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

S, Palaniammal; K, Kumar

doi:10.21595/mme.2023.23827

Mathematical Models in Engineering

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Published: 30 June 2024

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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 $M_{X} / G (a, b)$ 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 ( $π_{1}$ ) at any time, and during the server’s second phase of operation, the service channels will fail with a probability of ( $π_{2}$ ) 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/1queueing 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 $M_{X} / G (a, b) / 1$ 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 $M_{X} / G (a, b) / 1$ 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 $b \geq a$ , 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 $(π_{1})$ at any moment, and the second phase of service, which has a breakdown probability of $(π_{2})$ 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 $μ_{1}$ and $μ_{1}$ represent the service rate during peak periods in phases I and II respectively, $X (z_{1})$ 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 ${S^{0}}_{1} (t_{1})$ , ${S^{0}}_{2} (t_{1})$ , ${R^{0}}_{1} (t_{1})$ , ${R^{0}}_{2} (t_{1})$ and $V^{0} (t_{1})$ as the remaining service time in regular period in phase I of a batch time at time ‘ $t_{1}$ ’, remaining service time in regular period in phase II of a batch time at time $t_{1}$ , remaining renovation time in phase I and phase II respectively and denote ${\tilde{S}}_{1} (θ_{1})$ , ${\tilde{S}}_{2} (θ_{1})$ , ${\tilde{R}}_{1} (θ_{1})$ , ${\tilde{R}}_{2} (θ_{1})$ , and $\tilde{V} (θ_{1})$ the LST of $S_{1}$ , $S_{2}$ , $R_{1}$ , $R_{2}$ and $V$ respectively.

$N_{q_{1}} (t_{1})$ – size of the queue at time $t_{1}$ .

$N_{s_{1}} (t_{1})$ – Customers using the service at the moment $t_{1}$ .

$Y_{1} (t_{1}) =$ 0 – whenever the server is away.

$Y_{1} (t_{1}) =$ 1 if the server is performing phase I service while being busy.

$Y_{1} (t_{1}) =$ 2 if the server is performing phase II service while being busy.

$Y_{1} (t_{1}) =$ 3 if the server is undergoing the initial step of renovation.

$Y_{1} (t_{1}) =$ 4 if the server is undergoing the final step of renovation.

$Y_{1} (t_{1}) =$ 5 if the server is performing a shutdown task.

$Z_{1} (t_{1}) = j$ if the server is taking a vacation, it will begin during the idle period.

P_{i j}^{(1)} (x_{1}, t_{1}) {d t}_{1} = \Pr \{N_{s_{1}} (t_{1}) = i, N_{q_{1}} (t_{1}) = j, x_{1} \leq S_{1}^{0} (x_{1}) \leq x_{1} + {d t}_{1}, Y_{1} (t_{1}) = 1\},

(a \leq i \leq b), (j \geq 0),

P_{i j}^{(2)} (x_{1}, t_{1}) {d t}_{1} = \Pr \{N_{s_{2}} (t_{1}) = i, N_{q_{1}} (t_{1}) = j, x_{1} \leq S_{2}^{0} (x_{1}) \leq x_{1} + {d t}_{1}, Y_{1} (t_{1}) = 2\},

(j \geq 0),

Q_{j n} (x_{1}, t_{1}) {d t}_{1} = \Pr \{N_{q_{1}} (t_{1}) = n, x_{1} \leq V^{0} (t_{1}) \leq x_{1} + {d t}_{1}, Y_{1} (t_{1}) = 0, Z_{1} (t_{1}) = j\},

(n \geq 0), (j \geq 1),

R_{n}^{(1)} (x_{1}, t_{1}) {d t}_{1} = \Pr \{N_{q_{1}} (t_{1}) = n, x_{1} \leq R_{1}^{0} (t_{1}) \leq x_{1} + {d t}_{1}, Y_{1} (t_{1}) = 3\}, n \geq 0,

R_{n}^{(2)} (x_{1}, t_{1}) {d t}_{1} = \Pr \{N_{q_{1}} (t_{1}) = n, x_{1} \leq R_{2}^{0} (t_{1}) \leq x_{1} + {d t}_{1}, Y_{1} (t_{1}) = 4\}, n \geq 0,

C_{n} (x_{1}, t_{1}) {d t}_{1} = \Pr \{N_{q_{1}} (t_{1}) = n, x_{1} \leq C^{0} (t_{1}) \leq x_{1} + {d t}_{1}, Y_{1} (t_{1}) = 5\}, n \geq 0 .

2.2. Steady state equations

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

1

P_{i 0}^{(1)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = P_{i 0}^{(1)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ \sum_{m = a}^{b} P_{m i}^{(1)} (0, t_{1}) s_{1} (x_{1}) ∆ t_{1} + R_{i}^{(1)} (0, t_{1}) s_{1} (x_{1}) ∆ t_{1} + \sum_{i = 1}^{\infty} Q_{1 i} (0) s_{1} (x_{1}) ∆ t_{1}, a \leq i \leq b,

2

P_{i j}^{(1)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = P_{i j}^{(1)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ \sum_{k = 1}^{j} P_{i j - k}^{(1)} (x_{1}, t_{1}) ƛ g_{k} ∆ t_{1}, a \leq i \leq b - 1, j \geq 1),

3

P_{b j}^{(1)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = P_{b j}^{(1)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ \sum_{m = a}^{b} P_{m b + j}^{(1)} (0, t_{1}) s_{1} (x_{1}) ∆ t_{1} + R_{b + j}^{(1)} (0, t_{1}) s_{1} (x_{1}) ∆ t_{1} + \sum_{k = 1}^{j} P_{b j - k}^{(1)} (x_{1}, t_{1}) ƛ g_{k} ∆ t_{1}

+ \sum_{j = 1}^{\infty} Q_{1 b + j} (0) s_{1} (x_{1}) ∆ t_{1}, j \geq 1,

4

P_{i 0}^{(2)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = P_{i 0}^{(2)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ \sum_{m = a}^{b} P_{m i}^{(2)} (0, t_{1}) s_{2} (x_{1}) ∆ t_{1} + P_{i 0}^{(1)} (0, t_{1}) s_{2} (x_{1}) ∆ t_{1} + R_{0}^{(2)} (0, t_{1}) s_{1} (x_{1}) ∆ t_{1}, a \leq i \leq b,

5

P_{i j}^{(2)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = P_{i j}^{(2)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ \sum_{k = 1}^{j} P_{i j - k}^{(2)} (x_{1}, t_{1}) ƛ g_{k} ∆ t_{1}, a \leq i \leq b - 1, j \geq 1,

6

P_{b j}^{(2)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = P_{b j}^{(2)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ \sum_{m = a}^{b} P_{m b + j}^{(2)} (0, t_{1}) s_{2} (x_{1}) ∆ t_{1} + R_{b + j}^{(2)} (0, t_{1}) s_{2} (x_{1}) ∆ t_{1} + \sum_{k = 1}^{j} P_{b j - k}^{(2)} (x_{1}, t_{1}) ƛ g_{k} ∆ t_{1},

j \geq 1,

7

C_{n} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = C_{n} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ \sum_{m = a}^{b} P_{m n}^{(2)} (0, t_{1}) c (x_{1}) ∆ t_{1} + \sum_{k = 1}^{n} C_{n - k} (x_{1}, t_{1}) ƛ g_{k} ∆ t_{1}, n \leq a - 1,

