Showing papers on "Retrial queue published in 2020"
TL;DR: This paper derives the closed form solution for the stationary probabilities of the queue with working vacations and a constant retrial rate and concludes that the social benefit rate is best when customers know all information about the server.
Abstract: In this paper, we investigate the M/M/1 retrial queue with working vacations and a constant retrial rate. In the queue, customers decide about the entry based on the information upon their arrival instants. Scenarios regarding the availability of information (i.e., the server is occupied or not, and the server is on the vacation or not) for customers are compared. We derive the closed form solution for the stationary probabilities of the queue. Social optimizing and Nash equilibrium strategies for joining the system are investigated. Based on numerical results, the social benefit rate is best when customers know all information about the server.
20 citations
TL;DR: The dependencies of the basic performance measures of the system on the capacity of the pool and the threshold are obtained and numerical results are presented.
Abstract: A single-server retrial queue with a MAP flow, PH service times and a pool of finite capacity for accumulation of the customers and their group service is considered. Service to the next group is not provided until the number of customers in the pool will reach a certain predefined threshold value. The service time of a group depends on its size and it is less than the sum of the individual service times. The dependencies of the basic performance measures of the system on the capacity of the pool and the threshold are obtained. Numerical results are presented.
20 citations
03 Apr 2020
TL;DR: A multi-server retrial queue with a hyper-exponential service time and on the basis of the constructed diffusion process, approximations of stationary probability distributions of the number of customers in orbit and thenumber of busy servers are obtained.
Abstract: A multi-server retrial queue with a hyper-exponential service time is considered in this paper. The study is performed by the method of asymptotic diffusion analysis under the condition of long delay in orbit. On the basis of the constructed diffusion process, we obtain approximations of stationary probability distributions of the number of customers in orbit and the number of busy servers. Using simulations and numerical analysis, we estimate the accuracy and applicability area of the obtained approximations.
10 citations
TL;DR: In this paper, customers' strategic behavior and social optimisation in a constant retrial queue with setup time and the N-policy are investigated, and customers who find the server isn't idle either l...
Abstract: In this article, customers’ strategic behavior and social optimation in a constant retrial queue with setup time and the N-policy are investigated. Customers who find the server isn’t idle either l...
10 citations
TL;DR: A priority queueing-inventory problem with two types of customers that represents an accurate enough model of many real-world systems, including wireless sensor networks and system of cognitive radio with energy harvesting and healthcare systems is studied.
Abstract: In this paper, we study a priority queueing-inventory problem with two types of customers. Arrival of customers follows Marked Markovian arrival process and service times have phase-type distribution with parameters depending on the type of customer in service. For service of each type of customer, a certain number of additional items are needed. High priority customers do not have waiting space and so leave the system when on their arrival a priority 1 customer is in service or the number of available additional items is less than the required threshold. Preemptive priority is assumed. Type 2 customers, encountering a busy server or idle with the number of available additional items less than a threshold, go to an orbit of infinite capacity to retry for service. The customers in orbit are non-persistent: if on retrial the server is found to be busy/idle with the number of additional items less than the threshold, this customer abandons the system with certain probability. Such a system represents an accurate enough model of many real-world systems, including wireless sensor networks and system of cognitive radio with energy harvesting and healthcare systems. The probability distribution of the system states is computed, using which several of the characteristics are derived. A detailed numerical study of the system, including the analysis of the influence of the threshold, is performed.
9 citations
TL;DR: This paper deals with the analysis of MX/G/1 retrial queue with impatient customers, modified vacation policy and Bernoulli feedback, and approximated the analytical results using adaptive neuro-fuzzy interface system (ANFIS) soft computing technique, which can identify parameters using supervised learning methods.
Abstract: This paper deals with the analysis of MX/G/1 retrial queue with impatient customers, modified vacation policy and Bernoulli feedback. When the incoming customer finds the server busy, on vacation or in the state of breakdown, he joins the virtual queue called retrial orbit, otherwise the service is provided to the customer who is at the head of the queue. The service is provided in l(0 ≤ i ≤ l) phases where first is compulsory service and remaining services are optional. When the system becomes empty, server leaves for the vacation of arbitrary length and can take at most J number of vacations. When server comes back from the vacation and finds at least one customer in the queue, he starts providing service to the customer. Supplementary variable technique (SVT) and probability generating function (PGF) method is used to derive the system size distribution and other performance indices. We have also approximated the analytical results using adaptive neuro-fuzzy interface system (ANFIS) soft computing technique, which can identify parameters using supervised learning methods.
