A Feedback Retrial Queueing System with Two Types of Batch Arrivals

Abstract

A retrial queueing system with two types of batch arrivals, called type I and type II customers, is considered. Type I customers and type II customers arrive in batches of variable sizes according to two different Poisson processes. Service time distributions are identical and independent and are different for both types of customers. If the arriving customers are blocked due to the server being busy, type I customers are queued in a priority queue of infinite capacity, whereas type II customers enter into a retrial group in order to seek service again after a random amount of time. A type I customer who has received service departs the system with a preassigned probability or returns to the priority queue for reservice with the complement probability. A type II call who has received service leaves the system with a preassigned probability or rejoins the retrial group with complement probability. For this model, the joint distribution of the number of customers in the priority queue and in the retrial group is obtained in a closed form. Some particular models and operating characteristics are obtained. A numerical study is also carried out.

1. Introduction

In the last three decades there has been significant contribution in the area of retrial queueing theory. For detailed survey one can see Yang and Templeton [1], Falin [2] and Choi and Chang [3]. Choi and Park [4] investigated an 𝑀/𝐺/1 retrial queue with two type of customers in which the service time distribution for both types of customers are the same. Khalil et al. [5] investigated the above model at Markovian level in detail. Falin et al. [6] investigated a similar model, in which they assumed different service time distributions for both types of customers. In 1995, Choi et al. [7], studied an 𝑀/𝐺/1 retrial queue with two types of customers and finite capacity. Atenica and Moreno [8] has analyzed a single server retrial queuing system with infinite buffer, Poisson arrivals, general distribution of service time, and linear retrial policy. If an arriving customer finds the server occupied, he joins a retrial group (called orbit) with probability 𝑝 and with complementary probability 𝑞 a priority queue in order to be served. After the customer is served completely, he will decide either to return to the priority queue for another service with probability 𝜃 or to leave the system forever with probability 𝜃=1−𝜃, where 0≤𝜃<1. They proved the ergodicity of the embedded Markov chain and obtained its stationary distribution function and the joint generating function of the number of customers in both groups in the steady-state regime. Kalyanaraman and Srinivasan [9] studied an 𝑀/𝐺/1 retrial queue with geometric loss and with type I batch arrivals and type II single arrivals. Artalejo and Atenica [10] analyzed a single server retrial queue with batch arrivals. Atenica and Moreno [11] considered an 𝑀/𝐺/1 retrial queue with general retrial time where the blocked customers either join the infinite waiting room (called priority queue) with probability 𝑞 or with complementary probability 𝑝 leave the service area and enter the retrial group (called orbit) in accordance with an FCFS discipline. They assume that only the customers at the head of the orbit are allowed to retry for service. They studied the ergodicity of the embedded Markov chain, its stationary distribution function, and the joint generating function of the number of customers in both groups in the steady-state regime. In 2005, Lee [12] considered an 𝑀/𝐺/1 retrial queueing system with two types of customers and feedback and derived the joint generating function of the number of customers in two groups by using the supplementary variable method. Falin [13] considered a single server batch arrival queue with returning customers. In 2011, Thillaigovindan and Kalyanaraman [14] analyzed a feedback retrial queueing system with two types of arrivals and the type I arrival being in batches of fixed size 𝐾.

In this paper, we deal with a feedback retrial queue with two types of customers, in which both types of customers arrive in batches of variable sizes. In Section 2, we describe the system with stability condition. In Section 3, we obtain the joint probability generating function for the number of customers in the priority queue and in the retrial group when server is busy as well as idle. The expressions for some particular models are deduced in Section 4. Some operating characteristics are derived in Section 5 and a numerical study is carried out in Section 6.

2. The Model

A retrial queueing system with two types of customers is considered in this paper. Type I customers arrive in batches of size 𝑘 with probability 𝑐_𝑘 and type II customers arrive in batches of size 𝑘 with probability 𝑑_𝑘 according to two independent Poisson processes with rates 𝜆₁𝑐=𝜆₁∑^∞_𝑘=1𝑘𝑐_𝑘 and 𝜆₂𝑑=𝜆₂∑^∞_𝑘=1𝑘𝑑_𝑘, respectively. If type II customers, upon arrival, find the server busy, they enter in to an orbit of infinite capacity in order to seek service again after random amount of time. All the customers in the retrial group behave independently of each other. The retrial time is exponentially distributed with mean 1/𝛼. Type I customers are queued in a priority queue of infinite capacity after blocking, if the server is busy. As soon as the server is free, the customers in the priority queue are served using FCFS rule and the customers in the retrial group are served only if there are no customers in the priority queue. A type I call who has received service departs the system with probability 1−𝑞₁ or returns to the priority queue for reservice with probability 𝑞₁. A type II call who has received service leaves the system with probability 1−𝑞₂ or rejoins the retrial group with probability 𝑞₂.

The service time distributions for both type of customers are identically and independently distributed random variables and have different distributions. A supplementary variable technique is used for the analysis and the variable considered being the residual service time of a customer in service. The service time density function is 𝑏_𝑖(𝑥); 𝑖=1,2 and 𝐵^∗_𝑘(𝑠)=∫^∞₀𝑒^−𝑠𝑥𝑏_𝑘(𝑥)𝑑𝑥, 𝑘=1,2 is the Laplace transformation of the distribution function 𝑏_𝑘(𝑥).