8

C_{n} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = C_{n} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1}) + \sum_{k = 1}^{n} C_{n - k} (x_{1}, t_{1}) ƛ g_{k} ∆ t_{1}, n \geq a,

9

Q_{10} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = Q_{10} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1}) + C_{0} (0) v (x_{1}) ∆ t_{1},

10

Q_{1 n} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = Q_{1 n} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1}) + C_{n} (0) v (x_{1}) ∆ t_{1}

+ \sum_{k = 1}^{n} Q_{1, n - k} (x_{1}, t_{1}) ƛ g_{k} ∆ t_{1}, n \geq 1,

11

Q_{j 0} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = Q_{j 0} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1}) + Q_{j - 1,0} (0, t_{1}) v (x_{1}) ∆ t_{1}, j \geq 2,

12

Q_{j n} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = Q_{j n} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ \sum_{k = 1}^{n} Q_{j, n - k} (x_{1}, t_{1}) ƛ g_{k} ∆ t_{1} + Q_{j - 1, n} (0, t_{1}) v (x_{1}) ∆ t_{1}, 1 \leq n \leq a - 1, j \geq 2,

13

Q_{j n} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = Q_{j n} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1}) + \sum_{k = 1}^{n} Q_{j, n - k} (x_{1}, t_{1}) ƛ ∆ t_{1}, n \geq a, j \geq 2,

14

R_{i}^{(1)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = R_{i}^{(1)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ π_{1} r_{1} (x_{1}) \int_{0}^{\infty} P_{i 0}^{(1)} (y) d y ∆ t_{1}, a \leq i \leq b,

15

R_{i + j}^{(1)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = R_{i + j}^{(1)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ π_{1} r_{1} (x_{1}) \int_{0}^{\infty} P_{i j}^{(1)} (y) d y ∆ t_{1} + λ \sum_{k = 1}^{j} R_{i + j - k}^{(1)} (x_{1}, t_{1}) g_{k} ∆ t_{1}, a \leq i \leq b, j \geq 1,

16

R_{i}^{(2)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = R_{i}^{(2)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ π_{1} r_{1} (x_{1}) \int_{0}^{\infty} P_{i 0}^{(2)} (y) d y ∆ t_{1}, a \leq i \leq b,

17

R_{i + j}^{(2)} (x_{1} - ∆ t_{1}, t_{1} + ∆ t_{1}) = R_{i + j}^{(2)} (x_{1}, t_{1}) (1 - ƛ ∆ t_{1})

+ π_{1} r_{1} (x_{1}) \int_{0}^{\infty} P_{i j}^{(2)} (y) d y ∆ t_{1} + λ \sum_{k = 1}^{j} R_{i + j - k}^{(2)} (x_{1}, t_{1}) g_{k} ∆ t_{1}, a \leq i \leq b, j \geq 1 .

2.3. Steady state equations

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

18

{- P}_{i 0}^{{(1)}^{'}} (x_{1}) = {- ƛ P}_{i 0}^{(1)} (x_{1}) + \sum_{m = a}^{b} P_{m i}^{(1)} (0) s_{1} (x_{1}) + R_{i}^{(1)} (0) s_{1} (x_{1}) + \sum_{i = 1}^{\infty} Q_{1 i} (0) s_{1} (x_{1}),

a \leq i \leq b,

19

{- P}_{i j}^{{(1)}^{'}} (x_{1}) = {- λ P}_{i j}^{(1)} (x_{1}) + \sum_{k = 1}^{j} P_{i j - k}^{(1)} (x_{1}) λ g_{k}, a \leq i \leq b - 1, j \geq 1,

20

{- P}_{b j}^{{(1)}^{'}} (x_{1}) = {- λ P}_{b j}^{(1)} (x_{1}) + \sum_{m = a}^{b} P_{m b + j}^{(1)} (0) s_{1} (x_{1}) + \sum_{k = 1}^{j} P_{b j - k}^{(1)} (x_{1}) λ g_{k} + R_{b + j}^{(1)} (0) s_{1} (x_{1})

+ \sum_{j = 1}^{\infty} Q_{1 b + j} (0) s_{1} (x_{1}), j \geq 1,

21

{- P}_{i 0}^{{(2)}^{'}} (x_{1}) = {- ƛ P}_{i 0}^{(2)} (x_{1}) + \sum_{m = a}^{b} P_{m i}^{(2)} (0) s_{2} (x_{1}) + P_{i 0}^{(1)} (0) s_{2} (x_{1}) + R_{i}^{(2)} (0) s_{2} (x_{1}),

a \leq i \leq b,

22

{- P}_{i j}^{{(2)}^{'}} (x_{1}) = {- λ P}_{i j}^{(2)} (x_{1}) + \sum_{k = 1}^{j} P_{m j - k}^{(2)} (x_{1}) λ g_{k}, a \leq i \leq b - 1, j \geq 1,

23

{- P}_{b j}^{{(2)}^{'}} (x_{1}) = {- ƛ P}_{b j}^{(2)} (x_{1}) + \sum_{m = a}^{b} P_{m b + j}^{(2)} (0) s_{2} (x_{1}) + \sum_{k = 1}^{j} P_{b j - k}^{(2)} (x_{1}) ƛ g_{k}

+ R_{b + j}^{(2)} (0) s_{2} (x_{1}), j \geq 1,

24

{- C}_{n}^{'} (x_{1}) = - ƛ C_{n} (x_{1}) + \sum_{m = a}^{b} P_{m n}^{(2)} (0, t) C (x_{1}) + \sum_{k = 1}^{n} C_{n - k} (x_{1}, t_{1}) ƛ g_{k}, n \leq a - 1,

25

{- C}_{n}^{'} (x_{1}) = - ƛ C_{n} (x_{1}) + \sum_{k = 1}^{n} C_{n - k} (x_{1}, t_{1}) ƛ g_{k}, n \geq a,

26

{- Q}_{10}^{'} (x_{1}) = - ƛ Q_{10} (x_{1}) + C_{0} (0) v (x_{1}),

27

{- Q}_{1 n}^{'} (x_{1}) = - ƛ Q_{1 n} (x_{1}) + \sum_{k = 1}^{n} Q_{1 n - k} (x_{1}) ƛ g_{k} + C_{n} (0) v (x_{1}), n \geq 1,

28

{- Q}_{j 0}^{'} (x_{1}) = - ƛ Q_{j 0} (x_{1}) + Q_{j - 1,0} (0) v (x_{1}), j \geq 2,

29

{- Q}_{j n}^{'} (x_{1}) = - ƛ Q_{j n} (x_{1}) + Q_{j - 1,0} (0) v (x_{1}) + \sum_{k = 1}^{n} Q_{j n - k} (x_{1}) ƛ g_{k}, n \geq a - 1,

30

{- Q}_{j n}^{'} (x_{1}) = - ƛ Q_{j n} (x_{1}) + \sum_{k = 1}^{n} Q_{j n - k} (x_{1}) ƛ g_{k}, j \geq 2, n \geq a,