8 citations
TL;DR: This paper applies the generating function method to derive the joint distribution of the server state and the orbit length in the steady state, and obtains some performance measures.
Abstract: The present paper deals with the performance evaluation of an M/M/1 retrial queue with collisions, transmission errors and unreliable server. To the best of our knowledge, there are no works that h...
8 citations
TL;DR: This paper studies an M /M /1 retrial queue with a storage system driven by an (s ,S ) policy, and derives the stationary joint distribution of the queue length and the on-hand inventory in explicit product form and investigates long-run performance measures.
Abstract: The purpose of this paper is to obtain product form solution for retrial – queueing – inventory system. We study an M / M /1 retrial queue with a storage system driven by an ( s , S ) policy. When server is idle, external arrivals enter directly to an orbit. Inventory replenishment lead time is exponentially distributed. The interval between two successive retrials is exponentially distributed and only the customer at the head of the orbit is permitted to access the server. No customer is allowed to join the orbit when the storage system is empty and also when the serer is busy. We first derive the stationary joint distribution of the queue length and the on-hand inventory in explicit product form. Using the joint distribution, we investigate long-run performance measures such as distribution of number of customers served, number of arrivals, number of customers lost during an interval of random duration and a cost function. The optimal pair ( s , S ) is numerically investigated. FullText for HTML: https://doi.org/10.1051/ro/2018118
7 citations
16 Dec 2020
TL;DR: A multi-server retrial queueing system with setup time which is motivated by applications in power-saving data centers with the ON-OFF policy, where an idle server is immediately turned off and an off server is set up upon arrival of a customer is considered.
Abstract: In the paper, we consider a multi-server retrial queueing system with setup time which is motivated by applications in power-saving data centers with the ON-OFF policy, where an idle server is immediately turned off and an off server is set up upon arrival of a customer. Customers that find all the servers busy join the orbit and retry for service after an exponentially distributed time. For this model, we derive the stability condition which depends on the setup time and turns out to be more strict than that of the corresponding model with an infinite buffer which is independent of the setup time. We propose asymptotic methods to analyze the system under the condition that the delay in the orbit is extremely long. We show that the scaled-number of customers in the orbit converges to a diffusion process. Using this diffusion limit, we obtain approximations for the steady-state probability distribution of the number of busy servers and that of the number of customers in the orbit. We verify the accuracy of the approximations by simulations and numerical analysis. Numerical results show that the retrial system under the limiting condition consumes more energy than that with an infinite buffer in front of the servers.
7 citations
TL;DR: The heavy load of the system and long time patience of customers in the orbit are proposed as asymptotic conditions and theorem about the Gaussian form of the asymPTotic probability distribution of the number of customersin the orbit is formulated and proved.
Abstract: Abstract. The retrial queueing system of M/M/1 type with Poisson flow of arrivals, impatient customers, collisions and unreliable service device is considered in the paper. The novelty of our contribution is the inclusion of breakdowns and repairs of the service into our previous study to make the problem more realistic and hence more complicated. Retrial time of customers in the orbit, service time, impatience time of customers in the orbit, server lifetime (depending on whether it is idle or busy) and server recovery time are supposed to be exponentially distributed. An asymptotic analysis method is used to find the stationary distribution of the number of customers in the orbit. The heavy load of the system and long time patience of customers in the orbit are proposed as asymptotic conditions. Theorem about the Gaussian form of the asymptotic probability distribution of the number of customers in the orbit is formulated and proved. Numerical examples are given to show the accuracy and the area of feasibility of the proposed method.
7 citations
TL;DR: In this paper, the stationary characteristics of an M/G/1 retrial queue are investigated where the single server, subject to active failures, primarily attends incoming calls and directs outgoing calls only when idle.
Abstract: Efficient use of call center operators through technological innovations more often come at the expense of added operation management issues. In this paper, the stationary characteristics of an M/G/1 retrial queue is investigated where the single server, subject to active failures, primarily attends incoming calls and directs outgoing calls only when idle. The incoming calls arriving at the server follow a Poisson arrival process, while outgoing calls are made in an exponentially distributed time. On finding the server unavailable (either busy or temporarily broken down), incoming calls intrinsically join the virtual orbit from which they re-attempt for service at exponentially distributed time intervals. The system stability condition along with probability generating functions for the joint queue length distribution of the number of calls in the orbit and the state of the server are derived and evaluated numerically in the context of mean system size, server availability, failure frequency and orbit waiting time.
