Heavy outgoing call asymptotics for retrial queue with two way communication and multiple types of outgoing calls

In this paper, we consider a single server queueing model M |M |1|N with two types of calls: incoming calls and outgoing calls, where incoming calls arrive at the server according to a Poisson process. Upon arrival, an incoming call immediately occupies the server if it is idle or joins an orbit if the server is busy. From the orbit, an incoming call retries to occupy the server and behaves the same as a fresh incoming call. The server makes an outgoing calls after an exponentially distributed idle time. It can be interpreted as that outgoing calls arrive at the server according to a Poisson process. There are N types of outgoing calls whose durations follow N distinct exponential distributions. Our contribution is to derive the asymptotics of the number of incoming calls in retrial queue under the conditions of high rates of making outgoing calls and low rates of service time of each type of outgoing calls. Based on the obtained asymptotics, we have built the approximations of the probability distribution of the number of incoming calls in the system.


Introduction
Retrial queueing systems are characterized by the following distinctive feature: a customer who cannot receive service remains in the system and tries to occupy the server after some random delay. The pool of unsatisfied customers is called the orbit. Retrial queues have applications in telecommunication, computer networks and in daily life [1,2].
In retrial queues idle time of the server is the downtime and it should be reduced to increase the efficiency of the system. We consider systems where operator not only receives calls from outside but also makes outgoing calls Figure 1 shows the structure of the model.

Model description
We consider a single server retrial queue with two way communication and multiple types of outgoing calls. Incoming calls arrive at the system according to a Poisson process with rate λ and try to occupy the server for an exponentially distributed time with rate µ 1 . Incoming calls that find the server busy join the orbit and repeat their request for service after an exponentially distributed time with rate σ. When the server is idle it makes an outgoing call of type n in an exponentially distributed time with rate α n . There are N types of outgoing calls whose durations follow N distinct exponential distributions. We assume that the durations of outgoing calls of type n follow the exponential distribution with rate µ n .

Problem definition
Let k(t) denote the state of the server at the time t 0, n, if an outgoing call of type n is in service, n = 2, N + 1.
Let i(t) denote the number of incoming calls in the system at the time t. It is easy to see that process {k(t), i(t)} forms a continuous time Markov chain. We assume that the Markov chain is ergodic and the stationary distribution of {k(t), i(t)} exists.
Let P {k(t) = k, i(t) = i} = P k (i) denote the stationary probability distribution of the system state which is the unique solution of Kolmogorov system of equations: Let H k (u) denote the partial characteristic functions H k (u) = ∞ i=0 e jui P k (i), Multiplying equations of system (1) by e jui and taking the sum over i yields − (λ + µ n )H n (u) + λe ju H n (u) + α n H 0 (u) = 0, n = 2, N + 1. The characteristic function H(u) of the number of incoming calls in the retrial queue is expressed through partial characteristic functions H k (u) by The main content of this paper is the solution of system (2) by using an asymptotic analysis methods in two limit conditions: of the high rate of making outgoing calls and the low rate of service time of outgoing calls.

Prelimit analysis
In this section, we obtain expressions for the stationary distribution using the characteristic functions. First, we derive explicit expression for the characteristic function H(u) of the number of incoming calls in the system.
Substituting this equations into the first equation of the system (2), we find that α n e ju µ n + λ(1 − e ju ) H 0 (u). The solution of this differential equation is given by Substituting u = 0 into the system (2) yields: where expression for H 0 (u)| u=0 can be obtained substituting u = 0 into (5).
It follows from equations 2 and 3 of the system (7) that Furthermore, from the normalization condition: N +1 k=0 r k = 1, we obtain (3) and (4) and summing up results, we obtain

Asymptotic analysis of the model under the high rate of making outgoing calls
In this section, we will investigate system (2) by asymptotic analysis method under the high rate of making outgoing calls condition. In particular, we prove that asymptotic characteristic function of the number of incoming calls in the system corresponds to Gaussian distribution.

