scholarly journals Mathematical Model of Call Center in the Form of Multi-Server Queueing System

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2877
Author(s):  
Anatoly Nazarov ◽  
Alexander Moiseev ◽  
Svetlana Moiseeva
Keyword(s):  
Distribution Function ◽  
Stationary Distribution ◽  
Service Time ◽  
Call Center ◽  
Queueing System ◽  
Optimization Problems ◽  
Arbitrary Distribution ◽  
Hyperexponential Distribution ◽  
Poisson Arrivals ◽  
Multi Server

The paper considers the model of a call center in the form of a multi-server queueing system with Poisson arrivals and an unlimited waiting area. In the model under consideration, incoming calls do not differ in terms of service conditions, requested service, and interarrival periods. It is assumed that an incoming call can use any free server and they are all identical in terms of capabilities and quality. The goal problem is to find the stationary distribution of the number of calls in the system for an arbitrary recurrent service. This will allow us to evaluate the performance measures of such systems and solve various optimization problems for them. Considering models with non-exponential service times provides solutions for a wide class of mathematical models, making the results more adequate for real call centers. The solution is based on the approximation of the given distribution function of the service time by the hyperexponential distribution function. Therefore, first, the problem of studying a system with hyperexponential service is solved using the matrix-geometric method. Further, on the basis of this result, an approximation of the stationary distribution of the number of calls in a multi-server system with an arbitrary distribution function of the service time is constructed. Various issues in the application of this approximation are considered, and its accuracy is analyzed based on comparison with the known analytical result for a particular case, as well as with the results of the simulation.

A note on the queueing system GI/G/∞

1968 ◽  
Vol 64 (2) ◽  
pp. 477-479 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Distribution Function ◽  
Service Time ◽  
Queueing System ◽  
Expected Value ◽  
Arrival Times ◽  
Service Times

Consider a queueing system GI/G/∞ in which (i) the inter-arrival times are distributed with distribution function A(t) (A(O +) = 0) (ii) the service times have distribution function B(t) such that the expected value of the service time is β(>∞).

On the waiting time distribution in a generalized GI/G/1 queueing system

1971 ◽  
Vol 8 (2) ◽  
pp. 241-251 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Random Variables ◽  
Markovian Process ◽  
Random Time ◽  
Arbitrary Distribution ◽  
Waiting Time Distribution ◽  
Transient Behaviour ◽  
Set Up

The model considered in this paper describes a queueing system in which the station is dismantled at the end of a busy period and re-established on arrival of a new customer, in such a way that the closing-down process consists of N1 phases of random duration and that a customer 𝒞n who arrives while the station is being closed down must wait a random time idn(i = 1, ···, N1) if the ith phase is going on at the arrival instant. (For each fixed index i, the random variables idn are identically distributed.) A customer 𝒞n arriving when the closing-down of the station is already accomplished has to wait a random time (N1 + 1)dn corresponding to the set up time of the station. Besides, a customer 𝒞n who arrives when the station is busy has to wait an additional random time 0dn. We thus have (N1 + 2) types of “delay” (additional waiting time). Similarly, we consider (N2 + 2) types of service time and (N3 + 2) probabilities of joining the queue. This may be formulated as a model with (N + 2) types of triplets (delay, service time, probability of joining the queue). We consider the general case where the random variables defining the model all have an arbitrary distribution.The process {wn}, where wn denotes the waiting time of customer 𝒞n if he joins the queue at all, is not necessarily Markovian, so that we first study (by algebraic considerations) the transient behaviour of a Markovian process {vn} related to {wn}, and then derive the distribution of the variables wn.

scholarly journals MX/G/1 Vacation Queueing System with Two Types of Repair facilities and Server Timeout

2019 ◽  
Vol 8 (9) ◽  
pp. 2040-2043
Keyword(s):  
Size Distribution ◽  
Exponential Distribution ◽  
Service Time ◽  
Queueing System ◽  
Batch Size ◽  
Single Server ◽  
Poisson Arrivals ◽  
Server Failure ◽  
System Length ◽  
General Service

We consider a single server vacation queue with two types of repair facilities and server timeout. Here customers are in compound Poisson arrivals with general service time and the lifetime of the server follows an exponential distribution. The server find if the system is empty, then he will wait until the time ‘c’. At this time if no one customer arrives into the system, then the server takes vacation otherwise the server commence the service to the arrived customers exhaustively. If the system had broken down immediately, it is sent for repair. Here server failure can be rectified in two case types of repair facilities, case1, as failure happens during customer being served willstays in service facility with a probability of 1-q to complete the remaining service and in case2 it opts for new service also who joins in the head of the queue with probability q. Obtained an expression for the expected system length for different batch size distribution and also numerical results are shown

Approximations in finite-capacity multi-server queues by Poisson arrivals

1978 ◽  
Vol 15 (4) ◽  
pp. 826-834 ◽  
Author(s):  
Shirley A. Nozaki ◽  
Sheldon M. Ross
Keyword(s):  
Service Time ◽  
Joint Distribution ◽  
Finite Capacity ◽  
Queuing Model ◽  
Average Delay ◽  
Service Distribution ◽  
Poisson Arrivals ◽  
Multi Server

An approximation for the average delay in queue of an entering customer is presented for the M/G/K queuing model with finite capacity. The approximation is obtained by means of an approximation relating a joint distribution of remaining service time to the equilibrium service distribution.