TL;DR: An M/G/1 retrial queue with two types of breakdowns is studied and the steady-state joint queue length distribution by supplementary variable method is given, and some important performance measures and reliability indices are presented.
Abstract: This article studies an M/G/1 retrial queue with two types of breakdowns. When the server is idle, it is subject to breakdowns according to a Poisson process with rate $\delta $ and it cannot be repaired immediately. While when the server is busy, it may break down according to a Poisson process with rate $\theta $ and can be immediately repaired. Firstly, based on embedded Markov chain technique and probability generating function (PGF) method, we present the necessary and sufficient condition for the system to be stable and the PGF of the orbit size at the departure epochs. Secondly, we give the steady-state joint queue length distribution by supplementary variable method, and present some important performance measures and reliability indices. Thirdly, we provide the analysis of sojourn time of an arbitrary customer in the system when the system is in stable state. Finally, some numerical examples are presented to illustrate the effect of the some system parameters on important performance measures and reliability indices.
TL;DR: The offered load is defined and stability and instability conditions for this retrial queueing system are provided and the stability result can be obtained by introducing artificial primitive processes and using the fluid limit approach.
Abstract: We consider a multi-class multi-server retrial queueing system. Customers of each class arrive from outside the system according to a Poisson process. The service times for each class of customers are assumed to be exponentially distributed with service rates depending on both the customers’ class and the servers. We define the offered load and provide stability and instability conditions for this retrial queueing system. The stability result can be obtained by introducing artificial primitive processes and using the fluid limit approach.
TL;DR: It is found that there exist thresholds of system parameters such that the social planner would prefer revealing more information when the system parameter is greater than or less than the corresponding threshold.
Abstract: We study customers’ joining strategies in an M /M /1 constant retrial queue with a single vacation. There is no waiting space in front of the server and a vacation is triggered when the system is empty. If an arriving customer finds the server idle, he occupies the server immediately. Otherwise, if the server is found unavailable, the customer enters a retrial pool called orbit with infinite capacity and becomes a repeated customer. According to the different information provided for customers, we consider two situations, where we investigate system characteristics and customers’ joining or balk decisions based on a linear reward-cost structure. Furthermore, we establish the social welfare of the system and make comparisons between the two information levels. It is found that there exist thresholds of system parameters such that the social planner would prefer revealing more information when the system parameter is greater than or less than the corresponding threshold.
TL;DR: An exhaustive version of the stochastic decomposition approach is proposed for retrial queueing systems with priority, for the purpose of studying asymptotic behaviour of the tail probability of the number of customers in the steady state for this ret trial queue with two types of customers.
Abstract: Stochastic networks with complex structures are key modelling tools for many important applications. In this paper, we consider a specific type of network: retrial queueing systems with priority. This type of queueing system is important in various applications, including telecommunication and computer management networks with big data. The system considered here receives two types of customers, of which Type-1 customers (in a queue) have non-pre-emptive priority to receive service over Type-2 customers (in an orbit). For this type of system, we propose an exhaustive version of the stochastic decomposition approach, which is one of the main contributions made in this paper, for the purpose of studying asymptotic behaviour of the tail probability of the number of customers in the steady state for this retrial queue with two types of customers. Under the assumption that the service times of Type-1 customers have a regularly varying tail and the service times of Type-2 customers have a tail lighter than Type-1 customers, we obtain tail asymptotic properties for the numbers of customers in the queue and in the orbit, respectively, conditioning on the server’s status, in terms of the exhaustive stochastic decomposition results. These tail asymptotic results are new, which is another main contribution made in this paper. Tail asymptotic properties are very important, not only on their own merits but also often as key tools for approximating performance metrics and constructing numerical algorithms.
TL;DR: An equation is obtained for the joint transform of the number of ingoing calls in the orbit and the waiting time of an arbitrary ingoing call and it is obtained that the moments of the waitingTime distribution of an Arbitrary ingoingcall are obtained.
