Output Process of Retrial Queue with Two-Way Communication Under Low Rate of Retrials Limit Condition

2021 ◽  
pp. 341-352
Author(s):  
Ivan L. Lapatin ◽  
Anatoly A. Nazarov
Keyword(s):  
Retrial Queue ◽  
Output Process ◽  
Limit Condition ◽  
Low Rate

scholarly journals Output process of the M|GI|1 is an asymptotical renewal process

2021 ◽  
Vol 21 (1) ◽  
pp. 100-110
Author(s):  
Ivan L. Lapatin ◽  
◽  
Anatoly A. Nazarov ◽  
Keyword(s):  
Distribution Function ◽  
Service Time ◽  
Renewal Process ◽  
Retrial Queue ◽  
Arbitrary Distribution ◽  
Single Server ◽  
Output Process ◽  
Limit Condition ◽  
Repeated Calls ◽  
Number Of Customers

Most of the studies on models with retrials are devoted to the research of the number of applications in the system or in the source of repeated calls using asymptotic and numerical approaches or simulation. Although one of the main characteristics that determines the quality of the communication system is the number of applications served by the system per unit of time. Information on the characteristics of the output processes is of great practical interest, since the output process of one system may be incoming to another. The results of the study of the outgoing flows of queuing networks are widely used in the modeling of computer systems, in the design of data transmission networks and in the analysis of complex multi-stage production processes. In this paper, we have considered a single server system with redial, the input of which receives a stationary Poisson process. The service time in considered system is a random value with an arbitrary distribution function B(x). If the customer enters the system and finds the server busy, it instantly joins the orbit and carries out a random delay there during an exponentially distributed time. The object of study is the output process of this system. The output is characterized by the probability distribution of the number of customers that have completed service for time t. We have provided the study using asymptotic analysis method under low rate of retrials limit condition. We have shown in the paper that the output of retrial queue M|GI|1 is an asymptotical renewal process. Moreover, the lengths of the intervals in output process are the sum of an exponential random value with the parameter lambda + kappa and a random variable with the distribution function B(x). The results of a numerical experiment show that the probability distributions of the number of served customers in the system are practically the same for significantly different distribution laws B(x) of service time if the service times have the same first two moments.

scholarly journals Two-way communication retrial queue with unreliable server and multiple types of outgoing calls

2020 ◽  
Vol 28 (1) ◽  
pp. 49-61
Author(s):  
Anatoly A. Nazarov ◽  
Svetlana V. Paul ◽  
Olga D. Lizyura
Keyword(s):  
Probability Distribution ◽  
Call Center ◽  
Poisson Point Process ◽  
Retrial Queue ◽  
Idle Time ◽  
High Rate ◽  
Exponential Distributions ◽  
Limit Condition ◽  
Different Types ◽  
Incoming Call

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.

Two-way communication retrial queue with unreliable server and multiple types of outgoing calls

2020 ◽  
Vol 28 (1) ◽  
pp. 49-61
Author(s):  
Anatoly A. Nazarov ◽  
Svetlana V. Paul ◽  
Olga D. Lizyura
Keyword(s):  
Probability Distribution ◽  
Call Center ◽  
Poisson Point Process ◽  
Retrial Queue ◽  
Idle Time ◽  
High Rate ◽  
Exponential Distributions ◽  
Limit Condition ◽  
Different Types ◽  
Incoming Call

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.

Two-way communication retrial queue with unreliable server and multiple types of outgoing calls

2020 ◽  
Vol 28 (1) ◽  
pp. 49-61
Author(s):  
Anatoly A. Nazarov ◽  
Svetlana V. Paul ◽  
Olga D. Lizyura
Keyword(s):  
Probability Distribution ◽  
Call Center ◽  
Poisson Point Process ◽  
Retrial Queue ◽  
Idle Time ◽  
High Rate ◽  
Exponential Distributions ◽  
Limit Condition ◽  
Different Types ◽  
Incoming Call

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.

Sign in / Sign up

Export Citation Format

Share Document