First order asymptotic
Theorem 2. Suppose i(t) is the number of incoming calls in the system of the stationary M |M |1|N retrial queue with outgoing calls, then the (9) holds where Proof. We denote α = 1/ε in the system (8), and introduce the following notations in order to get the following system A. Nazarov, S. Paul, O. Lizyura, Heavy outgoing call asymptotics… 11 Summing up equations of system (11), we obtain Considering the limit as ε → 0 in the system (11) and equation (12), then we will get We propose to get the solution of the system (13) in the form of Here r k , k = 1, N + 1 is the probability of the server state k; r 0 has no sense of probability, since the probability that the server will be in the zero state as α → ∞ is zero: As the relation j Φ (w) Φ(w) does not depend on w, the function is obtained in the following form Φ(w) = exp{jwκ 1 }, which coincides with (9). The value of the parameter κ 1 will be defined below. We rewrite the system (15) in the form 12 DCM&ACS. 2019, 27 (1) 5-20 − µ n r n + γ n r 0 = 0, n = 2, N + 1, The normalization condition for stationary server state probability distribution is The solution of the system (17) is given by where ν 1 = N +1 n=2 γ n µ n . Substituting (18) into system (16), we obtain an equation for κ 1 , which coincides with (10). The first order asymptotic i.e. Theorem 2, only defines the mean asymptotic value κ 1 α of a number of incoming calls in the system in prelimit situation of α → ∞. For more detailed research of number i(t) of incoming calls in the system let's consider the second order asymptotic.

Second order asymptotic Theorem 3. In the context of Theorem 2 the following equation is true
where A. Nazarov, S. Paul, O. Lizyura, Heavy outgoing call asymptotics… 13 Proof. We introduce the following notations in the system (8) and we get Denoting α = 1/ε 2 , and introducing the following notations we obtain 0 (w, ε) = 0, n = 2, N + 1.

(24)
Summing up equations of the system (24), we obtain Our idea is to seek for a solution of the system (24) and equation (25) in the form Substituting (26) to (24) and (25), laying out the exhibitors in tailor series and taking (16) into account, dividing these equations by ε and taking the limit as ε → 0, we have −µ n f n + λr n + γ n f 0 = 0, n = 2, N + 1, This equations imply that Φ 2 (w) wΦ 2 (w) doesn't depend on w and thus the function Φ 2 (w) is given in the following form which coincides with (19). We have and then we obtain the system − µ n f n + γ n f 0 = −λr n , n = 2, N + 1, Substituting values (18) into the system (27), we have , n = 2, N + 1, Substituting this expressions into equation 2 of the system (27), we obtain an equation for κ 1 , which coincides with (20).
Second order asymptotic i.e. Theorem 3, shows that the asymptotic probability distribution of the number i(t) of incoming calls in the system is Gaussian with mean asymptotic κ 1 α and variance κ 2 α.

Asymptotic analysis of the model under the low rate of service time of outgoing calls
In this section, we will investigate system (2) by asymptotic analysis method under the low rate of service time of outgoing calls condition.

Theorem 4. Suppose i(t) is a number of incoming calls in a system of stationary M |M |1|N retrial queue with two way communication, then the following equation is true
Proof. We denote µ = ε, let's substitute the following in the system (28) u = wε, H 0 (u) = εF 0 (w, ε), H k (u) = F k (w, ε), k = 1, N + 1. We will get the system Considering the limit as ε → 0 in the system (30) then we will get Summing up equations of the system (30) we have Laying out the exhibitors in tailor series, dividing equations by ε and taking the limit as ε → 0, taking (31) into account, we obtain From the last system of equations we have Then Substituting (35) into (31), we obtain The solution of differential equation is given by where C is an integration constant and its value will be obtained later. We denote asymptotic characteristic function (33) and (34), we obtain , n = 2, N + 1.

Conclusions
In this paper, we have considered retrial queue with two way communication with multiple types of outgoing calls. We have found characteristic function of the number of incoming calls in the system. We have found the first and the second order asymptotics of the number of calls in the system under the condition of the high rate of making outgoing calls. Based on the obtained asymptotics we have built the Gaussian approximation of the probability distribution of the number of incoming calls in the system. We have found asymptotic characteristic function of the number of incoming calls in retrial queue under the condition of the low service rate of outgoing calls. In future we plan to consider this retrial queueing system under other asymptotic conditions.