The Stochastic process related to the model is 𝑋(𝑡)={(𝜉(𝑡),𝑁_𝑝(𝑡),𝑁_𝑟(𝑡),𝑆_𝑘(𝑡))∶𝑡≥0} where 𝑁_𝑝(𝑡) = number of customers in the priority queue at time 𝑡, 𝑁_𝑟(𝑡) = number of customers in the retrial group at time 𝑡, and 𝜉(𝑡) = the server state at time 𝑡 where 𝜉(𝑡)=0,whentheserverisidle1,whentheserverisbusywithtypeIcustomer2,whentheserverisbusywithtypeIIcustomer, (2.1)𝑆_𝑘(𝑡) = the residual service time of a type 𝑘 customer in service at time 𝑡. 𝑋(𝑡) is a Markov process with state space {0,1,2}×{0,1,2,3,…}×(0,∞) and the corresponding stationary process is {(𝜉,𝑁_𝑝,𝑁_𝑟,𝑆_𝑘)}.

The related probabilities are 𝑞_𝑗(𝑡)=Pr{𝜉(𝑡)=0,𝑁_𝑟(𝑡)=𝑗}𝑝𝑘,𝑖,𝑗;𝑥,𝑡𝑑𝑥=Pr𝜉(𝑡)=𝑘,𝑁_𝑝(𝑡)=𝑖,𝑁_𝑟(𝑡)=𝑗,𝑆_𝑘(𝑡)𝜖(𝑥,𝑥+𝑑𝑥),𝑘=1,2. (2.2)

In steady state, the corresponding probabilities are 𝑞_𝑗=lim_𝑡→∞𝑞_𝑗(𝑡),𝑝𝑘,𝑖,𝑗;𝑥=lim_𝑡→∞𝑝𝑘,𝑖,𝑗;𝑥,𝑡, (2.3) and the Laplace transformation of 𝑝(𝑘,𝑖,𝑗;𝑥) is 𝑝^∗𝑘,𝑖,𝑗;𝑠=^∞₀𝑒^−𝑠𝑥𝑝𝑘,𝑖,𝑗;𝑥𝑑𝑥,𝑖=1,2,𝑗≥0. (2.4)

It is clear that 𝑝𝑘,𝑖,𝑗;0=^∞₀𝑝𝑘,𝑖,𝑗;𝑥𝑑𝑥=Pr𝜉=𝑘,𝑁_𝑝=𝑖,𝑁_𝑟=𝑗 (2.5) is the steady-state probability that there are 𝑖 customers in the priority queue, 𝑗 customers in the retrial group, and the server is busy with a 𝑘th-type customer.

For |𝑍₁|,|𝑍₂|≤1, the following probability generating functions are defined for the analysis: 𝑄𝑍₂=^∞_𝑗=0𝑞_𝑗𝑍^𝑗₂,𝐶𝑍₁=^∞_𝑗=1𝑐_𝑗𝑍^𝑗₁,𝐷𝑍₂=^∞_𝑗=1𝑑_𝑗𝑍^𝑗₂,𝑃^∗𝑘,𝑖,𝑠,𝑍₂=^∞_𝑗=0𝑝^∗𝑘,𝑖,𝑗,𝑠𝑍^𝑗₂𝑃;𝑖=0,1,2,…,𝑘=1,2,_∗𝑘,𝑠,𝑍₁,𝑍₂=^∞_𝑖=0𝑃^∗𝑘,𝑖,𝑠,𝑍₂𝑍^𝑖₁𝑃;𝑘=1,2,𝑘,𝑖,0,𝑍₂=^∞_𝑗=0𝑝^∗𝑘,𝑖,𝑗,0𝑍^𝑗₂𝑃;𝑖=0,1,2,…,𝑘=1,2,𝑘,0,𝑍₁,𝑍₂=^∞_𝑖=0𝑃𝑘,𝑖,0,𝑍₂𝑍^𝑖₁. (2.6)

3. The Analysis

Using the mean drift argument of Falin [15], it can be shown that the system is stable if 𝜌₁+𝜌₂<1 where 𝜌₁=−𝜆₁𝑐𝐵^∗′₁(0),𝜌₂=−𝜆₂𝑑𝐵^∗′₂(0).

Now the mathematical equations that govern the system are obtained by employing the remaining service time (vacation time) as the supplementary variable. Relating the state of the system at time 𝑡 and 𝑡+𝑑𝑡, the following partial differential-difference equations are obtained.