31

{- R}_{i}^{{(1)}^{'}} (x_{1}) = - ƛ R_{i}^{(1)} (x_{1}) + π_{1} r_{1} (x_{1}) \int_{0}^{\infty} P_{i 0}^{(1)} (y) d y, 1 \leq i \leq b,

32

{- R}_{i + j}^{{(1)}^{'}} (x_{1}) = - ƛ R_{i + j}^{(1)} (x_{1}) + π_{1} r_{1} (x_{1}) \int_{0}^{\infty} P_{i j 0}^{(1)} (y) d y + ƛ \sum_{k = 1}^{j} R_{i + j - k}^{(1)} (x_{1}) g_{k}, j \leq 1,

33

{- R}_{i}^{{(2)}^{'}} (x_{1}) = - ƛ R_{i}^{(2)} (x_{1}) + π_{1} r_{1} (x_{1}) \int_{0}^{\infty} P_{i 0}^{(2)} (y) d y, 1 \leq i \leq b,

34

{- R}_{i + j}^{{(2)}^{'}} (x_{1}) = - ƛ R_{i + j}^{(2)} (x_{1}) + π_{1} r_{1} (x_{1}) \int_{0}^{\infty} P_{i j 0}^{(2)} (y) d y + ƛ \sum_{k = 1}^{j} R_{i + j - k}^{(2)} (x_{1}) g_{k}, j \geq 1 .

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

35

θ_{1} {\tilde{P}}_{i 0}^{(1)} (θ_{1}) {- P}_{i 0}^{(1)} (0) = ƛ {\tilde{P}}_{i 0}^{(1)} (θ_{1})

- \sum_{m = a}^{b} P_{m i}^{(1)} (0) {\tilde{S}}_{1} (θ_{1}) - R_{i}^{(1)} (0) {\tilde{S}}_{1} (θ_{1}) - \sum_{i = 1}^{\infty} Q_{1 i} (0) {\tilde{S}}_{1} (θ_{1}), a \leq i \leq b,

36

θ_{1} {\tilde{P}}_{i j}^{(1)} (θ_{1}) {- P}_{i j}^{(1)} (0) = ƛ {\tilde{P}}_{i j}^{(1)} (θ_{1}) - \sum_{k = 1}^{j} {\tilde{P}}_{i j - k} (θ_{1}) ƛ g_{k}, a \leq i \leq b - 1, j \geq 1,

37

θ_{1} {\tilde{P}}_{b j}^{(1)} (θ_{1}) {- P}_{b j}^{(1)} (0) = ƛ {\tilde{P}}_{b j}^{(1)} (θ_{1})

- \sum_{m = a}^{b} P_{m b + j}^{(1)} (0) {\tilde{S}}_{1} (θ_{1}) - R_{b + j}^{(1)} (0) {\tilde{S}}_{1} (θ_{1}) - \sum_{j = 1}^{\infty} Q_{1 b + j} (0) {\tilde{S}}_{1} (θ_{1}), a \leq i \leq b,

38

θ_{1} {\tilde{P}}_{i 0}^{(2)} (θ_{1}) {- P}_{i 0}^{(2)} (0) = ƛ {\tilde{P}}_{i 0}^{(2)} (θ_{1})

- \sum_{m = a}^{b} P_{m i}^{(2)} (0) {\tilde{S}}_{2} (θ_{1}) - P_{i 0}^{(1)} (0) {\tilde{S}}_{2} (θ_{1}) - R_{i}^{(2)} (0) {\tilde{S}}_{2} (θ_{1}), a \leq i \leq b,

39

θ_{1} {\tilde{P}}_{i j}^{(2)} (θ_{1}) {- P}_{i j}^{(2)} (0) = λ {\tilde{P}}_{i j}^{(2)} (θ_{1}) - \sum_{k = 1}^{j} {\tilde{P}}_{i j - k} (θ_{1}) λ g_{k}, a \leq i \leq b - 1, j \geq 1,

40

θ_{1} {\tilde{P}}_{b j}^{(2)} (θ_{1}) {- P}_{b j}^{(2)} (0) = ƛ {\tilde{P}}_{b j}^{(2)} (θ_{1})

- \sum_{m = a}^{b} P_{m b + j}^{(2)} (0) {\tilde{S}}_{2} (θ_{1}) - \sum_{k = 1}^{j} {\tilde{P}}_{b j - k} (θ_{1}) ƛ g_{k} - R_{b + j}^{(2)} (0) {\tilde{S}}_{2} (θ_{1}), a \leq i \leq b,

41

θ_{1} {\tilde{C}}_{n} (θ_{1}) - {\tilde{C}}_{n} (0) = ƛ {\tilde{C}}_{n} (θ_{1}) + \sum_{m = a}^{b} P_{m n}^{(2)} (0) \tilde{C} (θ_{1}) + \sum_{k = 1}^{j} C_{n - k} (x_{1}) ƛ g_{k}, n \leq a - 1,

42

θ_{1} {\tilde{C}}_{n} (θ_{1}) - {\tilde{C}}_{n} (0) = ƛ {\tilde{C}}_{n} (θ_{1}) + \sum_{k = 1}^{n} C_{n - k} (x_{1}) ƛ g_{k}, n \geq a,

43

θ_{1} {\tilde{Q}}_{10} (θ_{1}) - Q_{10} (0) = ƛ {\tilde{Q}}_{10} (θ_{1}) - C_{0} (0) \tilde{V} (θ_{1}),

44

θ_{1} {\tilde{Q}}_{1 n} (θ_{1}) - Q_{1 n} (0) = ƛ {\tilde{Q}}_{10} (θ_{1}) - ƛ \sum_{k = 1}^{n} {\tilde{Q}}_{1 n - k} (θ_{1}) g_{k} {- C}_{n} (0) \tilde{V} (θ_{1}), n \geq 1,

45

θ_{1} {\tilde{Q}}_{j 0} (θ_{1}) - Q_{j 0} (0) = ƛ {\tilde{Q}}_{j 0} (θ_{1}) - Q_{j - 10} (0) \tilde{V} (θ_{1}), j \geq 2,

46

θ_{1} {\tilde{Q}}_{j n} (θ_{1}) - Q_{j n} (0) = ƛ {\tilde{Q}}_{j n} (θ_{1}) - Q_{j - 1 n} (0) \tilde{V} (θ_{1}) - \sum_{k = 1}^{n} {\tilde{Q}}_{j n - k} (θ_{1}) g_{k},

n \leq a - 1, j \geq 2,

47

θ_{1} {\tilde{Q}}_{j n} (θ_{1}) - Q_{j n} (0) = ƛ {\tilde{Q}}_{j n} (θ_{1}) - \sum_{k = 1}^{n} {\tilde{Q}}_{j n - k} (θ_{1}) g_{k}, n \geq a, j \geq 2,

48

θ_{1} {\tilde{R}}_{i}^{(1)} (θ_{1}) {- R}_{i}^{(1)} (0) = ƛ {\tilde{R}}_{i}^{(1)} (θ_{1}) + π_{1} {\tilde{R}}_{1} (θ_{1}) \int_{0}^{\infty} P_{i 0}^{(1)} (y) d y, 1 \leq i \leq b,

