Equilibrium points and equilibrium sets of some $$\textit{GI}/M/1$$ GI / M
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Equilibrium points and equilibrium sets of some GI/M/1 queues N. Hemachandra1 · Kishor Patil2
· Sandhya Tripathi3
Received: 31 October 2019 / Revised: 7 November 2020 / Accepted: 12 November 2020 © The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature 2020
Abstract Queues can be seen as a service facility where quality of service (QoS) is an important measure for the performance of the system. In many cases, the queue implements the optimal admission control (either discounted or average) policy in the presence of holding/congestion cost and revenue collected from admitted customers. In this paper, users offer an arrival rate at stationarity that depends on the QoS they experience. We study the interaction between arriving customers and such a queue under two different QoS measures—the asymptotic rate of the customers lost and the fraction of customers lost in the long run. In particular, we investigate the behaviour of equilibrium points and equilibrium sets associated with this interaction and their interpretations in terms of business cycles. We provide sufficient conditions for existence of equilibrium sets for M/M/1 queue. These conditions further help us to identify the relationship among system parameters for which equilibrium sets exist. Next, we consider GI/M/1 queues and provide a sufficient condition for existence of multiple optimal revenue policies. We then specialize these results to study the equilibrium sets of (i) a D/M/1 queue and (ii) a queue where the arrival rate is locally continuous. The equilibrium behaviour in the latter case is more interesting as there may be multiple equilibrium points or sets. Motivated by such queues, we introduce a weaker version of monotonicity and investigate the existence of generalized equilibrium sets. Keywords Admission control of queues · Quality of service · Multiple optimal control limits · Fixed points · Threshold policies · Inter-arrival time distribution · Business cycles Mathematics Subject Classification 60K30 · 90B22 · 91A10 · 91A11 · 47H10 · 90C40 · 91A80
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Queueing Systems
1 Introduction We view a queue as a service-provider and consider a situation when the customer base (user-set) offers an arrival rate at stationarity that depends on the Quality of Service (QoS) they experience. The users are typically concerned about price and service levels they get. As a result, the queue uses a revenue-optimal policy based on the QoS experienced by the user-set. We are primarily interested in equilibrium behaviour associated with the interaction between such a queue and its user-set. Many present-day systems such as communication networks, transportation systems, etc., have congestion which is properly captured by such queueing models. The queue is generally interested in the revenue accrued to it, whereas the user-set is interested in the QoS it experiences. Both are independent entities making their own decisions. However, their rew
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