# A Feedback Retrial Queueing System with Two Types of Batch Arrivals

Authors: R. Kalyanaraman
Journal: International Journal of Stochastic Analysis
 (2012)

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## 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.


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## Body

## 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 Section2, 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 sizek with probability ck and type II customers arrive in batches of size k with probability dk according to two independent Poisson processes with rates λ1c-=λ1∑k=1∞kck and λ2d-=λ2∑k=1∞kdk, 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 isbi(x); i=1,2 and Bk*(s)=∫0∞e-sxbk(x)dx, k=1,2 is the Laplace transformation of the distribution function bk(x).The Stochastic process related to the model isX(t)={(ξ(t),Np(t),Nr(t),Sk(t)):t≥0} where Np(t) = number of customers in the priority queue at time t, Nr(t) = number of customers in the retrial group at time t, and ξ(t) = the server state at time t where
(2.1)ξ(t)={0,whentheserverisidle1,whentheserverisbusywithtypeIcustomer2,whentheserverisbusywithtypeIIcustomer,Sk(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,Np,Nr,Sk)}.The related probabilities areqj(t)=Pr{ξ(t)=0,Nr(t)=j}(2.2)p(k,i,j;x,t)dx=Pr{ξ(t)=k,Np(t)=i,Nr(t)=j,Sk(t)ϵ(x,x+dx)},k=1,2.In steady state, the corresponding probabilities are(2.3)qj=limt→∞qj(t),p(k,i,j;x)=limt→∞p(k,i,j;x,t),
and the Laplace transformation of p(k,i,j;x) is
(2.4)p*(k,i,j;s)=∫0∞‍e-sxp(k,i,j;x)dx,i=1,2,j≥0.It is clear that(2.5)p(k,i,j;0)=∫0∞‍p(k,i,j;x)dx=Pr{ξ=k,Np=i,Nr=j}
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:
(2.6)Q(Z2)=∑j=0∞‍qjZ2j,C(Z1)=∑j=1∞‍cjZ1j,D(Z2)=∑j=1∞‍djZ2j,P*(k,i,s,Z2)=∑j=0∞‍p*(k,i,j,s)Z2j;i=0,1,2,…,k=1,2,P*(k,s,Z1,Z2)=∑i=0∞‍P*(k,i,s,Z2)Z1i;k=1,2,P(k,i,0,Z2)=∑j=0∞‍p*(k,i,j,0)Z2j;i=0,1,2,…,k=1,2,P(k,0,Z1,Z2)=∑i=0∞‍P(k,i,0,Z2)Z1i.


## 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=-λ1c-B1*'(0),ρ2=-λ2d-B2*'(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 timet and t+dt, the following partial differential-difference  equations are obtained.Forj≥0,x≥0,i≥0(3.1)(λ+jα)ddtqj(t)=(1-q1)p(1,0,j;0,t)+(1-q2)p(2,0,j;0,t)+q2p(2,0,j-1;0,t),-∂p(1,0,j;x,t)∂x+-∂p(1,0,j;x,t)∂t=-λp(1,0,j;x,t)+q1b1(x)p(1,0,j;0,t)+λ1b1(x)qj(t)+(1-q1)b1(x)p(1,1,j;0,t)+λ2∑k=1j‍dkp(1,0,j-k;x,t),-∂p(1,i,j;x,t)∂x+-∂p(1,i,j;x,t)∂t=-λp(1,i,j;x,t)+q1b1(x)p(1,i,j;0,t)+(1-q1)b1(x)p(1,i+1,j;0,t)+λ1∑k=1i‍ckp(1,i-k,j;x,t)+λ2∑k=1j‍dkp(1,i,j-k;x,t),-∂p(2,0,j;x,t)∂x+-∂p(2,0,j;x,t)∂t=-λp(2,0,j;x,t)+λ2b2(x)∑k=0j‍dk+1qj-k(t)+(j+1)αb2(x)qj+1(t)+λ2∑k=1j‍dkp(2,0,j-k;x,t),-∂p(2,i,j;x,t)∂x+-∂p(2,i,j;x,t)∂t=-λp(2,i,j;x,t)+λ1∑k=1i‍ckp(2,i-k,j;x,t)+λ2∑k=1j‍dkp(2,i,j-k;x,t).
