a brief period of time while the server is functioning with the first phase of service, which has a breakdown probability of π1 at any moment, and the second phase of service, which has a breakdown probability of π2 at any moment. If a server breaks down during any phase of a batch of service, it is instantly sent for repair. The server can begin providing service to the remaining clients after it has been repaired. The server executes closedown work at its closedown time (C) once each SPS is finished. If the queue length is smaller than a the server then departs for a number of random-length vacations. Otherwise continues to serve for the next batch.

2.1. Notations and assumptions

This work employs the following notations.

ƛ is the arrival rate, let μ1 and μ1 represent the service rate during peak periods in phases I and II respectively, X(z1) is the PGF of X, which is the group size random variable, and is the probability that X=k. Assume that S1(.), S2(.), R1(.), R2(.), and V(.) represent the CDF of service time in phase I, services time in phase II, renovation time in phase I, renovation time in phase II, and vacation time, respectively, define S01t1, S02t1, R01t1, R02t1 and V0(t1) as the remaining service time in regular period in phase I of a batch time at time ‘t1’, remaining service time in regular period in phase II of a batch time at time t1, remaining renovation time in phase I and phase II respectively and denote S~1θ1, S~2θ1, R~1θ1, R~2θ1, and V~(θ1)the LST of S1, S2, R1, R2 and V respectively.

Nq1t1 – size of the queue at time t1.

Ns1t1 – Customers using the service at the moment t1.

Y1t1= 0 – whenever the server is away.

Y1t1= 1 if the server is performing phase I service while being busy.

Y1t1= 2 if the server is performing phase II service while being busy.

Y1t1= 3 if the server is undergoing the initial step of renovation.

Y1t1= 4 if the server is undergoing the final step of renovation.

Y1t1= 5 if the server is performing a shutdown task.

Z1t1=j if the server is taking a vacation, it will begin during the idle period.

Pi j 1x1, t1dt1=PrNs1t1=i,Nq1t1=j, x1S10x1x1+dt1 , Y1t1=1,
aib , j0,
Pi j 2x1, t1dt1=PrNs2t1=i ,Nq1t1=j, x1S20x1x1+dt1 , Y1t1=2,
j0,
Qj nx1, t1dt1=PrNq1t1=n, x1V0t1x1+dt1 , Y1t1=0 ,Z1t1=j ,
n0 , j1,
Rn1x1, t1dt1=PrNq1t1=n ,x1R10t1x1+dt1 , Y1t1=3 , n0,
Rn2x1, t1dt1=PrNq1t1=n ,x1R20t1x1+dt1 , Y1t1=4 , n0,
Cnx1, t1dt1=PrNq1t1=n ,x1C0t1x1+dt1 , Y1t1=5 , n0.

2.2. Steady state equations

Using the supplementary variable technique, the queueing system's equations are obtained as follows:

1
Pi 01x1-t1,t1+t1=Pi 0 1x1, t11-ƛt1
+m=abPmi 10,t1s1x1t1+Ri 10,t1s1x1t1+i=1Q1i0s1x1t1, aib,
2
Pij1x1-t1,t1+t1=Pij 1x1, t11-ƛt1
+k=1jPij-k 1x1,t1ƛgkt1, aib-1 , j1),
3
Pbj1x1-t1,t1+t1=Pbj 1x1, t11-ƛt1
+m=abPm b+j 10,t1s1x1t1+Rb+j 10,t1s1x1t1+k=1jPbj-k 1x1,t1ƛgkt1
+j=1Q1b+j0s1x1t1 , j1,
4
Pi 02x1-t1,t1+t1=Pi 0 2x1, t11-ƛt1
+m=abPmi 20,t1s2x1t1+Pi0 10,t1s2x1t1+R0 20,t1s1x1t1, aib,
5
Pij2x1-t1,t1+t1=Pij 2x1, t11-ƛt1
+k=1jPij-k 2x1,t1ƛgkt1, aib-1, j1,
6
Pbj2x1-t1,t1+t1=Pbj 2x1, t11-ƛt1
+m=abPm b+j 20,t1s2x1t1+Rb+j 20,t1s2x1t1+k=1jPbj-k 2x1,t1ƛgkt1,
j1,
7
Cnx1-t1,t1+t1=Cnx1, t11-ƛt1
+m=abPmn 20,t1cx1t1+k=1nCn-kx1,t1ƛgkt1, na-1,
8
Cnx1-t1,t1+t1=Cnx1, t11-ƛt1+k=1nCn-kx1,t1ƛgkt1, na,
9
Q10x1-t1,t1+t1=Q10x1, t11-ƛt1+C00vx1t1,
10
Q1nx1-t1,t1+t1=Q1nx1, t11-ƛt1+Cn0vx1t1
+k=1nQ1,n-kx1,t1ƛgkt1, n1,
11
Qj0x1-t1,t1+t1=Qj0x1, t11-ƛt1+Qj-1,00,t1vx1t1, j2,
12
Qjnx1-t1,t1+t1=Qjnx1, t11-ƛt1
+k=1nQj,n-kx1,t1ƛgkt1+Qj-1,n0,t1vx1t1, 1na-1, j2,
13
Qjnx1-t1,t1+t1=Qjnx1, t11-ƛt1+k=1nQj,n-kx1,t1ƛt1, na, j2,
14
Ri1x1-t1,t1+t1=Ri1x1, t11-ƛt1
+π1r1(x1)0Pi01ydyt1, aib,
15
Ri+j1x1-t1,t1+t1=Ri+j1x1, t11-ƛt1
+π1r1(x1)0Pij1ydyt1+ λk=1jRi+j-k1x1,t1gkt1, aib, j1,
16
Ri2x1-t1,t1+t1=Ri2x1, t11-ƛt1
+π1r1(x1)0Pi02ydyt1, aib,
17
Ri+j2x1-t1,t1+t1=Ri+j2x1, t11-ƛt1
+π1r1(x1)0Pij2ydyt1+ λk=1jRi+j-k2x1,t1gkt1, aib, j1.

2.3. Steady state equations

Using the supplementary variable technique, the queueing system's equations are obtained as follows:

18
-Pi 01'x1=-ƛPi0 1x1+m=abPmi 10s1x1+Ri 10s1x1+i=1Q1i0s1x1,
aib,
19
-Pij1'x1=-λPij 1x1+k=1jPij-k 1x1λgk, aib-1, j1,
20
-Pbj1'x1=-λPbj 1x1+m=abPm b+j 10s1x1+k=1jPbj-k 1x1λgk+R b+j 10s1x1
+j=1Q1b+j0s1x1, j1,
21
-Pi02'x1=-ƛPi0 2x1+m=abPmi 20s2x1+Pi0 10s2x1+Ri 20s2x1,
aib,
22
-Pij2'x1=-λPij 2x1+k=1jPmj-k 2x1λgk, aib-1, j1,
23
-Pbj2' x1=-ƛPbj 2x1+m=abPmb+j 20s2x1+k=1jPbj-k 2x1ƛgk
+R b+j 20s2x1, j1,
24
-Cn'x1=-ƛCnx1+m=abPmn 20,tCx1+k=1nCn-kx1,t1ƛgk , na-1,
25
-Cn'x1=-ƛCnx1+k=1nCn-kx1,t1ƛgk, na,
26
-Q10'x1=-ƛQ10x1+C00vx1,
27
-Q1n'x1=-ƛQ1nx1+k=1nQ1n-kx1ƛgk+Cn0vx1, n1,
28
-Qj0'x1=-ƛQj0x1+Qj-1,00vx1, j2,
29
-Qjn'x1=-ƛQjnx1+Qj-1,00vx1+k=1nQjn-kx1ƛgk, na-1,
30
-Qjn'x1=-ƛQjnx1+k=1nQjn-kx1ƛgk, j2, na,
31
-Ri(1)'x1=-ƛRi(1)x1+π1r1(x1)0Pi01ydy, 1ib,
32
-Ri+j(1)'x1=-ƛRi+j1x1+π1r1x10Pij 01ydy+ƛk=1jRi+j-k1x1gk, j1,
33
-Ri(2)'x1=-ƛRi(2)x1+π1r1(x1)0Pi02ydy, 1ib,
34
-Ri+j(2)'x1=-ƛRi+j2x1+π1r1x10Pij 02ydy+ƛk=1jRi+j-k2x1gk, j1.