For 𝑗≥0,𝑥≥0,𝑖≥0𝑑𝜆+𝑗𝛼𝑞𝑑𝑡^𝑗(𝑡)=1−𝑞₁𝑝1,0,𝑗;0,𝑡+1−𝑞₂𝑝2,0,𝑗;0,𝑡+𝑞₂𝑝2,0,𝑗−1;0,𝑡,−𝜕𝑝1,0,𝑗;𝑥,𝑡+𝜕𝑥−𝜕𝑝1,0,𝑗;𝑥,𝑡𝜕𝑡=−𝜆𝑝1,0,𝑗;𝑥,𝑡+𝑞₁𝑏₁(𝑥)𝑝1,0,𝑗;0,𝑡+𝜆₁𝑏₁(𝑥)𝑞_𝑗(𝑡)+1−𝑞₁𝑏₁(𝑥)𝑝1,1,𝑗;0,𝑡+𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑝1,0,𝑗−𝑘;𝑥,𝑡,−𝜕𝑝1,𝑖,𝑗;𝑥,𝑡+𝜕𝑥−𝜕𝑝1,𝑖,𝑗;𝑥,𝑡𝜕𝑡=−𝜆𝑝1,𝑖,𝑗;𝑥,𝑡+𝑞₁𝑏₁(𝑥)𝑝+1,𝑖,𝑗;0,𝑡1−𝑞₁𝑏₁(𝑥)𝑝1,𝑖+1,𝑗;0,𝑡+𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑝1,𝑖−𝑘,𝑗;𝑥,𝑡+𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑝1,𝑖,𝑗−𝑘;𝑥,𝑡,−𝜕𝑝2,0,𝑗;𝑥,𝑡+𝜕𝑥−𝜕𝑝2,0,𝑗;𝑥,𝑡𝜕𝑡=−𝜆𝑝2,0,𝑗;𝑥,𝑡+𝜆₂𝑏₂(𝑥)^𝑗_𝑘=0𝑑_𝑘+1𝑞_𝑗−𝑘(𝑡)+𝑗+1𝛼𝑏₂(𝑥)𝑞_𝑗+1(𝑡)+𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑝2,0,𝑗−𝑘;𝑥,𝑡,−𝜕𝑝2,𝑖,𝑗;𝑥,𝑡+𝜕𝑥−𝜕𝑝2,𝑖,𝑗;𝑥,𝑡𝜕𝑡=−𝜆𝑝2,𝑖,𝑗;𝑥,𝑡+𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑝2,𝑖−𝑘,𝑗;𝑥,𝑡+𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑝2,𝑖,𝑗−𝑘;𝑥,𝑡. (3.1) In steady state, (3.1) becomes 𝜆+𝑗𝛼𝑞_𝑗=1−𝑞₁𝑝1,0,𝑗;0+1−𝑞₂𝑝2,0,𝑗,0+𝑞₂𝑝2,0,𝑗−1,0,−𝑝^′1,0,𝑗;𝑥=−𝜆𝑝1,0,𝑗;𝑥+𝑞₁𝑏₁(𝑥)𝑝1,0,𝑗;0+𝜆₁𝑏₁(𝑥)𝑞_𝑗+1−𝑞₁𝑏₁(𝑥)𝑝1,1,𝑗;0+𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑝1,0,𝑗−𝑘;𝑥,−𝑝^′1,𝑖,𝑗;𝑥=−𝜆𝑝1,𝑖,𝑗;𝑥+𝑞₁𝑏₁(𝑥)𝑝1,𝑖,𝑗;0+1−𝑞₁𝑏₁(𝑥)𝑝1,𝑖+1,𝑗;0+𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑝1,𝑖−𝑘,𝑗;𝑥+𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑝1,𝑖,𝑗−𝑘;𝑥,−𝑝^′2,0,𝑗;𝑥=−𝜆𝑝2,0,𝑗;𝑥+𝜆₂𝑏₂(𝑥)^𝑗_𝑘=0𝑑_𝑘+1𝑞_𝑗−𝑘+𝑗+1𝛼𝑏₂(𝑥)𝑞_𝑗+1+𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑝2,0,𝑗−𝑘;𝑥,−𝑝^′2,𝑖,𝑗;𝑥=−𝜆𝑝2,𝑖,𝑗;𝑥+𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑝2,𝑖−𝑘,𝑗;𝑥+𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑝2,𝑖,𝑗−𝑘;𝑥, (3.2)(3.3)(3.4)(3.5)(3.6) and the normalization condition is ^∞∞_𝑗=0𝑖=0^∞₀𝑝1,𝑖,𝑗;𝑥𝑑𝑥+^∞∞_𝑖=0𝑗=0^∞₀𝑝2,𝑖,𝑗;𝑥𝑑𝑥+^∞_𝑗=0𝑞_𝑗=1, (3.7) where 𝜆=𝜆₁+𝜆₂.

By taking the Laplace Stieltjes transformation of (3.3) to (3.6), the following equations are obtained: (𝑠−𝜆)𝑃^∗1,0,𝑗;𝑠=𝑃1,0,𝑗;0−𝑞₁𝑝1,0,𝑗;0𝐵^∗₁(𝑠)−𝜆₁𝑞_𝑗𝐵^∗₁(𝑠)−1−𝑞₁𝑝1,1,𝑗;0𝐵^∗₁(𝑠)−𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑃^∗,()𝑃1,0,𝑗−𝑘;𝑠𝑠−𝜆^∗1,𝑖,𝑗;𝑠=𝑃1,𝑖,𝑗;0−𝑞₁𝑝1,𝑖,𝑗;0𝐵^∗₁(𝑠)−1−𝑞₁𝑝1,𝑖+1,𝑗;0𝐵^∗₁(𝑠)−𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑃^∗1,𝑖−𝑘,𝑗;𝑠−𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑃^∗,()𝑃1,𝑖,𝑗−𝑘;𝑠𝑠−𝜆^∗2,0,𝑗;𝑠=𝑃2,0,𝑗;0−𝜆₂𝐵^∗₂(𝑠)^𝑗_𝑘=0𝑑_𝑘+1𝑞_𝑗−𝑘−𝑗+1𝛼𝑞_𝑗+1𝐵^∗₂(𝑠)−𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑃^∗,()𝑃2,0,𝑗−𝑘;𝑠𝑠−𝜆^∗2,𝑖,𝑗;𝑠=−𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑃^∗2,𝑖−𝑘,𝑗;𝑥+𝜆₂^𝑗_𝑘=1𝑑_𝑘𝑃^∗,𝑃2,𝑖,𝑗−𝑘;𝑠2,𝑖,𝑗;0=0. (3.8)