In steady state, (3.1) becomes
(3.2)(λ+jα)qj=(1-q1)p(1,0,j;0)+(1-q2)p(2,0,j,0)+q2p(2,0,j-1,0),(3.3)-p′(1,0,j;x)=-λp(1,0,j;x)+q1b1(x)p(1,0,j;0)+λ1b1(x)qj+(1-q1)b1(x)p(1,1,j;0)+λ2∑k=1j‍dkp(1,0,j-k;x),(3.4)-p′(1,i,j;x)=-λp(1,i,j;x)+q1b1(x)p(1,i,j;0)+(1-q1)b1(x)p(1,i+1,j;0)+λ1∑k=1i‍ckp(1,i-k,j;x)+λ2∑k=1j‍dkp(1,i,j-k;x),(3.5)-p′(2,0,j;x)=-λp(2,0,j;x)+λ2b2(x)∑k=0j‍dk+1qj-k+(j+1)αb2(x)qj+1+λ2∑k=1j‍dkp(2,0,j-k;x),(3.6)-p′(2,i,j;x)=-λp(2,i,j;x)+λ1∑k=1i‍ckp(2,i-k,j;x)+λ2∑k=1j‍dkp(2,i,j-k;x),
and the normalization condition is
(3.7)∑j=0∞‍∑i=0∞‍∫0∞‍p(1,i,j;x)dx+∑i=0∞‍∑j=0∞‍∫0∞‍p(2,i,j;x)dx+∑j=0∞‍qj=1,
where λ=λ1+λ2.By taking the Laplace Stieltjes transformation of (3.3) to (3.6), the following equations are obtained:
(3.8)(s-λ)P*(1,0,j;s)=P(1,0,j;0)-q1p(1,0,j;0)B1*(s)-λ1qjB1*(s)-(1-q1)p(1,1,j;0)B1*(s)-λ2∑k=1j‍dkP*(1,0,j-k;s),(s-λ)P*(1,i,j;s)=P(1,i,j;0)-q1p(1,i,j;0)B1*(s)-(1-q1)p(1,i+1,j;0)B1*(s)-λ1∑k=1i‍ckP*(1,i-k,j;s)-λ2∑k=1j‍dkP*(1,i,j-k;s),(s-λ)P*(2,0,j;s)=P(2,0,j;0)-λ2B2*(s)∑k=0j‍dk+1qj-k-(j+1)αqj+1B2*(s)-λ2∑k=1j‍dkP*(2,0,j-k;s),(s-λ)P*(2,i,j;s)=-λ1∑k=1i‍ckP*(2,i-k,j;x)+λ2∑k=1j‍dkP*(2,i,j-k;s),P(2,i,j;0)=0.In (3.2) and (3.8), multiplying by Z2j and then summing over j, the following equations are obtained:
(3.9)λQ(Z2)+αZ2Q′(Z2)=(1-q1)P(1,0;0,Z2)+(1-q2)P(2,0;0,Z2)+q2Z2P(2,0;0,Z2),(3.10)(s-λ+λ2D(Z2))P*(1,0;s,Z2)=P(1,0;0,Z2)-λ1B1*(s)Q(Z2)-q1B1*(s)P(1,0;0,Z2)-(1-q1)B1*(s)P(1,1;0,Z2),(3.11)(s-λ+λ2D(Z2))P*(1,i;s,Z2)=P(1,i;0,Z2)-(1-q1)B1*(s)P(1,i+1;0,Z2)-q1B1*(s)P(1,i;0,Z2)-λ1∑k=1i‍ckP*(1,i-k;s,Z2),(3.12)(s-λ+λ2D(Z2))P*(2,0;s,Z2)=P(2,0;0,Z2)-λ2B2*(s)D(Z2)Z2Q(Z2)-αB2*(s)Q′(Z2),(3.13)(s-λ+λ2D(Z2))P*(2,i;s,Z2)=-λ1∑k=1i‍ckP*(2,i-k;s,Z2).Multiplying (3.11) and (3.13) by Z1i and summing over i=1,2,… and using (3.10) and (3.12) leads to