Abstract: We consider an M/G/1 retrial queue in which there are two types of calls: ingoing calls made by regular customers and outgoing calls made by the server in idle time. The service times of ingoing calls and outgoing calls have different arbitrary distributions. In this paper, we are interested in the analysis of the waiting time distribution. We obtain an equation for the joint transform of the number of ingoing calls in the orbit and the waiting time of an arbitrary ingoing call. Using this result, we can obtain the moments of the waiting time distribution of an arbitrary ingoing call.
TL;DR: The idea behind queueing modeling of the proposed work originates from its applicability in various real life service systems and stochastic decomposition laws have been demonstrated for this proposed model.
Abstract: The idea behind queueing modeling of the proposed work originates from its applicability in various real life service systems. The performance indices obtained for the proposed model can be used to enhance the quality-of-service of many computer networks (LANs, etc.), Internet mail servers and radio communication. In these systems, packets or emails or calls are transmitted from various sources to its respective destination. During this transmission, it may be possible that a packet or an email or a call may be lost due to the effect of an unwanted unit such as ‘virus’ in computer system or ‘collision of packets’ in LANs and others. These unwanted units which force the service provider to collapse immediately are referred as ‘damaging unit’ in terms of queueing theory. This collapsed service provider is then sent for becoming renewed again by the technician present in the system, who makes it perfect after renewed process and the service provider resumes its service. The positive units (which are normal units) and the damaging units arrive to the system in Poisson fashion. We consider that the service provider furnish initial stage of compulsory service referred as ‘first stage service’ to each incoming unit while it furnish non-compulsory services (up to the number l) referred as ‘second stage service’ to only those who urge for the same. On the finish of each service or renewed process, the service provider may abandon the service system and go for break referred as vacation with a random vacation time distribution. Moreover, stochastic decomposition laws have been demonstrated for this proposed model. In addition, numerical experiments and sensitivity analysis are also carried out.
TL;DR: A single-server, finite-source retrial queue where the server not only accepts incoming calls but after some exponentially distributed idle time makes outgoing calls, which implies that the model can be considered as a model with vacations or with customers of two types.
Abstract: In this paper we deal with a single-server, finite-source retrial queue where the server not only accepts incoming calls but after some exponentially distributed idle time makes outgoing calls. The service times of incoming and outgoing calls follow two distinct arbitrary distributions. The outgoing calls are directed not to the customers in the system but outside it, which implies that the model can be considered as a model with vacations or with customers of two types. Along with the standard retrial queue where all customers are allowed to join the orbit we consider also the corresponding queue with restriction on the orbit size. We derive formulas for computing the stationary system state distribution and investigate the influence of the system input parameters on the main macro characteristics of the system performance.
15 Dec 2020
TL;DR: This investigation considers an M/G/1 queue with general retrial times allowing balking and server subject to breakdowns and repairs, and derives insensitive bounds for the stationary distribution of the embedded Markov chain of the considered system.
Abstract: In this investigation, we consider an M/G/1 queue with general retrial times allowing balking and server subject to breakdowns and repairs. In addition, the customer whose service is interrupted can stay at the server waiting for repair or leave and return while the server is being repaired. The server is not allowed to begin service on other customers until the current customer has completed service, even if current customer is temporarily absent. This model has a potential application in various fields, such as in the cognitive radio network and the manufacturing systems, etc. The methodology is strongly based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary distribution of the embedded Markov chain of the considered system.
14 Sep 2020
TL;DR: In this paper, the authors considered a multiserver retrial queue with two-way communication and derived the diffusion limits of the number of calls in the orbit and the approximation of its stationary probability distribution.
Abstract: In this paper, we consider a multiserver retrial queue with two-way communication. Incoming calls arrive according to the stationary Poisson process and occupy the servers. Durations of incoming calls have an exponential distribution. If all the servers are busy upon arrival, the incoming call joins the orbit. The time spent by the call in the orbit is an exponential random variable. Idle servers also make outgoing calls whose durations follow an exponential distribution. We derive the diffusion limits of the number of calls in the orbit and the approximation of its stationary probability distribution.
TL;DR: In the present investigation, the supplementary variable technique and the method of generating function have been used and the sensitivity of various parameters on the system performance has been examined numerically by taking an illustration.