In (3.2) and (3.8), multiplying by 𝑍^𝑗₂ and then summing over 𝑗, the following equations are obtained: 𝜆𝑄𝑍₂+𝛼𝑍₂𝑄^′𝑍₂=1−𝑞₁𝑃1,0;0,𝑍₂+1−𝑞₂𝑃2,0;0,𝑍₂+𝑞₂𝑍₂𝑃2,0;0,𝑍₂,𝑠−𝜆+𝜆₂𝐷𝑍₂𝑃^∗1,0;𝑠,𝑍₂=𝑃1,0;0,𝑍₂−𝜆₁𝐵^∗₁(𝑠)𝑄𝑍₂−𝑞₁𝐵^∗₁(𝑠)𝑃1,0;0,𝑍₂−1−𝑞₁𝐵^∗₁(𝑠)𝑃1,1;0,𝑍₂,𝑠−𝜆+𝜆₂𝐷𝑍₂𝑃^∗1,𝑖;𝑠,𝑍₂=𝑃1,𝑖;0,𝑍₂−1−𝑞₁𝐵^∗₁(𝑠)𝑃1,𝑖+1;0,𝑍₂−𝑞₁𝐵^∗₁(𝑠)𝑃1,𝑖;0,𝑍₂−𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑃^∗1,𝑖−𝑘;𝑠,𝑍₂,𝑠−𝜆+𝜆₂𝐷𝑍₂𝑃^∗2,0;𝑠,𝑍₂=𝑃2,0;0,𝑍₂−𝜆₂𝐵^∗₂(𝑠)𝐷𝑍₂𝑍₂𝑄𝑍₂−𝛼𝐵^∗₂(𝑠)𝑄^′𝑍₂,𝑠−𝜆+𝜆₂𝐷𝑍₂𝑃^∗2,𝑖;𝑠,𝑍₂=−𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑃^∗2,𝑖−𝑘;𝑠,𝑍₂. (3.9)(3.10)(3.11)(3.12)(3.13)

Multiplying (3.11) and (3.13) by 𝑍^𝑖₁ and summing over 𝑖=1,2,… and using (3.10) and (3.12) leads to 𝑠−𝜆+𝜆₁𝐶𝑍₁+𝜆₂𝐷𝑍₂𝑃^∗1;𝑠,𝑍₁,𝑍₂=𝑃1;0,𝑍₁,𝑍₂−𝜆₁𝐵^∗₁(𝑠)𝑄𝑍₂−𝑞₁𝐵^∗₁(𝑠)𝑃1;0,𝑍₁,𝑍₂−1−𝑞₁𝐵^∗₁(𝑠)𝑍₁𝑃1;0,𝑍₁,𝑍₂+1−𝑞₁𝐵^∗₁(𝑠)𝑍₁𝑃1;0,0,𝑍₂,𝑠−𝜆+𝜆₁𝐶𝑍₁+𝜆₂𝐷𝑍₂𝑃^∗2;𝑠,𝑍₁,𝑍₂=𝑃2,0;0,𝑍₂−𝜆₂𝐵^∗₂(𝑠)𝐷𝑍₂𝑍₂𝑄𝑍₂−𝛼𝐵^∗₂(𝑠)𝑄^′𝑍₂. (3.14)(3.15)

By substituting 𝑠=𝜆−𝜆₁𝐶(𝑍₁)−𝜆₂𝐷(𝑍₂) in (3.14) and (3.15), we get 𝑃1,0;0,𝑍₂=𝜆₁𝑧₁1−𝑞₁𝑄𝑍₂−𝑍₁−𝐵^∗₁(𝑙)−𝑞₁𝑍₁−1𝐵^∗₁(𝑙)1−𝑞₁𝐵^∗₁(𝑙)𝑃1;0,𝑍₁,𝑍₂,𝑃2,0;0,𝑍₂=𝐵^∗₂(𝑙)𝜆₂𝐷𝑍₂𝑍₂𝑄𝑍₂+𝛼𝑄^′𝑍₂, (3.16)(3.17) where 𝑙=𝜆−𝜆₁𝐶(𝑍₁)−𝜆₂𝐷(𝑍₂).

Using (3.16) and (3.17) in (3.9) and on simplification, one can get the following equation: 𝛼𝑍₂−1−𝑞₂+𝑞₂𝑍₂𝐵^∗₂(𝑙)𝑄^′𝑍₂+𝜆−1−𝑞₁𝜆₁𝑍₁1−𝑞₁−1−𝑞₂+𝑞₂𝑍₂𝜆₂𝐵^∗₂(𝑙)𝐷𝑍₂𝑍₂𝑄𝑍₂=𝑞₁𝑍₁−1𝐵^∗₁(𝑙)−𝑍₁−𝐵^∗₁(𝑙)𝐵^∗₁(𝑙)𝑃1;0,𝑍₁,𝑍₂. (3.18)

Define 𝑓(𝑍₁,𝑍₂)=(𝑞₁(𝑍₁−1)𝐵^∗₁(𝑙)−(𝑍₁−𝐵^∗₁(𝑙)))/𝐵^∗₁(𝑙) for each fixed 𝑍₂, |𝑍₂|≤1. By Rouche’s theorem, there is a unique solution 𝑍₁=ℎ(𝑍₂) of the equation 𝑓(𝑍₁,𝑍₂)=0. Now (3.18) becomes 𝑄^′𝑍₂=1𝛼𝜆−𝜆₁ℎ𝑍₂−𝜆₂𝑈𝑍₂𝐷𝑍₂/𝑍₂𝑈𝑍₂−𝑍₂𝑄𝑍₂, (3.19) where ℎ(𝑍₂) is the root of the equation 𝑍₁=𝐵^∗₁(𝜆−𝜆₁𝐶(𝑍₁)−𝜆₂𝐷(𝑍₂)) and 𝑈(𝑍₂)=(1−𝑞₂+𝑞₂𝑍₂)𝐵^∗₂(𝑙).

Using (3.19) in (3.18), it can be seen that 𝑃1;0,𝑍₁,𝑍₂=𝐿𝑍₂−1−𝑞₂+𝑞₂𝑍₂𝐵^∗₂(𝑙)+𝑅𝑈𝑍₂−𝑍₂𝐵^∗₁(𝑙)𝑞₁𝑍₁−1𝐵^∗₁(𝑙)−𝑍₁−𝐵^∗₁(𝑙)𝑈𝑍₂−𝑍₂𝑄𝑍₂, (3.20) where 𝐿=𝜆−𝜆₁ℎ(𝑍₂)−𝜆₂𝑈(𝑍₂)(𝐷(𝑍₂)/𝑍₂), 𝑅=𝜆−𝜆₁𝑍₁−(1−𝑞₂+𝑞₂𝑍₂)𝜆₂𝐵^∗₂(𝑙)(𝐷(𝑍₂)/𝑍₂).