Approximations in finite-capacity multi-server queues by Poisson arrivals

1978 ◽  
Vol 15 (04) ◽  
pp. 826-834 ◽  
Author(s):  
Shirley A. Nozaki ◽  
Sheldon M. Ross
Keyword(s):  
Service Time ◽  
Joint Distribution ◽  
Finite Capacity ◽  
Queuing Model ◽  
Average Delay ◽  
Service Distribution ◽  
Poisson Arrivals ◽  
Multi Server

An approximation for the average delay in queue of an entering customer is presented for the M/G/K queuing model with finite capacity. The approximation is obtained by means of an approximation relating a joint distribution of remaining service time to the equilibrium service distribution.

scholarly journals Multi-server queueing system with reserve servers

2019 ◽  
pp. 57-70
Author(s):  
Valentina I. Klimenok
Keyword(s):  
Mathematical Model ◽  
Energy Consumption ◽  
Service Time ◽  
Stochastic Systems ◽  
Queueing System ◽  
Arrival Process ◽  
Batch Markovian Arrival Process ◽  
Finite Set ◽  
And Performance ◽  
Multi Server

In this paper, we investigate a multi-server queueing system with an unlimited buffer, which can be used in the design of energy consumption schemes and as a mathematical model of unreliable real stochastic systems. Customers arrive to the system in a batch Markovian arrival process, the service times are distributed according to the phase law. If the service time of the customer by the server exceeds a certain random value distributed according to the phase law, this server receives assistance from the reserve server from a finite set of reserve servers. In the paper, we calculate the stationary distribution and performance characteristics of the system.

Assigning Priorities (or Not) in Service Systems with Nonlinear Waiting Costs

2021 ◽  
Author(s):  
Huiyin Ouyang ◽  
Nilay Taník Argon ◽  
Serhan Ziya
Keyword(s):  
Service Time ◽  
Queueing System ◽  
Numerical Study ◽  
Arrival Rate ◽  
The Other ◽  
Cost Functions ◽  
Single Server ◽  
Dominant Type ◽  
Fixed Priority ◽  
Poisson Arrivals

For a queueing system with multiple customer types differing in service-time distributions and waiting costs, it is well known that the cµ-rule is optimal if costs for waiting are incurred linearly with time. In this paper, we seek to identify policies that minimize the long-run average cost under nonlinear waiting cost functions within the set of fixed priority policies that only use the type identities of customers independently of the system state. For a single-server queueing system with Poisson arrivals and two or more customer types, we first show that some form of the cµ-rule holds with the caveat that the indices are complex, depending on the arrival rate, higher moments of service time, and proportions of customer types. Under quadratic cost functions, we provide a set of conditions that determine whether to give priority to one type over the other or not to give priority but serve them according to first-come, first-served (FCFS). These conditions lead to useful insights into when strict (and fixed) priority policies should be preferred over FCFS and when they should be avoided. For example, we find that, when traffic is heavy, service times are highly variable, and the customer types are not heterogenous, so then prioritizing one type over the other (especially a proportionally dominant type) would be worse than not assigning any priority. By means of a numerical study, we generate further insights into more specific conditions under which fixed priority policies can be considered as an alternative to FCFS. This paper was accepted by Baris Ata, stochastic models and simulation.

On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1

1980 ◽  
Vol 17 (3) ◽  
pp. 802-813 ◽  
Author(s):  
A. De Meyer ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Service Time ◽  
Busy Period ◽  
Queueing System ◽  
Traffic Intensity

For the distribution function of the busy period in the M/G/l queueing system with traffic intensity less than one it is shown that the tail varies regularly at infinity iff the tail of the service time varies regularly at infinity.

On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1

1980 ◽  
Vol 17 (03) ◽  
pp. 802-813 ◽  
Author(s):  
A. De Meyer ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Service Time ◽  
Busy Period ◽  
Queueing System ◽  
Traffic Intensity

For the distribution function of the busy period in the M/G/l queueing system with traffic intensity less than one it is shown that the tail varies regularly at infinity iff the tail of the service time varies regularly at infinity.

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