Abstract: The study presented here deals with the steady state analysis of bulk arrival retrial queue with unreliable server and multi phase essential services. The server renders service to two types of customers viz. priority (type 1) and ordinary (type 2) customers. The ordinary customers are forced to join the orbit on nonavailability of the server whereas priority customers are served by the server as soon as they arrive. The service is provided in ‘l’ essential phases for both type of customers. Moreover, the server is not an ideal server and hence may stop working in between. As soon as the server fails or stops working, it is sent for repair immediately. The broken down server is also repaired in ‘d’ essential phases. In the present investigation, the supplementary variable technique and the method of generating function have been used. The sensitivity of various parameters on the system performance has been examined numerically by taking an illustration.
Posted Content•
TL;DR: To overcome inefficiency in a single-server first-come-first-served M/G/1 queue, an efficient method to compute a price function is included which internalizes the externalities created by customers.
Abstract: Consider a single-server first-come-first-served M/G/1 queue with the exception that service demand distribution is determined endogenously by the following mechanism: Each customer who enters into the service position faces a dynamic decision of when to terminate her service period and then to leave the system for good. In addition, all customers have random linear costs in their waiting times (not including service times) and each of them continuously observes the evolution of her marginal utility from service duration. It is assumed that the marginal utilities of the customers are general iid nonincreasing, right-continuous stochastic processes which are independent from the arrival process. Now, customers are acting according to their own self-interest and hence there is an overuse of the server. To overcome this inefficiency, this work includes an efficient method to compute a price function which implements a socially optimal resource allocation. Then, some examples of nonincreasing, right-continuous stochastic processes are considered. In addition, it is shown that this price function internalizes the externalities created by customers. Finally, similar results are derived for M/G/1 retrial queue. In particular, there is a conjecture regarding the expression for the expected externalities in such a retrial queueing system.
01 Jan 2020
TL;DR: Applying the supplementary variable method and the generating function technique, the Markov chain underlying the considered queueing model is analyzed and the stationary distributions under different system states, the generating functions for the number of customers in the orbit and in the system, as well as some crucial performance measures in steady state are derived.
Abstract: This paper deals with a discrete-time $Geo/G/1$ retrial queueing system with probabilistic preemptive priority and balking customers, in which the server is subject to starting failures and replacements in the repair times may occur with some probability. If the server is found busy at an arrival epoch, the newly arriving customer either interrupts the customer in service to begin its own service with probability $p$ or enters the orbit with probability $1-p$. When an arriving customer (external or repeated) finds the server free, he must turn on the server. If the server is activated successfully, the customer receives service immediately. Otherwise, the server undergoes a repair process. If an external arrival finds that the server is under repair, he decides either to join the orbit with probability $q$ or leaves the system completely (balking) with probability $1-q$. Applying the supplementary variable method and the generating function technique, we analyze the Markov chain underlying the considered queueing model and derive the stationary distributions under different system states, the generating functions for the number of customers in the orbit and in the system, as well as some crucial performance measures in steady state. Especially, some corresponding results under special cases are directly obtained by setting appropriate parameter values. Further, some numerical examples are provided to examine the effect of various system parameters on queueing characteristics. Finally, an operating cost function is formulated to discuss numerically a cost optimization problem.
TL;DR: This work considers a single-server retrial queue with a Poisson arrival process and exponential service times, where the server is unreliable, and investigates the system characteristics and study how the manager determines the price charging joining customers.
Abstract: We consider a single-server retrial queue with a Poisson arrival process and exponential service times, where the server is unreliable. Assume there is no waiting space in front of the server and the customer who finds the server unavailable joins an orbit to access the server some time later. We discuss two types of customers’ retrial behavior. One is that each customer in the orbit seeks for services independently and the total retrial rate of the system depends on the number of customers in the orbit. The other type of retrial discipline is called constant retrial policy and it arises from some situations in the computer and communication network where the retrial rate may be controlled by automatic mechanisms. An announced price charged by the server is imposed on customers joining the system and the actual demands for services depend on the price via a decreasing function. We investigate the system characteristics and study how the manager, whose goal is to maximize its own profit, determines the price charging joining customers. Finally, we present an application example to illustrate the obtained results and make comparisons between the two retrial policies from the perspective of customers’ expected waiting time.
02 Dec 2020
TL;DR: In this paper, the authors considered a priority multi-server retrial queue with two classes of customers: primary customers and secondary users, and they proved that a scaling version of the number of users in the orbit weakly converges to a diffusion process whose drift and diffusion coefficients are constructed.
