A two-class queueing system with constant retrial policy and general class dependent service times

# A two-class queueing system with constant retrial policy and general class dependent service times

Ioannis Dimitriou111 idimit@math.upatras.gr Department of Mathematics, University of Patras, P.O. Box 26500, Patras, Greece.
July 26, 2019
###### Abstract

A single server retrial queueing system with two-classes of orbiting customers, and general class dependent service times is considered. If an arriving customer finds the server unavailable, it enters a virtual queue, called the orbit, according to its type. The customers from the orbits retry independently to access the server according to the constant retrial policy. We derive the generating function of the stationary distribution of the number of orbiting customers at service completion epochs in terms of the solution of a Riemann boundary value problem. For the symmetrical system we also derived explicit expressions for the expected delay in an orbit without solving a boundary value problem. A simple numerical example is obtained to illustrate the system’s performance.
Keywords Queueing, Two-class retrial queue, Boundary value problem, Delay analysis, Embedded Markov chain.

## 1 Introduction

Queueing systems with retrial customers are characterized by the feature that an arriving customer who finds the server unavailable, departs temporarily from the system, and repeats its attempt to connect with the server after some random time according to a specific access policy. The so called repeated customers are temporarily stored in a pool of unsatisfied customers (called orbit or retrial group), and are superimposed on the normal stream of external arrivals. For a complete review of the main results, the interested reader is referred to the seminal books [21, 5], and in the detailed review papers [4, 36, 26].

### 1.1 Related work and applications

Single class retrial systems under constant retrial policy were investigated in [6, 8, 15, 16, 20, 23, 25, 38] (not exhaustive list). Clearly, there have been very limited results in retrial queueing literature with multiple classes of retrial customers. A two class retrial system with arbitrary distributed service requirements and classical retrial policy was firstly analyzed in [27], whereas the extension to an arbitrary number of classes of retrial customers was investigated in [22]. In [31] a non-preemptive priority mechanism was included in the work in [27, 22], while in [28] a multiclass retrial queue with many phases of service was also investigated. In all the above mentioned works, a classical retrial policy was used and the authors derived expressions for the expected number of customers in orbit queues. Recently, there has been a lot of attention to the application of polling retrial systems with glue periods on the modeling of optical networks [1, 2, 3, 11]. In [7], the authors studied a two-class system with common exponential service requirements and constant retrial policy. Their analysis led to a functional equation, which is solved with the aid of the theory of Riemann-Hilbert boundary value problems. Several generalizations of this model by considering coupled orbit queues, and simultaneous arrivals were considered in [18, 19]. A two class retrial system with common arbitrarily distributed paired service, and potential applications in wireless systems under network coding was investigated in [17].

In general, multiclass retrial systems with constant retrial policy serve as a model for competing job streams in a carrier sensing multiple access system, where the jobs, after a failed attempt to network access, wait in an orbit queue; e.g., a local area computer network with bus architecture where the different types of customers can be interpreted as customers with different priority requirements [35]. Under the constant retrial policy we are able to stabilize and control the multiple access system. Such a priority setting can also be applied to train or vehicular onboard networks. In such a case the high priority jobs correspond to critical system control signals, and the low priority jobs correspond to onboard passenger internet access traffic.

Other potential applications may be found in cooperative wireless systems. Such systems consist of a finite number of source users that transmit packets to a common destination node, and a finite number of assistant users, called relay nodes (i.e., the orbit queues) that assist them by retransmitting their failed packets; e.g., [33, 34, 18, 19]. Other applications can be found in telecommunication systems with call-back option in call centers [20, 37], where an operator (i.e. a server) calls-back an unsatisfied customer after some random time.

### 1.2 Our contribution

The important feature of this work is the two class setting under constant retrial policy, and arbitrarily distributed service requirements, which depend on the type of the job as well as the instant of its arrival. In particular, the service times of primary jobs that occupy upon arrival the server is different compared with the service times of the retrial jobs. Moreover, the service requirements of each class of retrial customers is also different. Besides its practical applicability in the modelling of relay assisted cooperative wireless networks, and in call centers with call-back option, our work is also theoretically oriented.

In particular, in this work we focus on the fundamental problem of investigating the queueing delay in multiclass retrial systems with constant retrial policy, and arbitrarily distributed class dependent service times, which remains an open problem. The only available results refer to the investigation of the stability conditions [9, 10, 29, 30]. More precisely, for the two orbit scenario, we generalize the seminal paper in [7] by allowing arbitrarily distributed class dependent service times, and obtain the generating function of the stationary joint orbit queue-length distribution in terms of a solution of a Riemann boundary value problem222In subsection 4.3 we also provided the way we can expressed it by solving a Fredholm integral equation of the second kind.. Our contribution provides a building block towards the generalization to the case of orbits; see also Section 8. For the completely symmetrical system, we also provide for the first time, explicit expressions for the expected number of customers at each orbit queue, without the need of solving a boundary value problem.