49

θ_{1} {\tilde{R}}_{i + j}^{(1)} (θ_{1}) {- R}_{i + j}^{(1)} (0) = ƛ {\tilde{R}}_{i + j}^{(1)} (θ_{1})

+ π_{1} {\tilde{R}}_{1} (θ_{1}) \int_{0}^{\infty} P_{i j}^{(1)} (y) d y + ƛ \sum_{k = 1}^{n} R_{i + j - k}^{(1)} (θ_{1}) g_{k}, j \geq 1,

50

θ_{1} {\tilde{R}}_{i}^{(2)} (θ_{1}) {- R}_{i}^{(2)} (0) = ƛ {\tilde{R}}_{i}^{(2)} (θ_{1}) + π_{1} {\tilde{R}}_{1} (θ_{1}) \int_{0}^{\infty} P_{i 0}^{(2)} (y) d y, 1 \leq i \leq b,

51

θ_{1} {\tilde{R}}_{i + j}^{(2)} (θ_{1}) {- R}_{i + j}^{(2)} (0) = ƛ {\tilde{R}}_{i + j}^{(2)} (θ_{1})

+ π_{1} {\tilde{R}}_{1} (θ_{1}) \int_{0}^{\infty} P_{i j}^{(2)} (y) d y + ƛ \sum_{k = 1}^{n} R_{i + j - k}^{(1)} (θ_{1}) g_{k}, j \geq 1 .

3. Queue size distribution

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

52

{\tilde{P}}_{i}^{(1)} (z_{1}, θ_{1}) = \sum_{j = 0}^{\infty} {\tilde{P}}_{i j}^{(1)} (θ_{1}) z_{1}^{n}, P_{i}^{(1)} (z_{1}, 0) = \sum_{j = 0}^{\infty} P_{i j}^{(1)} (0) z_{1}^{n}, a \leq i \leq b,

{\tilde{P}}_{i}^{(2)} (z_{1}, θ_{1}) = \sum_{j = 0}^{\infty} {\tilde{P}}_{i j}^{(2)} (θ_{1}) z_{1}^{n}, P_{i}^{(2)} (z_{1}, 0) = \sum_{j = 0}^{\infty} P_{i j}^{(2)} (0) z_{1}^{n}, a \leq i \leq b,

{\tilde{Q}}_{j} (z_{1}, θ_{1}) = \sum_{j = 1}^{\infty} {\tilde{Q}}_{1 j} (θ_{1}) z_{1}^{n}, Q_{j} (z_{1}, 0) = \sum_{j = 1}^{\infty} Q_{1 j} (0) z_{1}^{n}, j \geq 1,

\tilde{C} (z_{1}, θ_{1}) = \sum_{n = 0}^{\infty} {\tilde{C}}_{n} (θ_{1}) z_{1}^{n}, C (z_{1}, 0) = \sum_{n = 0}^{\infty} C_{n} (0) z_{1}^{n},

{\tilde{R}}_{i} (z_{1}, θ_{1}) = \sum_{n = a}^{\infty} {\tilde{R}}_{n} (θ_{1}) z_{1}^{n}, R_{i} (z_{1}, 0) = \sum_{n = a}^{\infty} R_{n} (0) z_{1}^{n} .

By multiplying the Eqs. (35-51) with suitable power of $z_{1}^{n}$ and summing over $n$ , $(n = 0$ to $\infty)$ and using Eq. (52):

53

{(θ}_{1} - ƛ - ƛ X (z_{1})) {\tilde{Q}}_{1} (z_{1}, θ_{1}) = Q_{1} (z_{1}, 0) - \tilde{V} (θ_{1}) C (z_{1}, 0),

54

{(θ}_{1} - ƛ + ƛ X (z_{1})) {\tilde{Q}}_{j} (z_{1}, θ_{1}) = Q_{j} (z_{1}, 0) - \tilde{V} (θ_{1}) \sum_{n = 0}^{a - 1} Q_{j - 1 n} (0) z_{1}^{n}, j \geq 2,

55

{(θ}_{1} - ƛ + ƛ X (z_{1})) \tilde{C} (z_{1}, θ_{1}) = C (z_{1}, 0) - \tilde{C} (θ_{1}) \sum_{n = 0}^{a - 1} \sum_{m = a}^{b} P_{m n}^{(2)} (0) z_{1}^{n},

56

{(θ}_{1} - ƛ + ƛ X (z_{1})) {\tilde{P}}_{i}^{(1)} (z_{1}, θ_{1}) = P_{i}^{(1)} (z_{1}, 0)

- {\tilde{S}}_{1} (θ_{1}) \sum_{m = a}^{b} P_{m i}^{(1)} (0) - {\tilde{S}}_{1} (θ_{1}) R_{i}^{(1)} (0) - {\tilde{S}}_{1} (θ_{1}) \sum_{i = 1}^{\infty} Q_{1 i} (0), a \leq i \leq b - 1

57

{(θ}_{1} - ƛ + ƛ X (z_{1})) {\tilde{P}}_{b}^{(1)} (z_{1}, θ_{1}) = P_{b}^{(1)} (z_{1}, 0)

- \frac{{\tilde{S}}_{1} (θ_{1})}{{z_{1}}^{b}} \{\sum_{m = a}^{b} (P_{m}^{(1)} (z_{1}, 0) - \sum_{j = 0}^{b - 1} P_{m j}^{(1)} (0) {z_{1}}^{j}) + \sum_{i = 1}^{\infty} (Q_{1} (z_{1}, 0) - \sum_{j = 0}^{b - 1} Q_{1 j} (0) {z_{1}}^{j}) + (R^{(1)} (z_{1}, 0) - \sum_{n = a}^{b - 1} R_{n}^{(1)} (0) {z_{1}}^{n})\},

58

{(θ}_{1} - ƛ + ƛ X (z_{1})) {\tilde{P}}_{i}^{(2)} (z_{1}, θ_{1}) = P_{i}^{(2)} (z_{1}, 0) - {\tilde{S}}_{2} (θ_{1}) P_{i 0}^{(1)} (0)

- {\tilde{S}}_{2} (θ_{1}) \sum_{m = a}^{b} P_{m i}^{(2)} (0) - {\tilde{S}}_{2} (θ_{1}) R_{i}^{(2)} (0), a \leq i \leq b - 1,

59

{(θ}_{1} - ƛ + ƛ X (z_{1})) {\tilde{P}}_{b}^{(2)} (z_{1}, θ_{1}) = P_{b}^{(2)} (z_{1}, 0)

- \frac{{\tilde{S}}_{2} (θ_{1})}{{z_{1}}^{b}} \{\sum_{m = a}^{b} (P_{m}^{(2)} (z_{1}, 0) - \sum_{j = 0}^{b - 1} P_{m j}^{(2)} (0) {z_{1}}^{j}) + P_{b 0}^{(2)} (0) + \sum_{i = 1}^{\infty} (R^{(2)} (z_{1}, 0) - \sum_{n = a}^{b - 1} R_{n}^{(2)} (0) {z_{1}}^{n})\}

60

{(θ}_{1} - ƛ + ƛ X (z_{1})) {\tilde{R}}^{(1)} (z_{1}, θ_{1}) = R^{(1)} (z_{1}, 0) - π_{1} {\tilde{R}}_{1} (θ_{1}) \sum_{i = a}^{b} {\tilde{P}}_{i}^{(1)} (z_{1}, 0),

61

{(θ}_{1} - ƛ + ƛ X (z_{1})) {\tilde{R}}^{(2)} (z_{1}, θ_{1}) = R^{(2)} (z_{1}, 0) - π_{2} {\tilde{R}}_{2} (θ_{1}) \sum_{i = a}^{b} {\tilde{P}}_{i}^{(2)} (z_{1}, 0) .