(3.14)(s-λ+λ1C(Z1)+λ2D(Z2))P*(1;s,Z1,Z2)=P(1;0,Z1,Z2)-λ1B1*(s)Q(Z2)-q1B1*(s)P(1;0,Z1,Z2)-(1-q1)B1*(s)Z1P(1;0,Z1,Z2)+(1-q1)B1*(s)Z1P(1;0,0,Z2),(3.15)(s-λ+λ1C(Z1)+λ2D(Z2))P*(2;s,Z1,Z2)=P(2,0;0,Z2)-λ2B2*(s)D(Z2)Z2Q(Z2)-αB2*(s)Q′(Z2).By substitutings=λ-λ1C(Z1)-λ2D(Z2) in (3.14) and (3.15), we get
(3.16)P(1,0;0,Z2)=λ1z11-q1Q(Z2)-(Z1-B1*(l))-q1(Z1-1)B1*(l)(1-q1)B1*(l)P(1;0,Z1,Z2),(3.17)P(2,0;0,Z2)=B2*(l)[λ2D(Z2)Z2Q(Z2)+αQ′(Z2)],
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:
(3.18)α[Z2-(1-q2+q2Z2)B2*(l)]Q′(Z2)+[λ-(1-q1)λ1Z11-q1-(1-q2+q2Z2)λ2B2*(l)D(Z2)Z2]Q(Z2)=q1(Z1-1)B1*(l)-(Z1-B1*(l))B1*(l)P(1;0,Z1,Z2).Definef(Z1,Z2)=(q1(Z1-1)B1*(l)-(Z1-B1*(l)))/B1*(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
(3.19)Q′(Z2)=1αλ-λ1h(Z2)-λ2U(Z2)(D(Z2)/Z2)U(Z2)-Z2Q(Z2),
where h(Z2) is the root of the equation Z1=B1*(λ-λ1C(Z1)-λ2D(Z2)) and U(Z2)=(1-q2+q2Z2)B2*(l).Using (3.19) in (3.18), it can be seen that
(3.20)P(1;0,Z1,Z2)={L[Z2-(1-q2+q2Z2)B2*(l)]+R[U(Z2)-Z2]}B1*(l)[q1(Z1-1)B1*(l)-(Z1-B1*(l))][U(Z2)-Z2]Q(Z2),
where L=λ-λ1h(Z2)-λ2U(Z2)(D(Z2)/Z2), R=λ-λ1Z1-(1-q2+q2Z2)λ2B2*(l)(D(Z2)/Z2).Using (3.20) in (3.16) leads to
(3.21)P(1;0,0,Z2)=[λ1Z1+R][U(Z2)-Z2]+L[Z2-(1-q2+q2Z2)B2*(l)](1-q1)[U(Z2)-Z2]Q(Z2).Using (3.19) in (3.17) leads to
(3.22)P(2;0,0,Z2)={λ2(D(Z2)/Z2)[U(Z2)-Z2]+L}B2*(l)[U(Z2)-Z2]Q(Z2).The general solution of the differential equation (3.19) is
(3.23)Q(Z2)=Q(1)exp{-1α∫Z21‍λ-λ1h(x)-λ2U(x)(D(x)/x)U(x)-xdx},
where Q(1) is a constant, which is the probability that the server is idle.Puttings=0 in (3.10) and in (3.11), we get
(3.24)(-λ+λ2D(Z2))P*(1,0;0,Z2)=P(1,0;0,Z2)-λ1Q(Z2)-q1P(1,0;0,Z2)-(1-q1)P(1,1;0,Z2),(3.25)(-λ+λ2D(Z2))P*(1,i;0,Z2)=P(1,i;0,Z2)-(1-q1)P(1,i+1;0,Z2)-q1P(1,i;0,Z2)-λ1∑k=1i‍ckP*(1,i-k;0,Z2).Summing (3.25) over i=1 to ∞ and then adding (3.24), we get
(3.26)λ2(D(Z2)-1)∑i=0∞‍P*(1,i;0,Z2)=(1-q1)P(1,0;0,Z2)-λ1Q(Z2).Puttings=0 in (3.12) and in (3.13) results in
