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

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.


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].

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.

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 problem 1 . Our contribution provides a building block towards the generalization to the case of N 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.

The model
Consider a single server queue accepting two types of customers, say P 1 , P 2 . P i , customers arrive according to Poisson process with rate λ i , and if upon arrival find the server unavailable, enter a dedicated virtual queue, called the orbit queue i, i = 1, 2. 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 P i customer are exponentially distributed with rate θ i /n i , given that there are n i customers in orbit i, i = 1, 2,. Upon a service completion, the server remains idle until either a primary or a retrial customer (of either type) arrives.

General results
Some interesting results can be deduced directly by the functional equation. Substituting z 1 = 1 in (1) and subsequently letting z 2 → 1, and vice versa yield the following linear relations between Π(0, 1), Π(1, 0) and Π(0, 0). where We proceed with an interesting interpretation for ρ j . Let S j , j = 1, 2 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 j given that both orbit queues are non-empty, and N i (S j ) the number of type i customers that join the orbit queue i during S j . Let also s (j) We restrict the analysis to the orbit queue 1. The analysis for the orbit queue 2 is similar. Then, and "*" means convolution. If

Preliminary analysis
We now follow the methodology given in [12]. Let, Clearly, , and it has two roots It is easy to check that the right hand side of (6) is the determinant of K(z 1 , 2δ − z 1 ) = 0, given by where " * " means convolution.
Next we introduce the following parametrization of G. Consider the function Using Rouche's theorem it can be proved that the function in (8) has exactly one zero, say (1 + e iφ ), Let Then, the following statements are readily verified: i) L 1 , and similarly L 2 , is a simple smooth contour, ii) The relations in (9) define a one to one mapping z 1 (δ(φ)) = ω 1 ( z 2 (δ(φ))) of L 2 onto L 1 .
Clearly, we can easily show starting by (11) that G(z) and also g(z) both possess a continuous derivative along |z| = 1 (note that L 1 , L 2 and L are all smooth contours) and consequently, they satisfy the Holder condition on |z| = 1.

Solution of a Riemann boundary value problem
In order to solve the Riemann boundary value problem formulated by (14) and conditions 1, 2, we have to compute the index of G(z) on |z| = 1. Note that Since L 1 and L 2 are simple contours with z 1 = 0 ∈ L + 1 , z 2 = 0 ∈ L + 2 , and f 1 (z) traverses L 1 counterclockwise, whereas f 2 (z) traverses L 2 clockwise we have ind |z|=1 f 1 (z) = 1, ind |z|=1 f 2 (z) = −1. The contours f j (z) − β(f 1 (z), f 2 (z)), are smooth and have only two real points of which one is negative and the other that corresponds to z = 1 is located at zero where the contours have vertical tangents.

Reduction to a Fredholm integral equation of the second kind
Another approach to cope with the solution of (1) is to reducing it to a Fredholm integral equation. Indeed, for the contours L 1 , L 2 , defined in (9) there exists a one-to-one map such as z 1 = ω 1 (z 2 ) : ) is a zero pair of the kernel. Let , Using results from subsection 4.2, ind z 1 ∈L 1 T (z 1 ) = 0. The functional equation (1) is now rewritten as, Ω 2 (ω 2 (z 1 )) = T (z 1 )Ω 1 (z 1 ) + t(z 1 ), z 1 ∈ L 1 .