The rest of the paper is organized as follows. In Section 2 we describe the model in detail and provide the fundamental functional equation. Some important preparatory results are given in Section 3. Sections 4, and 5 are devoted in the detailed analysis of the modified symmetrical and the asymmetrical system, respectively. In Section 6 we provide explicit expressions for the expected orbit delay for the completely symmetrical system without solving a boundary value problem, while in Section 7 a simple numerical example is presented.

## 2 The model

Consider a single server queue accepting two types of customers, say , . , customers arrive according to Poisson process with rate , and if upon arrival find the server unavailable, enter a dedicated virtual queue, called the orbit queue , . All the customers in each orbit behave independently of each other and try to access the server according to the constant retrial policy. More precisely, we assume that the retrial times for any orbiting customer are exponentially distributed with rate , given that there are customers in orbit , . Upon a service completion, the server remains idle until either a primary or a retrial customer (of either type) arrives.

The provided service time depends on the type (i.e., , ) and the state of the customer (i.e., either orbiting or primary). More precisely, the service times for orbiting customers of type , say is arbitrarily distributed with cumulative distribution function (cdf) , probability density function (pdf) , Laplace Stieltjes Transform (LST) , and moments , . An arriving primary customer of either type who finds the server idle will occupy it immediately and its service requirement, say , is arbitrarily distributed with cdf , pdf , LST , and moments , .

Let be the number of , orbiting customers, just after the end of the th service completion. Denote also by , the type of the th service time. Clearly forms an irreducible and aperiodic Markov chain. Define by , the number of customers that arrive during the th service service period if it is of type . Then,

 (X1(n+1),X2(n+1),ξ(n+1))=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩(X1(n)−1+A11(n+1),X2(n)+A21(n+1),1),w.p.θ1D(n)(X1(n)+A13(n+1),X2(n)+A23(n+1),3),w.p.λD(n),(X1(n)+A12(n+1),X2(n)−1+A22(n+1),2),w.p.θ2D(n),

where and . Denote,

 πm,l=limn→∞Pr((X1(n),X2(n))=(m,l)),Π(z1,z2)=∑∞m=0∑∞l=0πm,lzm1zl2,|z1|≤1,|z2|≤1,

and , . Clearly, for

 P(A1j=k,A2j=m)=d(j)km=∫∞0e−λ1t(λ1t)kk!e−λ2t(λ2t)km!dBj(x),d∗j(z1,z2)=∑∞m=0∑∞l=0d(j)kmzk1zm2=β∗j(λ(1−r1z1−r2z2)).

Let . Considering the transition probabilities at service completion epochs we obtain,

 πm,l=θ1λ+θ∑m+1k1=1∑lk2=1πk1,k2d(1)m+1−k1,l−k2+θ2λ+θ∑mk1=1∑l+1k2=1πk1,k2d(2)m−k1,l+1−k2λλ+θ∑mk1=1∑lk2=1πk1,k2d(3)m−k1,l−k2+θ1λ+θ1∑m+1k1=1πk1,0d(1)m+1−k1,l+π0,0dm,l+λλ+θ1∑mk1=1πk1,0d(3)m−k1,l+θ2λ+θ2∑l+1k2=1π0,k2d(2)m,l+1−k2+λλ+θ2∑lk2=1π0,k2d(2)m,l−k2.

Forming the generating functions we conclude that

 K(z1,z2)Π(z1,z2)=A(z1,z2)Π(z1,0)+B(z1,z2)Π(0,z2)+C(z1,z2)Π(0,0), (1)

where,

 K(z1,z2)=z1z2−˜K(z1,z2). (2)
 A(z1,z2)=z2˜A(z1,z2)−˜K(z1,z2),B(z1,z2)=z1˜B(z1,z2)−˜K(z1,z2),C(z1,z2)=˜K(z1,z2)+z2(r1z1β∗3(y)−˜A(z1,z2))+z1(r2z2β∗3(y)−˜B(z1,z2)),

and

 ˜K(z1,z2)=θ1λ+θz2β∗1(y)+θ2λ+θz1β∗2(y)+λz1z2λ+θβ∗3(y),˜A(z1,z2)=θ1β∗1(y)+λz1β∗3(y)λ+θ1,˜B(z1,z2)=θ2β∗2(y)+λz2β∗3(y)λ+θ2.

is called the kernel of the functional equation (1), and its investigation is of major importance for the fruitful analysis of (1). Contrary to [14, 17], is not a Poisson kernel.