Taking Laplace-Stieltjes transforms on both sides of the Eqs. (18-34) we get:

35
θ1 P~i01θ1-Pi010=ƛ P~i01θ1
-m=abPmi 10 S~1θ1-Ri 10 S~1θ1-i=1Q1i0 S~1θ1, aib,
36
θ1 P~ij1θ1-Pij10=ƛ P~ij1θ1-k=1j P~ij-kθ1ƛgk , aib-1, j1,
37
θ1 P~bj1θ1-Pbj10=ƛ P~bj1θ1
-m=abPmb+j 10 S~1θ1-Rb+j 10 S~1θ1-j=1Q1b+j0 S~1θ1, aib,
38
θ1 P~i02θ1-Pi020=ƛ P~i02θ1
-m=abPmi 20 S~2θ1-Pi0 10 S~2θ1-Ri 20 S~2θ1, aib,
39
θ1 P~ij2θ1-Pij20=λ P~ij2θ1-k=1j P~ij-kθ1λgk, aib-1, j1,
40
θ1 P~bj2θ1-Pbj20=ƛ P~bj2θ1
-m=abPmb+j 20 S~2θ1-k=1jP~bj-kθ1ƛgk-Rb+j 20 S~2θ1, aib,
41
θ1C~nθ1-C~n0=ƛC~nθ1+m=abPmn 20C~θ1+k=1jCn-kx1ƛgk, na-1,
42
θ1C~nθ1-C~n0=ƛC~nθ1+k=1nCn-kx1ƛgk, na,
43
θ1Q~10θ1-Q100=ƛQ~10θ1-C00V~θ1,
44
θ1Q~1nθ1-Q1n0=ƛQ~10θ1-ƛk=1nQ~1n-kθ1gk -Cn0V~θ1, n1,
45
θ1Q~j0θ1-Qj00=ƛQ~j0θ1-Qj-100V~θ1, j2,
46
θ1Q~jnθ1-Qjn0=ƛQ~jnθ1-Qj-1n0V~θ1-k=1nQ~jn-kθ1gk,
na-1, j2,
47
θ1Q~jnθ1-Qjn0=ƛQ~jnθ1-k=1nQ~jn-kθ1gk, na, j2,
48
θ1 R~i1θ1-Ri10=ƛ R~i1θ1+π1R~1θ10Pi01ydy, 1ib,
49
θ1 R~i+j1θ1-Ri+j10=ƛ R~i+j1θ1
+π1R~1θ10Pij1ydy+ƛk=1n Ri+j-k1θ1gk, j1,
50
θ1 R~i2θ1-Ri20=ƛ R~i2θ1+π1R~1θ10Pi02ydy, 1ib,
51
θ1 R~i+j2θ1-Ri+j20=ƛ R~i+j2θ1
+π1R~1θ10Pij2ydy+ƛk=1n Ri+j-k1θ1gk , j1.

3. Queue size distribution

Let's define the PGF as follows to determine the queue size distribution:

52
P~i1z1,θ1=j=0P~ij1θ1z1n , Pi1z1,0=j=0Pij10z1n, aib,
P~i2z1,θ1=j=0P~ij2θ1z1n , Pi2z1,0=j=0Pij20z1n, aib,
Q~jz1,θ1=j=1Q~1jθ1z1n , Qjz1,0=j=1Q1j0z1n, j1,
C~z1,θ1=n=0C~nθ1z1n , Cz1,0=n=0Cn0z1n,
R~iz1,θ1=n=aR~nθ1z1n , Riz1,0=n=aRn0z1n.