By substituting $θ_{1} = ƛ - ƛ X (z_{1})$ in the Eqs. (53-61), we get:

62

Q_{1} (z_{1}, 0) = \tilde{V} (ƛ - ƛ X (z_{1})) C (z_{1}, 0),

63

Q_{j} (z_{1}, 0) = \tilde{V} (ƛ - ƛ X (z_{1})) \sum_{n = 0}^{a - 1} Q_{j - 1 n} (0) z_{1}^{n}, j \geq 2,

64

C (z_{1}, 0) = \tilde{C} (ƛ - ƛ X (z_{1})) [\sum_{n = 0}^{a - 1} \sum_{m = a}^{b} P_{m n}^{(2)} (0) z_{1}^{n}],

65

P_{i}^{(1)} (z_{1}, 0) = {\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) [\sum_{m = a}^{b} P_{m i}^{(1)} (0) + R_{i}^{(1)} (0) + \sum_{i = 1}^{\infty} Q_{1 i} (0)], a \leq i \leq b - 1,

66

P_{b}^{(1)} (z_{1}, 0) = \frac{{\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) f_{1} (z_{1})}{{z_{1}}^{b} - {\tilde{S}}_{1} (ƛ - ƛ X (z_{1}))},

where:

f_{1} (z) = \sum_{m = a}^{b - 1} P_{m}^{(1)} (z, 0) + \sum_{l = 1}^{\infty} Q_{l} (z, 0) + R^{(1)} (z, 0) - [\sum_{i = 0}^{b - 1} (p_{i}^{(1)} + R_{i}^{(1)}) z^{i} + \sum_{i = 0}^{b - 1} q_{i} z^{i}],

67

P_{i}^{(2)} (z_{1}, 0) = {\tilde{S}}_{2} (ƛ - ƛ X (z_{1})) [\sum_{m = a}^{b} P_{m i}^{(2)} (0) + R_{i}^{(2)} (0) + P_{i 0}^{(2)} (0)], a \leq i \leq b - 1,

68

P_{b}^{(2)} (z_{1}, 0) = \frac{{\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) f_{2} (z_{1})}{{z_{1}}^{b} - {\tilde{S}}_{1} (ƛ - ƛ X (z_{1}))},

where:

f_{2} (z) = \sum_{m = a}^{b - 1} P_{m}^{(2)} (z, 0) + R^{(2)} (z, 0) - [\sum_{i = 0}^{b - 1} (p_{i}^{(2)} + R_{i}^{(2)}) z^{i}],

69

R^{(1)} (z_{1}, 0) = π_{1} {\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) \sum_{i = a}^{b} {\tilde{P}}_{i}^{(1)} (z_{1}, 0),

70

R^{(2)} (z_{1}, 0) = π_{1} {\tilde{R}}_{2} (ƛ - ƛ X (z_{1})) \sum_{i = a}^{b} {\tilde{P}}_{i}^{(2)} (z_{1}, 0),

here:

p_{i}^{(1)} = \sum_{m = a}^{b} P_{m i}^{(1)} (0), p_{i}^{(2)} = \sum_{m = a}^{b} P_{m i}^{(2)} (0), q_{i} = \sum_{i = 1}^{\infty} Q_{1 i} (0),

R_{i}^{(2)} = R_{i}^{(2)} (0), R_{i}^{(2)} = R_{i}^{(2)} (0) .

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

71

{\tilde{Q}}_{1} (z_{1}, θ_{1}) = \frac{[\tilde{V} (ƛ - ƛ X (z_{1})) - \tilde{V} (θ_{1})] C (z_{1}, θ_{1})}{(θ_{1} - ƛ + ƛ X (z_{1}))},

72

{\tilde{Q}}_{j} (z_{1}, θ_{1}) = \frac{[\tilde{V} (ƛ - ƛ X (z_{1})) - \tilde{V} (θ)] \sum_{n = 0}^{a - 1} Q_{j - 1 n} (0) z_{1}^{n}}{(θ_{1} - ƛ + ƛ X (z_{1}))}, j \geq 2,

73

\tilde{C} (z_{1}, 0) = \frac{[\tilde{C} (ƛ - ƛ X (z_{1})) - \tilde{C} (θ_{1})] [\sum_{n = 0}^{a - 1} \sum_{m = a}^{b} P_{m n}^{(2)} (0) z_{1}^{n}]}{(θ_{1} - ƛ + ƛ X (z_{1}))},

74

{\tilde{P}}_{i}^{(1)} (z_{1}, 0) = \frac{[{\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - {\tilde{S}}_{1} (θ_{1})] [\sum_{m = a}^{b} P_{m i}^{(1)} (0) + R_{i}^{(1)} (0) + \sum_{i = 1}^{\infty} Q_{1 i} (0)]}{(θ_{1} - ƛ + ƛ X (z_{1}))},

a \leq i \leq b - 1,

75

{\tilde{P}}_{b}^{(1)} (z_{1}, 0) = \frac{[{\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - {\tilde{S}}_{1} (θ_{1})] f_{1} (z_{1})}{(θ_{1} - ƛ + ƛ X (z_{1})) ({z_{1}}^{b} - {\tilde{S}}_{1} (ƛ - ƛ X (z_{1})))},

76

{\tilde{P}}_{i}^{(2)} (z_{1}, 0) = \frac{[{\tilde{S}}_{2} (ƛ - ƛ X (z_{1})) - {\tilde{S}}_{2} (θ_{1})] [\sum_{m = a}^{b} P_{m i}^{(2)} (0) + R_{i}^{(2)} (0) + P_{0}^{(1)} (0)]}{(θ_{1} - ƛ + ƛ X (z_{1}))},

a \leq i \leq b - 1,

77

{\tilde{P}}_{b}^{(2)} (z_{1}, 0) = \frac{[{\tilde{S}}_{2} (ƛ - ƛ X (z_{1})) - {\tilde{S}}_{2} (θ_{1})] f_{2} (z_{1})}{(θ_{1} - ƛ + ƛ X (z_{1})) ({z_{1}}^{b} - {\tilde{S}}_{2} (ƛ - ƛ X (z_{1})))},

78

{\tilde{R}}_{1} (z_{1}, θ_{1}) = \frac{[{\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - {\tilde{R}}_{1} (θ_{1})] π_{1} \sum_{i = a}^{b} {\tilde{P}}_{i}^{(1)} (z_{1}, 0)}{(θ_{1} - ƛ + ƛ X (z_{1}))},

79

{\tilde{R}}_{2} (z_{1}, θ_{1}) = \frac{[{\tilde{R}}_{2} (ƛ - ƛ X (z_{1})) - {\tilde{R}}_{2} (θ_{1})] π_{2} \sum_{i = a}^{b} {\tilde{P}}_{i}^{(2)} (z_{1}, 0)}{(θ_{1} - ƛ + ƛ X (z_{1}))} .