Using (3.20) in (3.16) leads to 𝑃1;0,0,𝑍₂=𝜆₁𝑍₁+𝑅𝑈𝑍₂−𝑍₂+𝐿𝑍₂−1−𝑞₂+𝑞₂𝑍₂𝐵^∗₂(𝑙)1−𝑞₁𝑈𝑍₂−𝑍₂𝑄𝑍₂. (3.21)

Using (3.19) in (3.17) leads to 𝑃2;0,0,𝑍₂=𝜆₂𝐷𝑍₂/𝑍₂𝑈𝑍₂−𝑍₂+𝐿𝐵^∗₂(𝑙)𝑈𝑍₂−𝑍₂𝑄𝑍₂. (3.22)

The general solution of the differential equation (3.19) is 𝑄𝑍₂=𝑄(1)exp−1𝛼¹_𝑍2𝜆−𝜆₁ℎ(𝑥)−𝜆₂𝑈(𝑥)(𝐷(𝑥)/𝑥)𝑈(𝑥),−𝑥𝑑𝑥 (3.23) where 𝑄(1) is a constant, which is the probability that the server is idle.

Putting 𝑠=0 in (3.10) and in (3.11), we get −𝜆+𝜆₂𝐷𝑍₂𝑃^∗1,0;0,𝑍₂=𝑃1,0;0,𝑍₂−𝜆₁𝑄𝑍₂−𝑞₁𝑃1,0;0,𝑍₂−1−𝑞₁𝑃1,1;0,𝑍₂,−𝜆+𝜆₂𝐷𝑍₂𝑃^∗1,𝑖;0,𝑍₂=𝑃1,𝑖;0,𝑍₂−1−𝑞₁𝑃1,𝑖+1;0,𝑍₂−𝑞₁𝑃1,𝑖;0,𝑍₂−𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑃^∗1,𝑖−𝑘;0,𝑍₂. (3.24)(3.25)

Summing (3.25) over 𝑖=1 to ∞ and then adding (3.24), we get 𝜆₂𝐷𝑍₂−1^∞_𝑖=0𝑃^∗1,𝑖;0,𝑍₂=1−𝑞₁𝑃1,0;0,𝑍₂−𝜆₁𝑄𝑍₂. (3.26)

Putting 𝑠=0 in (3.12) and in (3.13) results in −𝜆+𝜆₂𝐷𝑍₂𝑃^∗2,0;0,𝑍₂=𝑃2,0;0,𝑍₂−𝜆₂𝐷𝑍₂𝑍₂𝑄𝑍₂−𝛼𝑄^′𝑍₂,−𝜆+𝜆₂𝐷𝑍₂𝑃^∗2,𝑖;0,𝑍₂=−𝜆₁^𝑖_𝑘=1𝑐_𝑘𝑃^∗2,𝑖−𝑘;0,𝑍₂. (3.27)(3.28)

Summing (3.28) over 𝑖=1 to ∞ and then adding (3.27), we get 𝜆₂𝐷𝑍₂−1^∞_𝑖=0𝑃^∗2,𝑖;0,𝑍₂=𝑃2,0;0,𝑍₂−𝜆₂𝑄𝑍₂𝐷𝑍₂𝑍₂−𝛼𝑄^′𝑍₂. (3.29)

Adding (3.28) and (3.29) and using (3.9) leads to 𝜆₂𝐷𝑍₂−1^∞2_𝑖=0𝑘=1𝑃^∗𝑘,𝑖;0,𝑍₂=𝜆₂𝐷𝑍1−²𝑍₂−𝑞₂1−𝑍₂𝐵^∗₂(𝑙)𝐷𝑍₂𝑍₂𝑄𝑍₂+𝛼1−𝑍₂𝑞₂𝐵^∗₂(𝑙)−1𝑄^′𝑍₂. (3.30)

Evaluating at 𝑍₂=1 and using normalization condition we get 𝑄^′(1)=𝜆₂𝑑−𝜆₂1−𝑞₂𝑄(1)𝛼1−𝑞₂. (3.31)

Using (3.31) in (3.23) gives 𝑃_𝐼=𝑄(1)=𝑐1−𝜌₁1−𝑞₂−𝜌₂1−𝑞₂𝑐+𝜌₁−𝜌₁𝑐. (3.32)

In steady state, the probability generating function of number of customers in the orbit when the server is idle is obtained from (3.32) and (3.23).

Substituting 𝑠=0 in (3.14) 𝑃^∗1;0,𝑍₁,𝑍₂=1−𝑍₁1−𝑞₁𝑃1;0,𝑍₁,𝑍₂+𝜆₁𝑍₁𝑄𝑍₂−1−𝑞₁𝑃1,0,0;𝑍₂𝑍₁𝜆−𝜆₁𝐶𝑍₁−𝜆₂𝐷𝑍₂. (3.33)

Equation (3.33) together with (3.20) and (3.16) yields the joint probability generating function of the number of customers in the priority queue and in the orbit when the server is busy with type I customer as 𝑃^∗1;0,𝑍₁,𝑍₂=1−𝐵^∗₁(𝑙)𝐿𝑍₂−1−𝑞₂+𝑞₂𝑍₂𝐵^∗₂(𝑙)+𝑅𝑈𝑍₂−𝑍₂𝑙𝑞₁𝑍₁−1𝐵^∗₁(𝑙)−𝑍₁−𝐵^∗₁(𝑙)𝑈𝑍₂−𝑍₂𝑄𝑍₂. (3.34)