By multiplying the Eqs. (35-51) with suitable power of z1n and summing over n, (n=0 to ) and using Eq. (52):

53
(θ1-ƛ-ƛX(z1))Q~1z1,θ1=Q1z1,0-V~θ1Cz1,0,
54
(θ1-ƛ+ƛX(z1))Q~jz1,θ1=Qjz1,0-V~θ1n=0a-1Qj-1n0z1n, j2,
55
(θ1-ƛ+ƛX(z1))C~z1,θ1=Cz1,0-C~θ1n=0a-1m=ab Pmn20z1n,
56
(θ1-ƛ+ƛX(z1))P~i1z1,θ1=Pi1z1,0
-S~1θ1m=abPmi10-S~1θ1Ri10-S~1θ1i=1Q1i0, aib-1
57
(θ1-ƛ+ƛX(z1))P~b1z1,θ1=Pb1z1,0
-S~1θ1z1bm=abPm1z1,0-j=0b-1Pmj10z1j+i=1Q1z1,0-j=0b-1Q1j0z1j+R1z1,0-n=ab-1Rn10z1n,
58
(θ1-ƛ+ƛX(z1))P~i2z1,θ1=Pi2z1,0-S~2θ1Pi010
-S~2θ1m=abPmi20-S~2θ1Ri20, aib-1,
59
(θ1-ƛ+ƛX(z1))P~b2z1,θ1=Pb2z1,0
-S~2θ1z1bm=abPm2z1,0-j=0b-1Pmj20z1j+Pb020+i=1R2z1,0-n=ab-1Rn20z1n
60
(θ1-ƛ+ƛX(z1))R~1z1,θ1=R1z1,0-π1R~1θ1i=abP~i1z1,0,
61
(θ1-ƛ+ƛX(z1))R~2z1,θ1=R2z1,0-π2R~2θ1i=abP~i2z1,0.

By substituting θ1=ƛ-ƛX(z1) in the Eqs. (53-61), we get:

62
Q1z1,0=V~ƛ-ƛX(z1)Cz1,0,
63
Qjz1,0=V~ƛ-ƛX(z1)n=0a-1Qj-1n0z1n, j2,
64
Cz1,0=C~ƛ-ƛX(z1)n=0a-1m=ab Pmn(2)0z1n ,
65
Pi1z1,0=S~1ƛ-ƛX(z1)m=abPmi10+Ri10+i=1Q1i0, aib-1,
66
Pb1z1,0=S~1ƛ-ƛX(z1)f1(z1)z1b-S~1ƛ-ƛX(z1),

where:

f1z=m=ab-1Pm(1)(z,0)+l=1Ql(z,0)+R1z,0-i=0b-1pi1+Ri1zi+i=0b-1qizi,
67
Pi2z1,0=S~2ƛ-ƛX(z1)m=abPmi20+Ri20+Pi020, aib-1,
68
Pb2z1,0=S~1ƛ-ƛX(z1)f2(z1)z1b-S~1ƛ-ƛX(z1) ,

where:

f2z=m=ab-1Pm(2)(z,0)+R2z,0-i=0b-1pi2+Ri2zi,
69
R(1)z1,0=π1R~1ƛ-ƛX(z1)i=abP~i1z1,0 ,
70
R(2)z1,0=π1R~2ƛ-ƛX(z1)i=abP~i2z1,0,

here:

pi1=m=abPmi10, pi2=m=abPmi20, qi=i=1Q1i0,
Ri2=Ri20, Ri2=Ri20.

