Finding Expected Revenue of Open HM-Network with Limited Waiting Time and Unreliable Queueing Systems
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ORIGINAL RESEARCH
Finding Expected Revenue of Open HM‑Network with Limited Waiting Time and Unreliable Queueing Systems D. Kopats1 Received: 31 May 2020 / Accepted: 14 July 2020 © Springer Nature Singapore Pte Ltd 2020
Abstract The object of research in this paper is an open Markov queueing network with revenues, limited number of customers, random waiting time for them in the queue, and unreliable queueing systems. The service times of customers in systems and the recovery times of lines in them have an exponential distribution with parameters depending on the number of customers in the systems and the number of service lines in service, and the waiting times in queues and service hours are also distributed according to an exponential law distribution, the parameters of which depend on the number of customers in systems. Using an asymptotic analysis with a large number of customers, the total expected revenue of such network was found. To do this, first derived and solved the differential equation in partial derivatives of the density distribution of revenue, and then received and decided to ordinary differential equation for the expected revenue of the network. A model example is calculated. Keywords HM-queueing network · Expected revenues · Density distribution of revenues · Partial differential equation
Introduction QN with revenues, limited number of customers, random waiting time for them in the queue, and unreliable queueing systems apply in when modeling the system of interbank payments [1], collective income forecasting in information systems [2], workflow system modeling [2]. Consider an open Markov QN, total number of customers is limited by K, consist of n queueing systems (QS) S1 , ..., Sn . System Si consist of mi identical service lines, i = 1, n . The network state is described by the vector
𝛈(t) = (𝐝, 𝐤, t) = (d1 , d2 , ..., dn , k1 , k2 , ..., kn , t),
(1)
where di , ki - the number of serviceable lines and number of customers in the i-th QS respectively at time t,0 ≤ di (t) ≤ mi (t) , 0 ≤ ki (t) ≤ K, t = [0;∞), i = 1, n. In system arrive exponential customers flow with the rate 𝜆0i , i = 1, n. If customer in the system Sj finds at least one service line operable and free from the other customers it is immediately serviced. The time of service is a random * D. Kopats [email protected] 1
variable (RV) with parameter 𝜇i (di , ki , t), i = 1, n. In other case, customer begin servicing without limited of waiting time. Waiting time is limited by RV and have the exponential distribution with parameters 𝜃i (ki , t), i = 1, n. After finishing waiting time in queue Si , customer move to system Sj with probability qij , qii = 0, i = 1, n, j = 0, n. Service lines are subject to accidental breakdowns, and the uptime of each line in the system Si and have the exponential distribution with the parameter 𝛽i (ki , t), i = 1, n . After a breakdown, the line immediately begins to recover and the recovery time also has an exponential distribution with the parameter 𝛾i (di , ki , t), i = 1, n. After finishin
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