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

C (z_{1}) = \frac{(\tilde{C} (ƛ - ƛ X (z_{1})) - 1) \sum_{i = 0}^{a - 1} P_{i}^{(2)} {z_{1}}^{i}}{(- ƛ + ƛ X (z_{1}))} .

3.1.2. Vacation completion epoch

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

81

V (z_{1}) = \frac{(\tilde{V} (ƛ - ƛ X (z_{1})) - 1) [\tilde{C} (ƛ - ƛ X (z_{1})) \sum_{i = 0}^{a - 1} P_{i}^{(1)} z^{i} + \sum_{i = 0}^{a - 1} q_{i} {z_{1}}^{i}]}{(- ƛ + ƛ X (z_{1}))} .

3.1.3. Service in phase I completion epoch

82

P_{1} (z_{1}) = \frac{\{\begin{matrix} ({\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - 1) \sum_{i = a}^{b - 1} (P_{i}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i}) + R_{i}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i}) + q_{i} ({z_{1}}^{b} - {z_{1}}^{i})) \\ + ({\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - 1) \tilde{V} (ƛ - ƛ X (z_{1})) \tilde{C} ((ƛ - ƛ X (z_{1})) - 1) \sum_{i = 0}^{a - 1} P_{i}^{(1)} {z_{1}}^{i} \\ + ({\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - 1) (\tilde{V} (ƛ - ƛ X (z_{1})) - 1) \sum_{i = 0}^{a - 1} q_{i} {z_{1}}^{i} \end{matrix}\}}{\{\begin{matrix} (ƛ - ƛ X (z_{1})) ({1 + π}_{1} {\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - {π_{1} \tilde{S}}_{1} (ƛ - ƛ X (z_{1})) {\tilde{R}}_{1} (ƛ - ƛ X (z_{1}))) \\ {\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) ({z_{1}}^{b} - {\tilde{S}}_{1} (ƛ - ƛ X (z_{1}))) \end{matrix}\}} .

3.1.4. Service in phase II completion epoch

83

P_{2} (z_{1}) = \frac{({\tilde{S}}_{2} (ƛ - ƛ X (z_{1})) - 1) \sum_{i = a}^{b - 1} (P_{i}^{(2)} ({z_{1}}^{b} - {z_{1}}^{i}) + R_{i}^{(2)} ({z_{1}}^{b} - {z_{1}}^{i}) + P_{0}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i}))}{\{\begin{matrix} (ƛ - ƛ X (z_{1})) ({1 + π}_{2} {\tilde{R}}_{2} (ƛ - ƛ X (z_{1})) - {π_{2} \tilde{S}}_{2} (ƛ - ƛ X (z_{1})) {\tilde{R}}_{2} (ƛ - ƛ X (z_{1}))) \\ ({z_{1}}^{b} - {\tilde{S}}_{2} (ƛ - ƛ X (z_{1}))) \end{matrix}\}} .

3.1.5. Renovation in phase I completion epoch

84

R_{1} (z_{1}) = \frac{\{\begin{matrix} \begin{matrix} π_{1} ({\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - 1) ({\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - 1) \\ ∙ \sum_{i = a}^{b - 1} (P_{i}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i}) + R_{i}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i}) + q_{i} ({z_{1}}^{b} - {z_{1}}^{i})) \\ + π_{1} ({\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - 1) ({\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - 1) \end{matrix} \\ ∙ \tilde{V} (ƛ - ƛ X (z_{1})) \tilde{C} ((ƛ - ƛ X (z_{1})) - 1) \sum_{i = 0}^{a - 1} P_{i}^{(1)} {z_{1}}^{i} \\ + π_{1} ({\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - 1) ({\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - 1) (\tilde{V} (ƛ - ƛ X (z_{1})) - 1) \sum_{i = 0}^{a - 1} q_{i} {z_{1}}^{i} \end{matrix}\}}{\{\begin{matrix} (ƛ - ƛ X (z_{1})) ({1 + π}_{1} {\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - {π_{1} \tilde{S}}_{1} (ƛ - ƛ X (z_{1})) {\tilde{R}}_{1} (ƛ - ƛ X (z_{1}))) \\ {\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) ({z_{1}}^{b} - {\tilde{S}}_{1} (ƛ - ƛ X (z_{1}))) \end{matrix}\}} .

3.1.6. Renovation in phase II completion epoch

85

R_{2} (z_{1}) = \frac{\{\begin{matrix} π_{2} ({\tilde{R}}_{2} (ƛ - ƛ X (z_{1})) - 1) ({\tilde{S}}_{2} (ƛ - ƛ X (z_{1})) - 1) \\ \sum_{i = a}^{b - 1} (P_{i}^{(2)} ({z_{1}}^{b} - {z_{1}}^{i}) + R_{i}^{(2)} ({z_{1}}^{b} - {z_{1}}^{i}) + P_{0}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i})) \end{matrix}\}}{\{\begin{matrix} (ƛ - ƛ X (z_{1})) ({1 + π}_{1} {\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - {π_{1} \tilde{S}}_{1} (ƛ - ƛ X (z_{1})) {\tilde{R}}_{1} (ƛ - ƛ X (z_{1}))) \\ {\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) ({z_{1}}^{b} - {\tilde{S}}_{1} (ƛ - ƛ X (z_{1}))) \end{matrix}\}} .

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

P (z_{1}) = \tilde{C} (z_{1}, 0) + \sum_{m = a}^{b - 1} {\tilde{P}}_{m}^{(1)} (z_{1}, 0) + {\tilde{P}}_{b}^{(1)} (z_{1}, 0) + \sum_{m = a}^{b - 1} {\tilde{P}}_{m}^{(2)} (z_{1}, 0) + {\tilde{P}}_{b}^{(2)} (z_{1}, 0)

{+ \tilde{R}}_{1} (z_{1}, 0) + {\tilde{R}}_{2} (z_{1}, 0) .

