Browsing by Subject "Semi-Markov process"

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    Erlang Loss Formulas: An Elementary Derivation
    (Springer, 2021-08) Sarkar, Jyotirmoy; Mathematical Sciences, School of Science
    The celebrated Erlang loss formulas, which express the probability that exactly j of c available channels/servers are busy serving customers, were discovered about 100 years ago. Today we ask: “What is the simplest proof of these formulas?” As an alternative to more advanced methods, we derive the Erlang loss formulas using (1) an intuitive limit theorem of an alternating renewal process and (2) recursive relations that are solved using mathematical induction. Thus, we make the Erlang loss formulas comprehensible to beginning college mathematics students. We illustrate decision making in some practical problems using these formulas and other quantities derived from them.
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    A Repairable System Supported by Two Spare Units and Serviced by Two Types of Repairers
    (Atlantis Press, 2021-06) Andalib, Vahid; Sarkar, Jyotirmoy; Mathematical Sciences, School of Science
    We study a one-unit repairable system, supported by two identical spare units on cold standby, and serviced by two types of repairers. The model applies, for instance, to ANSI (American National Standard Institute) centrifugal pumps in a chemical plant, and hydraulic systems in aviation industry. The failed unit undergoes repair either by an in-house repairer within a random or deterministic patience time, or else by a visiting expert repairer. The expert repairs one or all failed units before leaving, and does so faster but at a higher cost rate than the regular repairer. Four models arise depending on the number of repairs done by the expert and the nature of the patience time. We compare these models based on the limiting availability , and the limiting profit per unit time , using semi-Markov processes, when all distributions are exponential. As anticipated, to maximize , the expert should repair all failed units. To maximize a suitably chosen deterministic patience time is better than a random patience time. Furthermore, given all cost parameters, we determine the optimum number of repairs the expert should complete, and the optimum patience time given to the regular repairer in order to maximize.