Putting 𝑠=0 in (3.15), −𝜆+𝜆₁𝐶𝑍₁+𝜆₂𝐷𝑍₂𝑃^∗2;0,𝑍₁,𝑍₂=𝑃2,0;0,𝑍₂−𝜆₂𝐷𝑍₂𝑍₂𝑄𝑍₂−𝛼𝑄^′𝑍₂. (3.35)

Substituting (3.19) in (3.35), we get 𝑃^∗2;0,𝑍₁,𝑍₂=1𝑙𝐿𝑈𝑍₂−𝑍₂+𝜆₂𝐷𝑍₂𝑍₂𝑄𝑍₂−𝑃2;0,0,𝑍₂. (3.36)

Equation (3.36) together with (3.19) and (3.17) yields the joint probability generating function of the number of customers in the priority queue and in the orbit when the server is busy with type II customer as 𝑃^∗2;0,𝑍₁,𝑍₂=1−𝐵^∗₂(𝑙)𝑙𝜆₂𝐷𝑍₂𝑍₂+𝐿𝑈𝑍₂−𝑍₂𝑄𝑍₂. (3.37)

Thus, we have the following theorem.

Theorem 3.1. The stationary distribution of {(𝜉,𝑁_𝑝,𝑁_𝑟,𝑆_𝑘)} has the following generating functions: 𝑄𝑍₂=𝑐1−𝜌₁1−𝑞₂−𝜌₂1−𝑞₂𝑐+𝜌₁−𝜌₁𝑐1exp𝛼^𝑍2₁𝜆−𝜆₁ℎ(𝑥)−𝜆₂𝑈(𝑥)(𝐷(𝑥)/𝑥)𝑈(𝑥),𝑃−𝑥𝑑𝑥_∗1;0,𝑍₁,𝑍₂=1−𝐵^∗₁(𝑙)𝐿𝑍₂−1−𝑞₂+𝑞₂𝑍₂𝐵^∗₂(𝑙)+𝑅𝑈𝑍₂−𝑍₂𝑙𝑞₁𝑍₁−1𝐵^∗₁(𝑙)−𝑍₁−𝐵^∗₁(𝑙)𝑈𝑍₂−𝑍₂𝑄𝑍₂,𝑃^∗2;0,𝑍₁,𝑍₂=1−𝐵^∗₂(𝑙)𝑙𝜆₂𝐷𝑍₂𝑍₂+𝐿𝑈𝑍₂−𝑍₂𝑄𝑍₂. (3.38)

Corollary 3.2. The probability that the server busy is 𝑃_𝐵=𝑃^∗(1;0,1,1)+𝑃^∗()=𝜌2;0,1,1¹1−𝑞₂+𝜌₂𝑐1−𝑞₂𝑐+𝜌₁−𝜌₁𝑐. (3.39)

4. Particular Models

By taking particular values to some parameters of the above model, the following models can be obtained.(i)When 𝑑_𝑘=0=𝑐_𝑘, 𝑘≠1, 𝑞₁=𝑞₂=0, and 𝐵₁(𝑥)=𝐵₂(𝑥)=𝐵(𝑥), the system coincides with that of Choi and Park [4].(ii)When 𝑑_𝑘=0=𝑐_𝑘, 𝑘≠1, and 𝑞₁=𝑞₂=0, the above results coincide with the results of Falin et al. [6].(iii)When 𝑑_𝑘=0, 𝑘≠1, the system coincides with that of Thillaigovindan and Kalyanaraman [14].

5. Operating Characteristics

The operating characteristics like the mean number of customers in the priority queue (𝑁_𝑝) and the mean number of customers in the orbit (𝑁_𝑟) have been calculated using the formulas 𝑁_𝑝=lim_𝑍1→1𝑃^∗′(1;0,𝑍₁,1)+lim_𝑍1→1𝑃^∗′(2;0,𝑍₁,1) and 𝑁_𝑟=lim_𝑍2→1𝑃^∗′(1;0,1,𝑍₂)+lim_𝑍2→1𝑃^∗′(2;0,1,𝑍₂)+𝑄^′(1). After putting 𝑍₂=1 in (3.34) and in (3.37), we get 𝑃^∗1;0,𝑍₁,1=1−𝐵^∗₁𝜆₁1−𝐶𝑍₁𝐴₁+𝐴₂𝑄(1)𝜆₁1−𝐶𝑍₁,𝑃^∗2;0,𝑍₁,1=1−𝐵^∗₂𝜆₁1−𝐶𝑍₁𝜆₁ℎ^′(1)+𝜆₂𝑑𝑄(1)𝜆₁1−𝐶𝑍₁1−𝑈^′(1,) (5.1)where 𝐴₁=([1−𝐵^∗₂[𝜆₁(1−𝐶(𝑍₁))]][−𝜆₁ℎ^′(1)−𝜆₂𝑈^′(1)−𝜆₂𝑑+𝜆₂])/((1−𝐶(𝑍₁))[𝑈^′(1)−1]), 𝐴₂=𝜆−𝜆₁𝑍₁−𝜆₂𝐵^∗₂[𝜆₁(1−𝐶(𝑍₁))], 𝐴₃=𝑞₁(𝑍₁−1)𝐵^∗₁[𝜆₁(1−𝐶(𝑍₁))]−(𝑍₁−𝐵^∗₁[𝜆₁(1−𝐶(𝑍₁))]), ℎ^′(1)=𝜆₂𝑑𝜌₁/𝜆₁𝑐(1−𝜌₁), 𝑈^′(1)=𝑞₂𝜌₂/(1−𝜌₁).