## 3 General results

Some interesting results can be deduced directly by the functional equation. Substituting in (1) and subsequently letting , and vice versa yield the following linear relations between , and .

 1−ˆρ2=Π(1,0)λ+θ1+λ2[λ(¯b3−¯b2)+θ1(¯b1−¯b2)]λ+θ1+Π(0,1)θ1θ2[λ2[λ(¯b3−¯b1)+θ2(¯b2−¯b1)]−θ2λ+θ2]+Π(0,0)θ1[λ2(¯b3−¯b1)λ+θ1+1+λ2(¯b3−¯b2)λ+θ2],1−ˆρ1=Π(1,0)θ2θ1[λ1[λ(¯b3−¯b2)+θ1(¯b1−¯b2)]−θ1λ+θ1]+Π(0,1)λ+θ2+λ1[λ(¯b3−¯b1)+θ2(¯b2−¯b1)]λ+θ2+Π(0,0)θ2[λ1(¯b3−¯b2)λ+θ2+1+λ1(¯b3−¯b1)λ+θ1], (3)

where ,

We proceed with an interesting interpretation for . Let , be the time elapsed form the epoch a service is initiated until the epoch the server becomes idle after a service completion of a retrial customer of type given that both orbit queues are non-empty, and the number of type customers that join the orbit queue during . Let also . We restrict the analysis to the orbit queue 1. The analysis for the orbit queue 2 is similar. Then,

 s(1)k1,k2(t)=θ1λ+θhk1,k2(t)b1(t)+λλ+θ∑k1m1=0∑k2m2=0hk1,k2(t)b3(t)∗s(j)k1−m1,k2−m2(t)+θ2λ+θ∑k1m1=0∑k2m2=0hk1,k2(t)b2(t)∗s(j)k1−m1,k2−m2(t),

where and “*” means convolution. If

 ˜s∗j(z1,z2,s)=∫∞0e−st∑∞k1=0∑∞k2=0s(j)k1,k2(t)zk11zk22dt,

then,

 ˜s∗1(z1,z2,s)=θ1β∗1(s+y)s+θ1+θ2(1−β∗2(s+y))+λ(1−β∗3(s+y)),˜s∗2(z1,z2,s)=θ2β∗2(s+y)s+θ2+θ1(1−β∗2(s+y))+λ(1−β∗3(s+y)),

and

 ˆρ1=∂∂z1˜s∗1(z1,1,0)|z1=1,ˆρ2=∂∂z2˜s∗2(1,z2,0)|z2=1.

That said, is the expected number of customers that join the orbit queue during this special service time . Therefore, we expect that , , which is consistent with the results regarding the stability conditions derived in [9].

### 3.1 Special cases

#### The modified symmetrical model

Consider the modified symmetrical model where, (i.e., ), and and . Then, (3) becomes

 1−λ(θ¯b+λ¯b3)θ=Π(1,0)θ+λ(2+λ(¯b3−¯b))2λ+θ+Π(0,1)λ2(¯b3−¯b)−θ2λ+θ+Π(0,0)θ(1+λ(¯b3−¯b))2λ+θ1−λ(θ¯b+λ¯b3)θ=Π(1,0)λ2(¯b3−¯b)−θ2λ+θ+Π(0,1)θ+λ(2+λ(¯b3−¯b))2λ+θ+Π(0,0)θ(1+λ(¯b3−¯b))2λ+θ. (4)

By subtracting the above equations we conclude that and substituting back we derive,

 (1−ˆρ)(2λ+θ)1+λ(¯b3−¯b)=2λΠ(1,0)+θΠ(0,0). (5)

Since the right hand side of the above equation is positive, it is straightforward that is the ergodicity condition.

#### The completely symmetrical model

Let (i.e., ), , , . Equations (4), (5) remain valid with slight modifications, and the stability condition is

## 4 Detailed analysis of the modified symmetrical model

Consider the modified symmetrical model, where (i.e., ), , and , and assume that .

### 4.1 Preliminary analysis

We now follow the methodology given in [12]. Let,

 wi=2rizi=zi,i=1,2,δ=12(w1+w2)=12(z1+z2).