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

87

P (z_{1}) = \frac{\{\begin{matrix} ({\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - 1) [π_{1} ({\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - 1) + 1] D_{2} \sum_{i = a}^{b - 1} C_{i}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i}) \\ + ({\tilde{S}}_{2} (ƛ - ƛ X (z_{1})) - 1) [π_{2} ({\tilde{R}}_{2} (ƛ - ƛ X (z_{1})) - 1) + 1] D_{1} \sum_{i = a}^{b - 1} C_{i}^{(2)} ({z_{1}}^{b} - {z_{1}}^{i}) \\ + ({\tilde{S}}_{2} (ƛ - ƛ X (z_{1})) - 1) [π_{2} ({\tilde{R}}_{2} (ƛ - ƛ X (z_{1})) - 1) + 1] D_{1} \sum_{i = a}^{b - 1} P_{0}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i}) \\ + ({\tilde{S}}_{2} (ƛ - ƛ X (z_{1})) - 1) [π_{2} ({\tilde{R}}_{2} (ƛ - ƛ X (z_{1})) - 1) + 1] \tilde{V} (ƛ - ƛ X (z_{1})) \\ \tilde{C} ((ƛ - ƛ X (z_{1})) - 1) D_{1} D_{2} \sum_{i = 0}^{a - 1} P_{i}^{(1)} {z_{1}}^{i} + ({\tilde{S}}_{1} (ƛ - ƛ X (z_{1})) - 1) \\ [π_{1} ({\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - 1) + 1] \tilde{V} (ƛ - ƛ X (z_{1}) - 1) D_{2} + \tilde{V} (ƛ - ƛ X (z_{1}) - 1) D_{1} D_{2} \\ \sum_{i = 0}^{a - 1} q_{i} {z_{1}}^{i} + \tilde{C} ((ƛ - ƛ X (z_{1})) - 1) D_{1} D_{2} \sum_{i = 0}^{a - 1} P_{i}^{(2)} {z_{1}}^{i} \end{matrix}\}}{(- ƛ + ƛ X (z_{1})) D_{1} D_{2}},

where:

D_{1} = ({1 + π}_{1} {\tilde{R}}_{1} (ƛ - ƛ X (z_{1})) - {π_{1} \tilde{S}}_{1} (ƛ - ƛ X (z_{1})) {\tilde{R}}_{1} (ƛ - ƛ X (z_{1}))) ({z_{1}}^{b} - {\tilde{S}}_{1} (ƛ - ƛ X (z_{1}))),

D_{2} = ({1 + π}_{2} {\tilde{R}}_{2} (ƛ - ƛ X (z_{1})) - {π_{2} \tilde{S}}_{2} (ƛ - ƛ X (z_{1})) {\tilde{R}}_{2} (ƛ - ƛ X (z_{1}))) ({z_{1}}^{b} - {\tilde{S}}_{2} (ƛ - ƛ X (z_{1}))) .

3.3. Steady state condition

$P (1) =$ 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

E (S_{1}) (b - ƛ E (X) E (S_{2})) \sum_{i = a}^{b - 1} C_{i}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i}) + E (S_{2}) (b - ƛ E (X) E (S_{1})) \sum_{i = a}^{b - 1} C_{i}^{(1)} ({z_{1}}^{b} - {z_{1}}^{i}) + E (S_{2}) (b - ƛ E (X) E (S_{1})) \sum_{i = a}^{b - 1} P_{0} ({z_{1}}^{b} - {z_{1}}^{i}) + E (S_{1}) (b - ƛ E (X) E (S_{2})) + (b - ƛ E (X) E (S_{1})) (b - ƛ E (X) E (S_{2})) [ƛ E (X) E (V) + ƛ E (X) E (C)] \sum_{i = 0}^{a - 1} P_{i}^{(1)} + (b - ƛ E (X) E (S_{1})) + (b - ƛ E (X) E (S_{2})) ƛ E (X) E (C) \sum_{i = 0}^{a - 1} P_{i}^{(2)}

+ E (S_{1}) (b - ƛ E (X) E (S_{2})) + (b - ƛ E (X) E (S_{1})) (b - ƛ E (X) E (S_{2})) ƛ E (X) E (V) \sum_{i = 0}^{a - 1} q_{i}

= (b - ƛ E (X) E (S_{1})) (b - ƛ E (X) E (S_{2})) .

As a result $P (1) =$ 1 satisfied if:

({z_{1}}^{b} - {\tilde{S}}_{1} (ƛ - ƛ X (z_{1}))) ({z_{1}}^{b} - {\tilde{S}}_{2} (ƛ - ƛ X (z_{1}))) > 0, ρ = ƛ^{2} \frac{{(E (X))}^{2} E (S_{1}) E (S_{2})}{b} .

Theorem 1.

q_{n} = \sum_{i = 0}^{n} K_{i} P_{n - 1}^{(2)}, n = 0, 1, 2, 3, \dots . . a - 1,

where $K_{n} = \frac{h_{n} + \sum_{i = 0}^{n} a_{i} K_{n - i}}{1 - α_{0}}$ , $n = 0, 1, 2, 3, \dots . . a - 1$ with $K_{0} = \frac{α_{0} β_{0}}{1 - α_{0}}$ , $h_{n} = \sum_{i = 0}^{n} α_{0} β_{n - i}$ .

The possibilities that a customer will arrive during a holiday or a shut-down period are $α_{i}^{'} s$ and $β_{i}^{'} s$ respectively.

Proof.

Using the Eqs. (44)-(46), $\sum_{j = 1}^{\infty} Q_{j} (z_{1}, 0)$ simplifies to:

89

\sum_{n = 0}^{\infty} q_{n} {z_{1}}^{n} = \tilde{V} (ƛ - ƛ X (z_{1})) [\tilde{C} (ƛ - ƛ X (z_{1})) \sum_{n = 0}^{a - 1} {P_{n}^{(2)} z}_{1}^{n} \sum_{n = 0}^{a - 1} q_{n} {z_{1}}^{n}] = (\sum_{n = 0}^{\infty} α_{n} {z_{1}}^{n}) [\sum_{j = 0}^{\infty} β_{j} {z_{1}}^{j} \sum_{n = 0}^{a - 1} {P_{n}^{(2)} z}_{1}^{n} + \sum_{n = 0}^{a - 1} q_{n} {z_{1}}^{n}] \sum_{j = 0}^{n} [(\sum_{i = 0}^{n - j} {α_{i} β}_{n - i} {p_{j}}^{(2)} + \sum_{i = 0}^{n} α_{n - i} q_{i})] {z_{1}}^{n} .

Equating the coefficients of ${z_{1}}^{n}$ on both sides of the above equation for $n =$ 0, 1, 2, 3, …, $a - 1$ we have:

q_{n} = \sum_{j = 0}^{n} (\sum_{i = 0}^{n - j} α_{i} β_{n - i - j}) P_{j}^{(2)} + \sum_{i = 0}^{n - j} α_{n - i} q_{i},

q_{n} = \sum_{j = 0}^{n} (\sum_{i = 0}^{n - j} α_{i} β_{n - i - j}) P_{j}^{(2)} + \sum_{i = 0}^{n} α_{n - i} q_{i} .

On solving for $q_{n},$ we get:

q_{n} = \frac{\sum_{j = 0}^{n} (\sum_{i = 0}^{n - j} α_{i} β_{n - i - j}) P_{j}^{(2)} + \sum_{i = 0}^{n - 1} α_{n - i} q_{i}}{(1 - α_{0})} .

Coefficient of $P_{n}^{(2)}$ in $q_{n}$ is $\frac{α_{0} β_{0}}{1 - α_{0}} = K_{0}$ .

Coefficient of $P_{n - 1}^{(2)}$ in $q_{n - 1}$ is $(h_{1} + α_{1})$ .

Coefficient of $\frac{P_{n - 1}^{(2)}}{1 - α_{0}}$ in $q_{n - 1}$ is $\frac{h_{1} + α_{1} K_{0}}{1 - α_{0}} = K_{1}$ .