Differentiating (5.1) with respect to 𝑍₁ and then taking the limit as 𝑍₁→1, we get lim_𝑍1→1𝑃^∗′1;0,𝑍₁,1=1−𝜌₁1−𝑞₂−𝜌₂(1−𝑐)𝐴₄+𝜆₁𝜆₂𝛽₂𝜌₁𝑐𝑑21−𝜌₁−𝑞₁1−𝑞₂−𝑐₂𝜌₁1−𝜌₁1−𝑞₂−𝜌₂2𝑐1−𝑞₂1−𝜌₁−𝑞₁𝜌₁+𝑐1−𝜌₁, (5.2) where 𝐴₄=((1−𝑞₁)[𝜌₁𝑐₂+𝜆²₁𝑐³𝛽₁]+2𝜌²₁𝑞₁𝑐)/(2𝑐(1−𝑞₂)(1−𝜌₁−𝑞₁)²[𝜌₁+𝑐(1−𝜌₁)])lim_𝑍1→1𝑃^∗′2;0,𝑍₁=𝜆,1¹𝜆₂𝑐𝑑𝛽₂21−𝑞₂. (5.3)

After putting 𝑍₁=1 in (3.34) and in (3.37), we get 𝑃^∗1;0,1,𝑍₂=𝐿𝑍₂−1−𝑞₂+𝑞₂𝑍₂𝐵^∗₂𝜆₂1−𝐷𝑍₂+𝑅𝑈𝑍₂−𝑍₂−𝜆₂1−𝐷𝑍₂𝑈𝑍₂−𝑍₂𝑄(1),𝑃^∗2;0,1,𝑍₂=1−𝐵^∗₂𝜆₂1−𝐷𝑍₂𝐿𝑍₂+𝜆₂𝐷𝑍₂𝑈𝑍₂−𝑍₂𝜆₂1−𝐷𝑍₂𝑍₂𝑈𝑍₂−𝑍₂𝑄(1). (5.4)

Differentiating (5.4) with respect to 𝑍₂ and then taking limit as 𝑍₂→1, we get lim_𝑍2→1𝑃^∗′1;0,1,𝑍₂=𝐷₁𝜆₁𝑐²1−𝜌₁²𝐷₂−1−𝑞₂−𝜌₂𝐷₃2𝜆₁𝑐²𝑑²1−𝜌₁²𝑞₂−1𝜌₁+𝑐1−𝜌₁𝐷₀+1−𝑞₂−𝜌₂𝐷₄+𝜆₂𝑐𝐷₅−𝐷₆−𝐷₇2𝜆₁𝜆₂𝑐²𝑑²1−𝜌₁²1−𝑞₂𝜌₁+𝑐1−𝜌₁+𝐷₈𝑐1−𝜌₁1−𝑞₂−𝜌₂2𝑑²1−𝑞₂𝜌₁+𝑐1−𝜌₁+𝜌₁𝜆₂𝐷₁1−𝑞₂−𝜌₂−𝜌₂𝑐𝛼𝐷₀1−𝑞₂𝜌₁+𝑐1−𝜌₁,𝐷₀=1−𝑞₂1−𝜌₁−𝜌₂,𝐷₁=𝜌₁𝑑+𝜌₂𝑐+𝑐1−𝜌₁𝑑−1+𝑞₂,𝐷₂=𝐷₀2𝑞₂𝜌₂𝑑+𝜌₂𝑑₂+𝜆²₂𝑑³𝛽₂,𝐷₃=2𝜆₁𝑐²𝑑𝑞₂𝜌₂1−𝜌₁²+𝜆₁𝜆²₂𝑐²𝛽₂𝑑³1−𝜌₁+𝜆₂𝑐₂𝑑²𝜌²₁𝜌₂+𝜆²₁𝜆₂𝑐³𝑑²𝛽₁𝜌₂+𝜆₁𝑐²𝑑₂𝜌₂1−𝜌₁²,𝐷₄=𝜆²₁𝜆²₂𝑐³𝑑³𝛽₁+𝜆²₂𝑑³𝜌³₁𝑐₂+𝜆₁𝜆₂𝑑₂𝑑𝜌₁𝑐²1−𝜌₁²+2𝜆₁𝜆₂𝑐³𝑑𝑞₂𝜌₂1−𝜌₁²,𝐷₅=𝜆₁𝜆²₂𝑐²𝑑³𝛽₂1−𝜌₁+𝜆₂𝑐₂𝑑²𝜌²₁𝜌₂+𝜆²₁𝜆₂𝑐³𝑑²𝛽₁𝜌₂+𝜆₁𝑐²𝑑₂𝜌₂1−𝜌₁²,𝐷₆=2𝜆₁𝜆₂𝑐³𝑑𝑑−11−𝜌₁²1−𝜌₁−𝜌₂−𝑞₂1−𝜌₁,𝐷₇=𝜆₁𝜆₂𝑐²𝑑₂1−𝜌₁²𝜌₁𝑑+𝑐𝑞₂1−𝜌₁+𝜌₂−𝑐1−𝜌₁,𝐷₈=2𝑑1−𝑑𝑞₂+𝜌₂−1−𝜆²₂𝑑³𝛽₂−𝑑₂+𝑞₂−2𝜌₂𝑑+𝑑₂,lim_𝑍2→1𝑃^∗′2;0,1,𝑍₂=𝜆²₂𝑑²𝛽₂21−𝑞₂+𝜆₂𝜌₂𝐷₁𝛼𝑐1−𝑞₂𝐷₀+𝜌²₂𝑞₂1−𝑞₂1−𝜌₁1−𝑞₂−𝜌₂+𝜌₂1−𝜌₁1−𝑞₂−𝜌₂𝐷₉+𝑐+𝜌₁−𝜌₁𝑐𝐷₁₀2𝜆₁𝑑𝑐²1−𝜌₁²𝐷₀𝑐+𝜌₁−𝜌₁𝑐1−𝑞₂,𝐷₉=𝜆²₁𝜆₂𝑐³𝑑²𝛽₁+𝜆₂𝑑²𝜌³₁𝑐₂+𝜆₁𝑐²𝑑₂1−𝜌₁𝑐+𝜌₁−𝜌₁𝑐,𝐷₁₀=𝜆₁𝜆²₂𝑐²𝑑³𝛽₂1−𝜌₁+𝜆₂𝑑²𝜌²₁𝜌₂𝑐₂+𝜆²₁𝜆₂𝑐³𝑑²𝛽₁𝜌₂+𝜆₁𝑐²𝑑₂𝜌₂1−𝜌₁². (5.5)