Clearly,

 K(z1,z2)=z1z2−θβ∗(y)2(λ+θ)(z1+z2)−λz1z2λ+θβ∗3(y),

is well defined for with and is a quadratic equation in for every fixed with . Particularly,

 K(z1,2δ−z1)=0⇒z21−2δz1+δˇβ∗(δ)=0,

where , and it has two roots , . The equation can also be written as

 (z1−δ)2=(δ−ˇβ∗(δ)2)2−(ˇβ∗(δ)2)2. (6)

It is easy to check that the right hand side of (6) is the determinant of , given by . Clearly, has two roots in , viz. and , since has exactly one zero (i.e., ) in when .

Note now that has a very intuitive probabilistic interpretation. Indeed, let be the time elapsed from the epoch a service is initiated, until the service completion of a retrial customer of either type, given that both orbit queues are non-empty. Let is the number of newly arriving type customers during . Then,

 ˇbl1,l2(t)dt=P(t<ˇB≤t+dt,Ni(ˇB)=li,i=1,2),ˇbl1,l2(t)=θλ+θhl1,l2(t)b(t)+λλ+θ∑l1m1=0∑l2m2=0hm1,m2(t)b(t)∗ˇbl1−m1,l2−m2(t), (7)

where “” means convolution. If , , we have .

Put , and consider the two-bladed Riemann surface composed of two semi-planes slitted along , then and also constitute analytic functions on for .

Next we introduce the following parametrization of . Consider the function

 δ−ˇβ∗(δ)2(1+cosϕ),R(δ)≤1,ϕ∈[0,2π]. (8)

Using Rouche’s theorem it can be proved that the function in (8) has exactly one zero, say in for , which is real. Thus, . Therefore, for , substitute the zero of (8) in (6), we have

 ˆz1=ˆz1(δ(ϕ))=ˇβ∗(δ(ϕ))2(1+eiϕ),ˆz2=ˆz2(δ(ϕ))=ˇβ∗(δ(ϕ))2(1+e−iϕ). (9)

Let . Then, the following statements are readily verified: , and similarly , is a simple smooth contour, , , , , , , The relations in (9) define a one to one mapping of onto .

Clearly, the contours , satisfy the conditions of Theorem 1.1 in [13], p. 101. Put, for ,

 ρ(ϕ)=|ˆz1(δ(ϕ))|,ω(ϕ)=arg(ˆz1(δ(ϕ)),

so we can write:

We proceed by applying Theorem 1.1 in [13], and thus there exists a unique simple contour in the plane with , and functions , such that: is a simple zero of , is a simple zero of , i.e., , is regular and univalent for , is regular and univalent for , , .

Therefore, should be regular for and continuous for , and should be regular for and continuous for . Let , with , be a real function with . Then,

 (f1(z)z)+log(zf2(z))d=log(ρ2(ψ(z)))d,z∈L.

If satisfies the Holder condition on , the equation above represent a simple Riemann boundary value problem and following [12], [13],

 f1(z)=zexp{12iπ∫ζ∈Llog(ρ(ψ(ζ)))[ζ+zζ−z−ζ+1ζ−1]dζζ},z∈L+,f2(z)=z−1exp{−12iπ∫ζ∈Llog(ρ(ψ(ζ)))[ζ+zζ−z−ζ+1ζ−1]dζζ},z∈L−. (10)

By applying the Plemelj-Sokhotski formulas we obtain,

 f1(z)=zρ(ψ(z))exp{12iπ∫ζ∈Llog(ρ(ψ(ζ)))[ζ+zζ−z−ζ+1ζ−1]dζζ},z∈L,f2(z)=z−1ρ(ψ(z))exp{−12iπ∫ζ∈Llog(ρ(ψ(ζ)))[ζ+zζ−z−ζ+1ζ−1]dζζ},z∈L. (11)

From these expressions it is seen (see [12], p. 99) that should satisfy,

 eiω(ψ(z))=zexp{12iπ∫ζ∈Llog(ρ(ψ(ζ)))[ζ+zζ−z−ζ+1ζ−1]dζζ},z∈L. (12)

The solution of the Riemann problem above depends on the value of the constant , which is chosen such that . Thus, corresponds , and to .

We proceed with the solution of the functional equation. Since with , , 333This is due to the symmetry of the model. (see Theorem 4.1 in [12] or Section 4 in[13]), is a zerotuple of the kernel with , , , as constructed above, it follows that for ,

 Π(f1(z),0)A(f1(z),f2(z))+Π(0,f2(z))B(f1(z),f2(z))+Π(0,0)C(f1(z),f2(z))=0. (13)

Moreover, it follows from the regularity of , , , , that , is regular for and continuous for , , is regular for and continuous for , , .

Note also that for , so that the regularity of for implies by means of the maximum modulus theorem that for , so that is well defined, analogously for