Analysis of Multi-server Loss Queueing System with the Batch Marked Markov Arrival Process

2021 ◽  
pp. 182-195
Author(s):  
Chesoong Kim ◽  
Alexander Dudin ◽  
Sergey Dudin ◽  
Olga Dudina
Keyword(s):  
Queueing System ◽  
Arrival Process ◽  
Markov Arrival ◽  
Markov Arrival Process ◽  
Multi Server

scholarly journals Analysis of Multi-Server Queue with Self-Sustained Servers

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2134
Author(s):  
Alexander Dudin ◽  
Olga Dudina ◽  
Sergei Dudin ◽  
Konstantin Samouylov
Keyword(s):  
Three Dimensional ◽  
Optimal Choice ◽  
Arrival Process ◽  
Queuing Model ◽  
Numerical Result ◽  
Self Sufficiency ◽  
Server Queue ◽  
Markov Arrival ◽  
Markov Arrival Process ◽  
Multi Server

A novel multi-server vacation queuing model is considered. The distinguishing feature of the model, compared to the standard queues, is the self-sufficiency of servers. A server can terminate service and go on vacation independently of the system manager and the overall situation in the system. The system manager can make decisions whether to allow the server to start work after vacation completion and when to try returning some server from a vacation to process customers. The arrival flow is defined by a general batch Markov arrival process. The problem of optimal choice of the total number of servers and the thresholds defining decisions of the manager arises. To solve this problem, the behavior of the system is described by the three-dimensional Markov chain with the special block structure of the generator. Conditions for the ergodicity of this chain are derived, the problem of computation of the steady-state distribution of the chain is discussed. Expressions for the key performance indicators of the system in terms of the distribution of the chain states are derived. An illustrative numerical result is presented.

Two-Phase Queueing System with a Markov Arrival Process and Blocking

2006 ◽  
Vol 132 (5) ◽  
pp. 578-589 ◽  
Author(s):  
P. P. Bocharov ◽  
R. Manzo ◽  
A. V. Pechinkin
Keyword(s):  
Queueing System ◽  
Arrival Process ◽  
Two Phase ◽  
Markov Arrival ◽  
Markov Arrival Process

Analysis of a Two-Phase Queueing System with a Markov Arrival Process and Losses

2005 ◽  
Vol 131 (3) ◽  
pp. 5606-5613 ◽  
Author(s):  
P. P. Bocharov ◽  
R. Manzo ◽  
A. V. Pechinkin
Keyword(s):  
Queueing System ◽  
Arrival Process ◽  
Two Phase ◽  
Markov Arrival ◽  
Markov Arrival Process

COMPARISON OF QUEUES WITH DIFFERENT DISCRETE-TIME ARRIVAL PROCESSES

2001 ◽  
Vol 15 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Arie Hordijk
Keyword(s):  
Lower Bound ◽  
Discrete Time ◽  
Sojourn Time ◽  
Arrival Process ◽  
Convex Ordering ◽  
Arrival Processes ◽  
Markov Arrival ◽  
Markov Arrival Process ◽  
Special Case ◽  
Stochastic Event

Traveling times in a FIFO-stochastic event graph are compared in increasing convex ordering for different arrival processes. As a special case, a stochastic lower bound is obtained for the sojourn time in a tandem network of FIFO queues with a Markov arrival process. A counterexample shows that the extended Ross conjecture is not true for discrete-time arrival processes.

scholarly journals Priority Multi-Server Queueing System with Heterogeneous Customers

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1501
Author(s):  
Valentina Klimenok ◽  
Alexander Dudin ◽  
Vladimir Vishnevsky
Keyword(s):  
Information Transmission ◽  
Performance Measures ◽  
Queueing System ◽  
Real Life ◽  
Telecommunication Networks ◽  
Arrival Process ◽  
Phase Type Distribution ◽  
Arrival Times ◽  
Process Measurements ◽  
Multi Server

In this paper, we analyze a multi-server queueing system with heterogeneous customers that arrive according to a marked Markovian arrival process. Customers of two types differ in priorities and parameters of phase type distribution of their service time. The queue under consideration can be used to model the processes of information transmission in telecommunication networks in which often the flow of information is the superposition of several types of flows with correlation of inter-arrival times within each flow and cross-correlation. We define the process of information transmission as the multi-dimensional Markov chain, derive the generator of this chain and compute its stationary distribution. Expressions for computation of various performance measures of the system, including the probabilities of loss of customers of different types, are presented. Output flow from the system is characterized. The presented numerical results confirm the high importance of account of correlation in the arrival process. The values of important performance measures for the systems with the correlated arrival process are essentially different from the corresponding values for the systems with the stationary Poisson arrival process. Measurements in many real world systems show poor approximation of real flows by such an arrival process. However, this process is still popular among the telecommunication engineers due to the evident existing gap between the needs of adequately modeling the real life systems and the current state of the theory of algorithmic methods of queueing theory.

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.

scholarly journals Analysis of an MMAP/PH1, PH2/N/∞ queueing system operating in a random environment

2014 ◽  
Vol 24 (3) ◽  
pp. 485-501 ◽  
Author(s):  
Chesoong Kim ◽  
Alexander Dudin ◽  
Sergey Dudin ◽  
Olga Dudina
Keyword(s):  
Random Environment ◽  
Queueing System ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Contact Center ◽  
Phase Type Distribution ◽  
Phase Type ◽  
Multi Server

Abstract A multi-server queueing system with two types of customers and an infinite buffer operating in a random environment as a model of a contact center is investigated. The arrival flow of customers is described by a marked Markovian arrival process. Type 1 customers have a non-preemptive priority over type 2 customers and can leave the buffer due to a lack of service. The service times of different type customers have a phase-type distribution with different parameters. To facilitate the investigation of the system we use a generalized phase-type service time distribution. The criterion of ergodicity for a multi-dimensional Markov chain describing the behavior of the system and the algorithm for computation of its steady-state distribution are outlined. Some key performance measures are calculated. The Laplace-Stieltjes transforms of the sojourn and waiting time distributions of priority and non-priority customers are derived. A numerical example illustrating the importance of taking into account the correlation in the arrival process is presented

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