Using the Eqs. (62-70) in (53-61), after simplification we get:

71
Q~1z1,θ1=V~ƛ-ƛX(z1)-V~(θ1)Cz1,θ1 θ1-ƛ+ƛX(z1),
72
Q~jz1,θ1=V~ƛ-ƛXz1-V~θn=0a-1Qj-1n0z1n θ1-ƛ+ƛXz1, j 2,
73
C~z1,0=C~ƛ-ƛXz1-C~θ1n=0a-1m=ab Pmn20z1n θ1-ƛ+ƛXz1,
74
P~i1z1,0= S~1ƛ-ƛXz1-S~1θ1m=abPmi10+Ri10+i=1Q1i0θ1-ƛ+ƛXz1,
aib-1,
75
P~b1z1,0=S~1ƛ-ƛX(z1)-S~1(θ1)f1(z1)θ1-ƛ+ƛXz1z1b-S~1ƛ-ƛX(z1),
76
P~i2z1,0= S~2ƛ-ƛXz1-S~2θ1m=abPmi20+Ri20+P010θ1-ƛ+ƛXz1,
aib-1,
77
P~b2z1,0=S~2ƛ-ƛX(z1)-S~2(θ1)f2(z1)θ1-ƛ+ƛXz1z1b-S~2ƛ-ƛX(z1),
78
R~1z1,θ1=R~1ƛ-ƛX(z1)-R~1θ1π1i=abP~i1z1,0 θ1-ƛ+ƛXz1,
79
R~2z1,θ1=R~2ƛ-ƛX(z1)-R~2θ1π2i=abP~i2z1,0 θ1-ƛ+ƛXz1.

3.1. PGF of the queue size at different epochs

3.1.1. Close down completion epoch

Using the Eqs. (24) to (25) and substituting θ=0 and after some algebra, we get:

80
Cz1=C~ƛ-ƛXz1-1i=0a-1Pi2z1i -ƛ+ƛXz1.

3.1.2. Vacation completion epoch

Using the Eqs. (26) to (30) and substituting θ=0 and after some computation, we get:

81
Vz1=V~ƛ-ƛXz1-1C~ƛ-ƛXz1i=0a-1Pi1zi +i=0a-1qiz1i -ƛ+ƛXz1.

3.1.3. Service in phase I completion epoch

82
P1z1=S~1ƛ-ƛXz1-1i=ab-1Pi1z1b-z1i+Ri1z1b-z1i+qiz1b-z1i+S~1ƛ-ƛXz1-1V~ƛ-ƛXz1C~ƛ-ƛXz1-1i=0a-1Pi1z1i +S~1ƛ-ƛXz1-1V~ƛ-ƛXz1-1i=0a-1qiz1i ƛ-ƛXz11+π1R~1ƛ-ƛXz1-π1S~1ƛ-ƛXz1R~1ƛ-ƛXz1R~1ƛ-ƛXz1z1b-S~1ƛ-ƛXz1.

3.1.4. Service in phase II completion epoch

83
P2z1=S~2ƛ-ƛXz1-1i=ab-1Pi2z1b-z1i+Ri2z1b-z1i+P01z1b-z1iƛ-ƛXz11+π2R~2ƛ-ƛXz1-π2S~2ƛ-ƛXz1R~2ƛ-ƛXz1z1b-S~2ƛ-ƛXz1.

3.1.5. Renovation in phase I completion epoch

84
R1z1 =π1R~1ƛ-ƛXz1-1S~1ƛ-ƛXz1-1i=ab-1Pi1z1b-z1i+Ri1z1b-z1i+qiz1b-z1i+π1R~1ƛ-ƛXz1-1S~1ƛ-ƛXz1-1V~ƛ-ƛXz1C~ƛ-ƛXz1-1i=0a-1Pi1z1i +π1R~1ƛ-ƛXz1-1S~1ƛ-ƛXz1-1V~ƛ-ƛXz1-1i=0a-1qiz1i ƛ-ƛXz11+π1R~1ƛ-ƛXz1-π1S~1ƛ-ƛXz1R~1ƛ-ƛXz1R~1ƛ-ƛXz1z1b-S~1ƛ-ƛXz1.