Abstract: This paper considers a priority multi-server retrial queue with two classes of customers. Primary customers have preemptive priority over secondary users. The dynamics of primary customers is the same as that of an Erlang loss system with Poisson input and exponential service time distribution. Secondary users can cognitively use the channels when they are not used by primary users. Secondary users that see all the channels occupied upon arrival join the orbit and retry later. Upon arrival, if a primary user is lost if it sees all the channels occupied by other primary users. Upon the arrival of a primary customer, if all the channels are occupied but some channels are occupied by secondary users, one of these ongoing secondary users is interrupted by the primary user and the interrupted secondary user enters the orbit. Secondary users from the orbit retry to occupy an idle server until they are successfully occupying one. For this model, we consider an asymptotic regime in which the retrial rate is extremely low. While the number of secondary users in the orbit explodes in this regime, we prove that a scaling version of the number of users in the orbit weakly converges to a diffusion process whose drift and diffusion coefficients are constructed.
01 Jan 2020
TL;DR: This analysis is devoted to model retrial queue with, Bernoulli feedback, preferred as well as impatient customers or units in discrete environment, where server follows state-dependent policy and may leave for single vacation whenever it is idle.
Abstract: This analysis is devoted to model retrial queue with, Bernoulli feedback, preferred as well as impatient customers or units in discrete environment. Here, server follows state-dependent policy and may leave for single vacation whenever it is idle. This investigation is motivated by the increasing impact of discrete-time retrial queues in real-world scenarios. For instance, it is widely used in multiplexing of voice data, digital communication (ATMs, BISDN), switching modules and networks, etc. In such types of queueing system, time is a random variable of discrete type, and we calculate it in equally sized data units. We have considered a system with early arrival pattern and studied the Markov’s chain underlying this model. Along with this, the marginal generating function (mgf) for the total units present in the orbit depends on the server state. Few performance measures like average units in the orbit are also calculated by applying probability generating function method. Further, a practical example is numerically illustrated as well as sensitivity analysis is provided.
TL;DR: In this paper, a feedback retrial queue with customer balking and unreliable servers is studied, where a customer will either join a retraced queue or leave the retrace queue.
Abstract: Motivated by some real stochastic service systems, we study a feedback retrial queue with customer balking and unreliable servers (servers subject to breakdowns). A customer will either join a retr...
TL;DR: In this paper, an M/G/1 retrial queue with general retrial times and single working vacation is considered, and the generating functions of the server state and the number of customers in the orbit are dealt with.
Abstract: In this paper, an M/G/1 retrial queue with general retrial times and single working vacation is considered. We assume that the customers who find the server busy are queued in the orbit in accordance with a first-come-first-served (FCFS) discipline and only the customer at the head of the queue is allowed access to the server. During the normal period, if the orbit queue is not empty at a service completion instant, the server begins a working vacation with specified probability q (0 ≤ q ≤ 1), and with probability 1 − q , he waits for serving the next customer. During the working vacation period, customers can be served at a lower service rate. We first present the necessary and sufficient condition for the system to be stable. Using the supplementary variable method, we deal with the generating functions of the server state and the number of customers in the orbit. Various interesting performance measures are also derived. Finally, some numerical examples and cost optimization analysis are presented.
TL;DR: Based on the obtained asymptotics, the approximations of the probability distribution of the number of calls in the orbit under high rate of making outgoing calls limit condition are built.
Abstract: Retrial queue under consideration is the model of call center operator switching between input and outgoing calls. Incoming calls form a Poisson point process. Upon arrival, an incoming call occupies the server for an exponentially distributed service time if the server is idle. If the server if busy, an incoming call joins the orbit to make a delay before the next attempt to take the server. The probability distribution of the length of delay is an exponential distribution. Otherwise, the server makes outgoing calls in its idle time. There are multiple types of outgoing calls in the system. Outgoing call rates are different for each type of outgoing call. Durations of different types of outgoing calls follow distinct exponential distributions. Unsteadiness is that the server crashes after an exponentially distributed time and needs recovery. The rates of breakdowns and restorations are different and depend on server state. Our contribution is to obtain the probability distribution of the number of calls in the orbit under high rate of making outgoing calls limit condition. Based on the obtained asymptotics, we have built the approximations of the probability distribution of the number of calls in the orbit.