(3.27)(-λ+λ2D(Z2))P*(2,0;0,Z2)=P(2,0;0,Z2)-λ2D(Z2)Z2Q(Z2)-αQ′(Z2),(3.28)(-λ+λ2D(Z2))P*(2,i;0,Z2)=-λ1∑k=1i‍ckP*(2,i-k;0,Z2).Summing (3.28) over i=1 to ∞ and then adding (3.27), we get
(3.29)λ2(D(Z2)-1)∑i=0∞‍P*(2,i;0,Z2)=P(2,0;0,Z2)-λ2Q(Z2)D(Z2)Z2-αQ′(Z2).Adding (3.28) and (3.29) and using (3.9) leads to
(3.30)λ2(D(Z2)-1)∑i=0∞‍∑k=12‍P*(k,i;0,Z2)=λ2(1-D(Z2)Z2-q2(1-Z2)B2*(l)D(Z2)Z2)Q(Z2)+α(1-Z2)[q2B2*(l)-1]Q′(Z2).Evaluating atZ2=1 and using normalization condition we get
(3.31)Q′(1)=λ2d--λ2(1-q2)Q(1)α(1-q2).Using (3.31) in (3.23) gives
(3.32)PI=Q(1)=c-[(1-ρ1)(1-q2)-ρ2](1-q2)(c-+ρ1-ρ1c-).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).Substitutings=0 in (3.14)
(3.33)P*(1;0,Z1,Z2)=(1-Z1)(1-q1)P(1;0,Z1,Z2)+λ1Z1Q(Z2)-(1-q1)P(1,0,0;Z2)Z1[λ-λ1C(Z1)-λ2D(Z2)].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
(3.34)P*(1;0,Z1,Z2)=[1-B1*(l)]{L[Z2-(1-q2+q2Z2)B2*(l)]+R[U(Z2)-Z2]}l[q1(Z1-1)B1*(l)-(Z1-B1*(l))][U(Z2)-Z2]Q(Z2).Puttings=0 in (3.15),
(3.35)(-λ+λ1C(Z1)+λ2D(Z2))P*(2;0,Z1,Z2)=P(2,0;0,Z2)-λ2D(Z2)Z2Q(Z2)-αQ′(Z2).Substituting (3.19) in (3.35), we get
(3.36)P*(2;0,Z1,Z2)=1l{[L[U(Z2)-Z2]+λ2D(Z2)Z2]Q(Z2)-P(2;0,0,Z2)}.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
(3.37)P*(2;0,Z1,Z2)=[1-B2*(l)]l{λ2D(Z2)Z2+LU(Z2)-Z2}Q(Z2).Thus, we have the following theorem.Theorem 3.1.
The stationary distribution of{(ξ,Np,Nr,Sk)} has the following generating functions:
(3.38)Q(Z2)=c-[(1-ρ1)(1-q2)-ρ2](1-q2)(c-+ρ1-ρ1c-)exp{1α∫1Z2‍λ-λ1h(x)-λ2U(x)(D(x)/x)U(x)-xdx},P*(1;0,Z1,Z2)=[1-B1*(l)]{L[Z2-(1-q2+q2Z2)B2*(l)]+R[U(Z2)-Z2]}l[q1(Z1-1)B1*(l)-(Z1-B1*(l))][U(Z2)-Z2]Q(Z2),P*(2;0,Z1,Z2)=[1-B2*(l)]l{λ2D(Z2)Z2+LU(Z2)-Z2}Q(Z2).Corollary 3.2.
The probability that the server busy is(3.39)PB=P*(1;0,1,1)+P*(2;0,1,1)=[ρ1(1-q2)+ρ2c-](1-q2)(c-+ρ1-ρ1c-).