3.1.6. Renovation in phase II completion epoch

85
R2z1=π2R~2ƛ-ƛXz1-1S~2ƛ-ƛXz1-1i=ab-1Pi2z1b-z1i+Ri2z1b-z1i+P01z1b-z1iƛ-ƛXz11+π1R~1ƛ-ƛXz1-π1S~1ƛ-ƛXz1R~1ƛ-ƛXz1R~1ƛ-ƛXz1z1b-S~1ƛ-ƛXz1.

3.2. PGF of queue size at an arbitrary time epoch

Obtaining the PGF of the queue size at any given time epoch is as follows:

86
Pz1=C~z1,0+m=ab-1P~m1z1,0+P~b1z1,0+m=ab-1P~m2z1,0+P~b2z1,0
+R~1z1,0+R~2z1,0.

Substitute Eqs. (58-66) in Eq. (73) with θ=0 we get:

87
Pz1=S~1ƛ-ƛXz1-1π1R~1ƛ-ƛXz1-1+1D2i=ab-1Ci1z1b-z1i+S~2ƛ-ƛXz1-1π2R~2ƛ-ƛXz1-1+1D1i=ab-1Ci2z1b-z1i+S~2ƛ-ƛXz1-1π2R~2ƛ-ƛXz1-1+1D1i=ab-1P01z1b-z1i+S~2ƛ-ƛXz1-1π2R~2ƛ-ƛXz1-1+1V~ƛ-ƛXz1C~ƛ-ƛXz1-1D1D2i=0a-1Pi1z1i+S~1ƛ-ƛXz1-1π1R~1ƛ-ƛXz1-1+1V~ƛ-ƛXz1-1D2+V~ƛ-ƛXz1-1D1D2i=0a-1qiz1i +C~ƛ-ƛX(z1)-1D1D2i=0a-1Pi2z1i-ƛ+ƛX(z1)D1D2,

where:

D1=1+π1R~1ƛ-ƛXz1-π1S~1ƛ-ƛXz1R~1ƛ-ƛXz1z1b-S~1ƛ-ƛXz1,
D2=1+π2R~2ƛ-ƛXz1-π2S~2ƛ-ƛXz1R~2ƛ-ƛXz1z1b-S~2ƛ-ƛXz1.

3.3. Steady state condition

P1= 1 must be satisfied by the probability generating function. Applying L’Hospital principles and equating the expression to 1 will satisfy this requirement. Consecutively:

88
ES1b-ƛEXES2i=ab-1Ci1z1b-z1i+ES2b-ƛEXES1i=ab-1Ci1z1b-z1i+ES2b-ƛEXES1i=ab-1P0z1b-z1i+ES1b-ƛEXES2+b-ƛEXES1b-ƛEXES2ƛEXEV+ƛEXECi=0a-1Pi1+b-ƛEXES1+b-ƛEXES2 ƛEXECi=0a-1Pi2
+ES1b-ƛEXES2+b-ƛEXES1b-ƛEXES2ƛEXEVi=0a-1qi
=b-ƛEXES1b-ƛEXES2.

As a result P1= 1 satisfied if:

z1b-S~1ƛ-ƛXz1z1b-S~2ƛ-ƛXz1>0, ρ=ƛ2EX2ES1ES2b.

Theorem 1.

qn=i=0nKiPn-12 , n= 0 , 1 , 2 , 3 , .. a-1,

where Kn=hn+i=0naiKn-i1-α0, n= 0 , 1 , 2 , 3 , .. a-1 with K0=α0β01-α0, hn=i=0nα0βn-i .

The possibilities that a customer will arrive during a holiday or a shut-down period are αi's and βi's respectively.

Proof.

Using the Eqs. (44)-(46), j=1Qj(z1,0) simplifies to:

89
n=0qnz1n=V~ƛ-ƛXz1C~ƛ-ƛXz1n=0a-1Pn2z1nn=0a-1qnz1n=n=0αnz1nj=0βjz1jn=0a-1Pn2z1n+n=0a-1qnz1nj=0ni=0n-jαiβn-ipj(2)+i=0nαn-iqiz1n.