Basic performance metrics and numerical issues
In the following we derive expressions for the mean number of customers in orbits at a departure instant. Substituting z 2 = 1, and using the in the functional equation yields 2λ+θ , where AC(F (z)) represents the analytic continuation of a function F (z). When Analogous calculations can be made for d dz 2 Π(0, z 2 )| z 2 =1 , which lead to the derivation of E(X 2 ).
The solution of (1) as described in subsection 4.2 is based on the properties of th conformal mappings In equations (10), (11) we derived integral expressions for these mappings. These expressions contain the function ψ(z), z ∈ L, which is determined as the solution of an integral equation (12). Such an integral equation cannot be solved explicitly, but numerically.
There are a lot of existing techniques to solve numerically such integrals (e.g. trapezoid rule), and standard iteration procedures show rapid convergence based on the values of the parameters. For a detailed treatment of how you can treat numerically (10), (11), (12), see [12], Ch. IV.2.
Clearly, the numerical computation of the exact conformal mappings is generally time consuming. Since L 1 , L 2 are close to ellipses, alternatively, we can approximate them by conformal mappings that map the interior of ellipses to L + [32]. In particular, we can approximate the contour L j by ellipse E j with semi-axes ρ(0), ρ(π/2), j = 1, 2. Then, (z j ) maps E + j to L + [32], where , where sn(w; l) is the Jacobian elliptic function. Our approximation for f −1 j (z j ) is (z j ), z j ∈ L j ∪ L + j .

The asymmetrical system
In the following we investigate the general asymmetric model where We proceed with the analysis of the kernel.
Proof 2 See Appendix.
Thus, we have the following problem on the unit circle L: Find two functions Π 1 (w), Π 2 (w) such that: i) Π 1 (w) is regular for w ∈ L + , continuous for w ∈ L ∪ L + , ii) Π 2 (w) is regular for w ∈ L − , continuous for w ∈ L ∪ L − , iii) for w ∈ L, Π 1 (w) = G(w) Π 2 (w) + g(w), and In order to derive a solution for the Riemann boundary value problem, we need to investigate properties of the functions G(w), g(w), w ∈ L. In particular, i) 0 < G(w) < ∞, 0 < g(w) < ∞, w ∈ L, ii) G(w), g(w) should satisfy the Holder condition on L. Instead of them, we will consider the following equivalent functions for φ ∈ [0, 2π]: Let χ be the index of the function G(w), w ∈ L, Lemma 3 Let r 2 θ 1 ≤ r 1 θ 2 and r 1 θ 2 ≥ r 2 c 2 / θ 1 . Under the assumption the index χ = 1.

A numerical example
In this section we provide a numerical example regarding the performance of the completely symmetrical system. Assume that the service time is an Erlang(2, µ) distributed random variable withb = 2 µ , b (2) = 6 µ 2 . In Figure 1 (left) we can observe the effect of λ and θ on the average delay obtained in (38). As expected, the increase in λ will cause the increase in E(D). That increase becomes more apparent for small values of θ since in such a case the orbiting customers retry in a "slow" fashion. Moreover, if the service rate increases, the average delay in an orbit will decrease. Figure 1 (right) shows the way E(D) is affected for increasing values of µ, θ. Clearly, the increase in µ will result in the decrease of the average delay. However, we can easily observe how sensitive is E(D) when we slightly increase λ, and especially when µ, and θ take small values.

Conclusion and future work
As already mentioned this paper aims to provide a general framework for the fundamental problem of the analysis of multiclass retrial queueing systems with constant retrial policy and general class dependent service times. For the two-orbit case, we generalize the model in [7], and provided a compact methodological approach in order to obtain the generating function of the joint orbit queue length distribution in terms of a solution of a Riemann boundary value problem.
Our results serve as a building block to obtain expressions for the delay in the case of N orbit queues. Clearly, the delay analysis for the general case of N orbit queues under constant retrial policy is highly non-trivial and still remains an open problem. We are currently working towards this direction, and we intent to provide bounds for the queueing delay in an orbit in a general topology with N orbit queues. Another point of interest is to explore the possibility to study the heavy traffic behavior of such a model, when the arrivals λ j are such that ρ j → 1, j = 1, 2. Our approach is also valid for the modelling of even general systems that include vacations, server failures, feedback and arbitrarily distributed retrial times.
Since a(δ(φ)) and b(δ(φ)) are the differentiable functions of δ, we only need to show that δ(φ) is a continuous differentiable function of φ. By differentiating (39) in φ, we can show after some algebra that under the stability conditions L 1 is smooth and non-self intersecting.