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:

E (B) = \frac{(E (T_{1}) + E (T_{2}))}{\sum_{i = 0}^{a - 1} d_{i}} + π_{1} E (R_{1}), E (T_{2}) = E (S_{2}) + π_{2} E (R_{2}), where E (T_{1}) = E (S_{1}) .

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, $E (I) = E (I_{1}) + E (C) :$

E (I_{1}) = \frac{E (V)}{1 - \sum_{n = 0}^{a - 1} Q_{1 n} (0)} = \frac{E (V)}{1 - \sum_{n = 0}^{a - 1} \sum_{i = 0}^{n} \{\sum_{j = 0}^{n - 1} α_{j} β_{n - i - j}\} P_{i}^{(1)}},

where $I_{1}$ is the random variable denoting the “Idle period due to multiple vacation process”, $E (C)$ is the expected closedown time.

By theorem the expected idle period $E (I)$ is obtained as $E (I) = E (I_{1}) + E (C) .$

3.6. Expected queue length at an arbitrary time epoch

90

E (Q) = \frac{\{\begin{matrix} ({(b - s_{21})}^{2} H_{11} + {(b - s_{21})}^{2} {(b - s_{11})}^{2} V_{11}) \sum_{i = 0}^{a - 1} P_{i}^{(1)} \\ + ({(b - s_{21})}^{2} H_{12} - {(b - s_{21})}^{2} {(b - s_{11})}^{2} V_{12}) \sum_{n = 0}^{a - 1} i P_{i}^{(1)} \\ + {(b - s_{21})}^{2} {(b - s_{11})}^{2} L_{1} \sum_{i = 0}^{a - 1} P_{i}^{(2)} + {(b - s_{21})}^{2} {(b - s_{11})}^{2} L_{2} \sum_{n = 0}^{a - 1} {i P}_{i}^{(2)} \\ + ({(b - s_{21})}^{2} H_{13} + {(b - s_{21})}^{2} {(b - s_{11})}^{2} V_{13}) \sum_{i = 0}^{a - 1} q_{i} \\ + ({(b - s_{21})}^{2} H_{14} - {(b - s_{21})}^{2} {(b - s_{11})}^{2} V_{14}) \sum_{n = 0}^{a - 1} i q_{i} \\ + {(b - s_{21})}^{2} H_{15} \sum_{i = a}^{b - 1} (b - i) C_{i}^{(1)} + {(b - s_{21})}^{2} H_{16} \sum_{i = a}^{b - 1} (b (b - 1) - i (i - 1)) C_{i}^{(1)} \\ {+ (b - s_{21})}^{2} H_{25} \sum_{i = a}^{b - 1} (b - i) (C_{i}^{(2)} + P_{0}) \\ + {(b - s_{11})}^{2} H_{26} \sum_{i = a}^{b - 1} (b (b - 1) - i (i - 1)) (C_{i}^{(2)} + P_{0}) \end{matrix}\}}{2 {(ƛ X_{1})}^{2} {(b - s_{11})}^{2} {(b - s_{21})}^{2}} .

H_{11} = 2 π_{1} T_{11} S_{11} R_{11} (V_{1} + C_{1}) + T_{11} S_{11} (V_{2} + C_{2} + 2 V_{1} C_{1}) - 2 T_{11} S_{12} (V_{1} + C_{1}),

H_{12} = T_{11} S_{11} (V_{1} + C_{1}), H_{13} = 2 π_{1} T_{11} S_{11} R_{11} V_{1} + T_{11} S_{11} V_{2} - 2 T_{11} S_{12} V_{1},

H_{14} = T_{11} S_{11} V_{1}, H_{15} = 2 π_{1} T_{11} S_{11} R_{11} - 2 T_{11} S_{11},

H_{16} = T_{11} S_{11}, T_{11} = (ƛ X_{1}) (b - s_{11}),

H_{25} = 2 π_{1} T_{21} S_{21} R_{21} - 2 T_{21} S_{22}, H_{26} = T_{21} S_{21}, T_{21} = (ƛ X_{1}) (b - s_{21}),

V_{11} = (V_{1} (ƛ X_{2}) - 2 V_{2} (ƛ X_{1})), V_{12} = 2 V_{1} (ƛ X_{1}), V_{13} = V_{1} (ƛ X_{2}), V_{14} = 2 V_{1} (ƛ X_{1}),

L_{1} = ((ƛ X_{1}) C_{2} - (ƛ X_{2}) C_{1}), L_{1} = 2 (ƛ X_{1}) C_{1} .

3.7. Expected waiting time

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

91

E (W) = \frac{E (Q)}{ƛ E (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 $M_{X} / G (a, b) / 1$ 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

DOI

https://doi.org/10.21595/mme.2023.23827

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.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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P. S and K. K, “A bulk queue’s analysis with two-stage heterogeneous services, multiple vacations, closedown with server breakdown, and two types of renovation,” Mathematical Models in Engineering, Vol. 10, No. 2, pp. 49–64, Jun. 2024, https://doi.org/10.21595/mme.2023.23827

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TY  - JOUR
DO  - 10.21595/mme.2023.23827
UR  - https://doi.org/10.21595/mme.2023.23827
TI  - A bulk queue’s analysis with two-stage heterogeneous services, multiple vacations, closedown with server breakdown, and two types of renovation
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 @article{S_2024, title={A bulk queue’s analysis with two-stage heterogeneous services, multiple vacations, closedown with server breakdown, and two types of renovation}, volume={10}, ISSN={2424-4627}, url={https://doi.org/10.21595/mme.2023.23827}, DOI={10.21595/mme.2023.23827}, number={2}, journal={Mathematical Models in Engineering}, publisher={JVE International Ltd.}, author={S, Palaniammal and K, Kumar}, year={2024}, month=jun, pages={49–64} }

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[1]P. S and K. K, “A bulk queue’s analysis with two-stage heterogeneous services, multiple vacations, closedown with server breakdown, and two types of renovation,” Mathematical Models in Engineering, vol. 10, no. 2, pp. 49–64, Jun. 2024, doi: 10.21595/mme.2023.23827.

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S, Palaniammal, and Kumar K. “A Bulk Queue’s Analysis with Two-Stage Heterogeneous Services, Multiple Vacations, Closedown with Server Breakdown, and Two Types of Renovation.” Mathematical Models in Engineering 10, no. 2 (June 30, 2024): 49–64. https://doi.org/10.21595/mme.2023.23827.

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

Abstract

Highlights

1. Introduction

2. Model description and system equations

2.1. Notations and assumptions

2.2. Steady state equations

2.3. Steady state equations

3. Queue size distribution

3.1. PGF of the queue size at different epochs

3.1.1. Close down completion epoch

3.1.2. Vacation completion epoch

3.1.3. Service in phase I completion epoch

3.1.4. Service in phase II completion epoch

3.1.5. Renovation in phase I completion epoch

3.1.6. Renovation in phase II completion epoch

3.2. PGF of queue size at an arbitrary time epoch

3.3. Steady state condition

3.4. Expected length of busy period

3.5. Expected length of the idle period

3.6. Expected queue length at an arbitrary time epoch

3.7. Expected waiting time

4. Conclusions

References

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