Priority Multi-Server Queueing System with Heterogeneous Customers
Valentina Klimenok,Alexander Dudin,Vladimir Vishnevsky +2 more
- 04 Sep 2020
- Vol. 8, Iss: 9, pp 1501
TL;DR: A multi-server queueing system with heterogeneous customers that arrive according to a marked Markovian arrival process, which defines the process of information transmission as the multi-dimensional Markov chain, derive the generator of this chain and compute its stationary distribution.
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Abstract: 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.
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Citations
A Two-Server Queue with Interdependence between Arrival and Service Processes
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TL;DR: A two-server queue with interdependent arrival and service processes is analyzed. The system is described by a Markov chain and the stability condition and stationary distribution are derived.
1
Randomized Threshold Strategy for Providing Flexible Priority in Multi-Server Queueing System with a Marked Markov Arrival Process and Phase-Type Distribution of Service Time
TL;DR: In this paper , the authors analyzed a multi-server queueing system with a marked Markov arrival process of two types of customers and a phase-type distribution of service time depending on the type of customer.
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Abstract: Single queueing nodes are usually described using Kendall’s notation in the form A/S/C where A describes the time between arrivals to the queue, S the size of jobs and C the number of servers at the node.[5][6] Many theorems in queueing theory can be proved by reducing queues to mathematical systems known as Markov chains, first described by Andrey Markov in his 1906 paper.[7] Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory in 1909.[8][9][10] He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920.[11] In Kendall’s notation:
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