Equating the coefficients of z1n on both sides of the above equation for n= 0, 1, 2, 3, …, a-1 we have:

qn=j=0ni=0n-jαiβn-i-j Pj2+ i=0n-jαn-iqi,
qn=j=0ni=0n-jαiβn-i-j Pj2+ i=0nαn-iqi.

On solving for qn , we get:

qn=j=0ni=0n-jαiβn-i-j Pj2+ i=0n-1αn-iqi1-α0.

Coefficient of Pn2 in qn is α0β01-α0=K0.

Coefficient of Pn-12 in qn-1 is h1+α1.

Coefficient of Pn-121-α0 in qn-1 is h1+α1K01-α0=K1.

3.4. Expected length of busy period

Let B be the random variable for the busy period. Then estimated duration of the busy season is:

EB=ET1+ET2i=0a-1di +π1ER1, ET2=ES2+π2ER2, where ET1=ES1.

3.5. Expected length of the idle period

According to Arumuganathan and Jeyakumar’s method [1], it is obtained.

Theorem 2.

Let I be the random variable. Then the duration of the waiting period as anticipated is given by, EI=EI1+EC:

EI1=EV1-n=0a-1Q1n0=EV1-n=0a-1i=0nj=0n-1αjβn-i-jPi1,

where I1 is the random variable denoting the “Idle period due to multiple vacation process”, EC is the expected closedown time.

By theorem the expected idle period EI is obtained as EI=EI1+EC.

3.6. Expected queue length at an arbitrary time epoch

90
EQ=b-s212H11+b-s212b-s112V11i=0a-1Pi1+b-s212H12-b-s212b-s112V12n=0a-1iPi1+b-s212b-s112L1i=0a-1Pi2+b-s212b-s112L2n=0a-1iPi2+b-s212H13+b-s212b-s112V13i=0a-1qi+b-s212H14-b-s212b-s112V14n=0a-1iqi+b-s212H15i=ab-1(b-i)Ci1+b-s212H16i=ab-1(bb-1-i(i-1))Ci1+b-s212H25i=ab-1b-iCi2+P0+b-s112H26i=ab-1(bb-1-i(i-1))Ci2+P02ƛX12b-s112b-s212.
H11=2π1T11S11R11V1+C1+T11S11V2+C2+2V1C1-2T11S12V1+C1 ,
H12=T11S11V1+C1, H13=2π1T11S11R11V1+T11S11V2-2T11S12V1,
H14=T11S11V1, H15=2π1T11S11R11-2T11S11,
H16=T11S11, T11= ƛX1b-s11,
H25=2π1T21S21R21-2T21S22, H26=T21S21, T21= ƛX1b-s21,
V11= V1ƛX2-2V2ƛX1, V12= 2V1ƛX1, V13= V1ƛX2, V14= 2V1ƛX1,
L1= ƛX1C2-ƛX2C1, L1= 2ƛX1C1.

3.7. Expected waiting time

The Little’s formula is used to calculate the expected waiting time as follows:

91
EW=E(Q)ƛE(X),

where E(Q) is given in Eq. (89).

4. Conclusions

In this study, examine the behaviour of the server failure without interruption in a queueing system with MX/G(a , b)/1 and two phases of heterogeneous service. We obtain the probability generating function of the queue size at any time epoch, Close down completion epoch, Vacation completion epoch, Service in phase I completion epoch, Service in phase II completion epoch, Renovation in phase I completion epoch and Renovation in phase II completion epoch. In future a cost model is discuss with numerical examples.

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About this article

Received
27 November 2023
Accepted
18 December 2023
Published
30 June 2024
Keywords
bulk service
heterogeneous services
multiple vacations
closedown
queue size distribution
server breakdown
Acknowledgements

The authors have not disclosed any funding.

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Author Contributions

Palaniammal S and Kumar K established the PGF of queue size and various time epoch of the model. Palaniammal S: project administration, supervision, validation. Kumar K: conceptualization, formal analysis, investigation, methodology, visualization, writing – review and editing.

Conflict of interest

The authors declare that they have no conflict of interest.