From (3.31) and (3.32) 𝑄^′(1)=𝜆₂𝛼𝑑1−𝑞₂−𝑐1−𝜌₁1−𝑞₂−𝜌₂1−𝑞₂𝑐+𝜌₁−𝜌₁𝑐. (5.6)(i)Mean number of customers in the priority queue is 𝑁_𝑝=lim_𝑍1→1𝑃^∗′1;0,𝑍₁,1+lim_𝑍1→1𝑃^∗′2;0,𝑍₁,1. (5.7)(ii)Mean number of customers in the orbit is 𝑁_𝑟=lim_𝑍2→1𝑃^∗′1;0,1,𝑍₂+lim_𝑍2→1𝑃^∗′2;0,1,𝑍₂+𝑄^′(1). (5.8)(iii)Mean busy period.

Busy period 𝑇_𝑏 is the length of the time interval that keeps the server busy continuously and this continues till the instant server becomes free again and let 𝑇₀ be the length of the idle period. For this model, 𝑇_𝑏 and 𝑇₀ generates an alternating renewal process and therefore 𝐸𝑇_𝑏𝐸𝑇₀=𝑃𝑟𝑇_𝑏1−𝑃𝑟𝑇_𝑏=𝑃_𝐵1−𝑃_𝐵. (5.9)

But 𝐸(𝑇₀)=1/𝜆𝐸𝑇_𝑏=𝑃_𝐵𝜆1−𝑃_𝐵. (5.10)

Using (3.39) on (5.10), we get 𝐸𝑇_𝑏=𝜌₁1−𝑞₂+𝜌₂𝑐𝜆𝑐1−𝜌₁1−𝑞₂−𝜌₂. (5.11)

6. Numerical Study

In this section, some numerical examples related to the model analyzed in this paper are given. By varying type I arrival rate, type II arrival rate, and the retrial rate, the mean number of customers in the priority queue, the mean number of customers in the orbit, the mean busy period, the probability that the server is idle, and the probability that the server is busy are calculated. For the analysis, the parameters 𝑞₁, 𝑞₂, 𝑐, 𝑐^′′(1), 𝑑, 𝑑^′′(1), 𝛽₁, 𝛽₂, 𝐵^∗₁⁽¹⁾₍₀₎ and 𝐵^∗₂⁽¹⁾₍₀₎ are fixed. In Figures 1 and 2, the retrial rate is taken as 0.9 and type II arrival rate is taken as 0.8 and 0.9, respectively; by varying values of type I arrival rate the graphs of the mean number of customers in the priority queue and the mean number of customers in the orbit are drawn. In Figures 3 and 4, the retrial rate has been fixed as 0.9 but type I arrival rates are 0.8 and 0.9, respectively. The graphs of the mean number of customers in the priority queue and the mean number of customers in the orbit are drawn against varying values of type II arrival rate. In Figures 5 and 6, the same graphs with respect to varying retrial rate are drawn for fixed values of type I and type II arrival rates (𝜆₁=0.9, 𝜆₂=0.8 and 0.9). Figures 7, 8, and 9 show the graphs of mean busy period for varying values of type I arrival rate and type II arrival rate and retrial rate, respectively. From Figures 7 and 8, it is seen that the mean busy period is an decreasing function with respect to type I arrival rate and the mean busy period is an increasing function with respect to type II arrival rate. Whereas from Figure 9, the mean busy period is a constant function with respect to retrial rate. In Tables 1 and 2, the probability that the server is idle (𝑃_𝐼) and the probability that the server is busy (𝑃_𝐵) are presented. From the graphs, it is clear that as type I arrival rate (type II arrival rate) increases the mean number of customers in the priority queue and the mean number of customers in the orbit also increase, whereas as the retrial rate increases the mean number of customers in the orbit decreases and the mean number of customers in the priority queue is a constant function.

Table 1 

The probabilities 𝑃_𝐼 and 𝑃_𝐵.

Table 2 

The probabilities 𝑃_𝐼 and 𝑃_𝐵.

Figure 1 

Type I arrival rate versus mean number of customers.

Figure 2 

Type I arrival rate versus mean number of customers.

Figure 3 

Type II arrival rate versus mean number of customers.

Figure 4 

Type II arrival rate versus mean number of customers.

Figure 5 

Retrial rate versus mean number of customers.

Figure 6 

Retrial rate versus mean number of customers.

Figure 7 

Type I arrival rate versus mean busy period.

Figure 8 

Type I arrival rate versus mean busy period.

Figure 9 

Retrial rate versus mean busy period.

7. Conclusion

In the foregoing analysis, an 𝑀/𝐺/1 queue with retrial queueing system with two types of batch arrivals is considered to obtain queue length distribution and mean queue length. Extensive numerical work has been carried out to observe the trends of the operating characters of the system.

Acknowledgments

The author is thankful to the referees and the editor for their valuable comments and suggestions which have helped in improving the quality of this paper.