## 4. Particular Models

By taking particular values to some parameters of the above model, the following models can be obtained.(i)
Whendk=0=ck, k≠1, q1=q2=0, and B1(x)=B2(x)=B(x), the system coincides with that of Choi and Park [4].(ii)
Whendk=0=ck, k≠1, and q1=q2=0, the above results coincide with the results of Falin et al. [6].(iii)
Whendk=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(Np) and the mean number of customers in the orbit (Nr) have been calculated using the formulas Np=limZ1→1P*′(1;0,Z1,1)+limZ1→1P*′(2;0,Z1,1) and Nr=limZ2→1P*′(1;0,1,Z2)+limZ2→1P*′(2;0,1,Z2)+Q′(1). After putting Z2=1 in (3.34) and in (3.37), we get
(5.1)P*(1;0,Z1,1)={1-B1*[λ1(1-C(Z1))]}(A1+A2)Q(1)[λ1(1-C(Z1))],P*(2;0,Z1,1)={1-B2*[λ1(1-C(Z1))]}[λ1h′(1)+λ2d-]Q(1)[λ1(1-C(Z1))][1-U′(1)],where A1=([1-B2*[λ1(1-C(Z1))]][-λ1h′(1)-λ2U′(1)-λ2d-+λ2])/((1-C(Z1))[U′(1)-1]), A2=λ-λ1Z1-λ2B2*[λ1(1-C(Z1))], A3=q1(Z1-1)B1*[λ1(1-C(Z1))]-(Z1-B1*[λ1(1-C(Z1))]), h′(1)=λ2d-ρ1/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
(5.2)limZ1→1P*′(1;0,Z1,1)=(1-ρ1)[(1-q2)-ρ2(1-c-)]A4+λ1λ2β2ρ1c-d-2(1-ρ1-q1)(1-q2)-c2ρ1[(1-ρ1)(1-q2)-ρ2]2c-(1-q2)(1-ρ1-q1)[ρ1+c-(1-ρ1)],
where A4=((1-q1)[ρ1c2+λ12c-3β1]+2ρ12q1c-)/(2c-(1-q2)(1-ρ1-q1)2[ρ1+c-(1-ρ1)])(5.3)limZ1→1P*′(2;0,Z1,1)=λ1λ2c-d-β22(1-q2).After puttingZ1=1 in (3.34) and in (3.37), we get
(5.4)P*(1;0,1,Z2)={L[Z2-(1-q2+q2Z2)B2*[λ2[1-D(Z2)]]]+R[U(Z2)-Z2]}-λ2[1-D(Z2)][U(Z2)-Z2]Q(1),P*(2;0,1,Z2)=[1-B2*[λ2[1-D(Z2)]]]{LZ2+λ2D(Z2)[U(Z2)-Z2]}λ2[1-D(Z2)]Z2[U(Z2)-Z2]Q(1).Differentiating (5.4) with respect to Z2 and then taking limit as Z2→1, we get
(5.5)limZ2→1P*′(1;0,1,Z2)=D1[λ1c-2(1-ρ1)2D2-(1-q2-ρ2)D3]2λ1c-2d-2(1-ρ1)2(q2-1)[ρ1+c-(1-ρ1)]D0+(1-q2-ρ2)(D4+λ2c-D5-D6-D7)2λ1λ2c-2d-2(1-ρ1)2(1-q2)[ρ1+c-(1-ρ1)]+D8c-[(1-ρ1)(1-q2)-ρ2]2d-2(1-q2)[ρ1+c-(1-ρ1)]+ρ1λ2D1(1-q2-ρ2-ρ2c-)αD0(1-q2)[ρ1+c-(1-ρ1)],D0=[(1-q2)(1-ρ1)-ρ2],D1=ρ1d-+ρ2c-+c-(1-ρ1)(d--1+q2),D2=D0[2q2ρ2d-+ρ2d2+λ22d-3β2],D3=2λ1c-2d-q2ρ2(1-ρ1)2+λ1λ22c-2β2d-3(1-ρ1)+λ2c2d-2ρ12ρ2+λ12λ2c-3d-2β1ρ2+λ1c-2d2ρ2(1-ρ1)2,D4=λ12λ22c-3d-3β1+λ22d-3ρ13c2+λ1λ2d2d-ρ1c-2(1-ρ1)2+2λ1λ2c-3d-q2ρ2(1-ρ1)2,D5=λ1λ22c-2d-3β2(1-ρ1)+λ2c2d-2ρ12ρ2+λ12λ2c-3d-2β1ρ2+λ1c-2d2ρ2(1-ρ1)2,D6=2λ1λ2c-3d-(d--1)(1-ρ1)2[1-ρ1-ρ2-q2(1-ρ1)],D7=λ1λ2c-2d2(1-ρ1)2[ρ1d-+c-[q2(1-ρ1)+ρ2]-c-(1-ρ1)],D8=2d-(1-d-)(q2+ρ2-1)-λ22d-3β2-d2+q2[-2ρ2d-+d2],limZ2→1P*′(2;0,1,Z2)=λ22d-2β22(1-q2)+λ2ρ2D1αc-(1-q2)D0+ρ22q2(1-q2)[(1-ρ1)(1-q2)-ρ2]+ρ2[((1-ρ1)(1-q2)-ρ2)D9+(c-+ρ1-ρ1c-)D10]2λ1d-c-2(1-ρ1)2D0(c-+ρ1-ρ1c-)(1-q2),D9=λ12λ2c-3d-2β1+λ2d-2ρ13c2+λ1c-2d2(1-ρ1)(c-+ρ1-ρ1c-),D10=λ1λ22c-2d-3β2(1-ρ1)+λ2d-2ρ12ρ2c2+λ12λ2c-3d-2β1ρ2+λ1c-2d2ρ2(1-ρ1)2.From (3.31) and (3.32)
(5.6)Q′(1)=λ2α[d-1-q2-c-[(1-ρ1)(1-q2)-ρ2](1-q2)(c-+ρ1-ρ1c-)].(i)
Mean number of customers in the priority queue is(5.7)Np=limZ1→1P*′(1;0,Z1,1)+limZ1→1P*′(2;0,Z1,1).(ii)
Mean number of customers in the orbit is(5.8)Nr=limZ2→1P*′(1;0,1,Z2)+limZ2→1P*′(2;0,1,Z2)+Q′(1).(iii)
Mean busy period.Busy periodTb 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, Tb and T0 generates an alternating renewal process and therefore
(5.9)E(Tb)E(T0)=Pr{Tb}1-Pr{Tb}=PB1-PB.ButE(T0)=1/λ(5.10)E(Tb)=PBλ(1-PB).Using (3.39) on (5.10), we get
(5.11)E(Tb)=[ρ1(1-q2)+ρ2c-]λc-[(1-ρ1)(1-q2)-ρ2].


## 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 parametersq1, q2, c-, c′′(1), d-, d′′(1), β1, β2, B1*(0)(1) and B2*(0)(1) 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 (PI) and the probability that the server is busy (PB) 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 probabilitiesPI and PB.
λ
1
α
=
0.9,  λ2=0.8
α
=
0.9,  λ2=0.9
P
I
P
B
P
I
P
B
0.1
0.4511
0.0089
0.3837
0.0088
0.2
0.4422
0.0178
0.3750
0.0175
0.3
0.4334
0.0266
0.3663
0.0262
0.4
0.4246
0.0354
0.3576
0.0349
0.5
0.4158
0.0442
0.3490
0.0435
0.6
0.4071
0.0529
0.3404
0.0521
0.7
0.3984
0.0616
0.3319
0.0606
0.8
0.3898
0.0702
0.3233
0.0692
0.9
0.3811
0.0789
0.3148
0.0777
1.0
0.3725
0.0875
0.3064
0.0861Table 2
The probabilitiesPI and PB.
λ
2
α
=
0.9,  λ1=0.8
α
=
0.9,  λ1=0.9
P
I
P
B
P
I
P
B
0.1
0.8548
0.0777
0.8453
0.0872
0.2
0.7884
0.0766
0.7790
0.0860
0.3
0.7219
0.0756
0.7127
0.0848
0.4
0.6555
0.0745
0.6464
0.0836
0.5
0.5891
0.0734
0.5801
0.0824
0.6
0.5226
0.0724
0.5138
0.0812
0.7
0.4562
0.0713
0.4474
0.0801
0.8
0.3898
0.0702
0.3811
0.0789
0.9
0.3233
0.0692
0.3148
0.0777
1.0
0.2569
0.0681
0.2485
0.0765Figure 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, anM/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.



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*Source: 673642-2012-12-10.xml*