Equilibrium in a finite capacity M / M /1 queue with unknown service rates consisting of strategic and non-strategic cus
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Equilibrium in a finite capacity M/M/1 queue with unknown service rates consisting of strategic and non-strategic customers S. Srivatsa Srinivas1
· Rahul R. Marathe1
Received: 29 September 2019 / Revised: 26 September 2020 / Accepted: 29 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider an M/M/1/N observable non-customer-intensive service queueing system with unknown service rates consisting of strategic impatient customers who make balking decisions and non-strategic patient customers who do not make any decision. In the queueing game amongst the impatient customers, we show that there exists at least one pure threshold strategy equilibrium in the presence of patient customers. As multiple pure threshold strategy equilibria exist in certain cases, we consider the minimal pure threshold strategy equilibrium in our sensitivity analysis. We find that the likelihood ratio of a fast server to a slow server in an empty queue is monotonically decreasing in the proportion of impatient customers and monotonically increasing in the waiting area capacity. Further, we find that the minimal pure threshold strategy equilibrium is non-increasing in the proportion of impatient customers and non-decreasing in the waiting area capacity. We also show that at least one pure threshold strategy equilibrium exists when the waiting area capacity is infinite. Keywords Queueing game · Service operations · Strategic behavior · Threshold strategy equilibrium Mathematics Subject Classification 60K25 · 90B22 · 91A40
1 Introduction We consider a finite capacity observable M/M/1 queueing system, which provides non-customer-intensive service and follows a first-in-first-out (FIFO) queue discipline.
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Rahul R. Marathe [email protected] S. Srivatsa Srinivas [email protected]
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Department of Management Studies, Indian Institute of Technology Madras, Chennai, India
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Queueing Systems
In contrast to customer-intensive services where service value increases with service time, service value decreases with service time in non-customer-intensive services. The arriving customers are of two types—(1) impatient customers who are strategic and make balking decisions, (2) patient customers who are non-strategic, always join the queue, and do not make any decision at all. Nature first probabilistically selects the service rate (fast or slow). The impatient customers then update their beliefs on the service rate by observing the queue and then make balking decisions based on the net utility from the service. A classic example of our setting is the queue in front of a ticket purchase counter in a railway station on the Mumbai suburban railway [22]. Consider the passengers waiting in a queue to purchase a ticket from a human server at the Churchgate station on the Western line. A passenger traveling from Churchgate to Dadar (approximately 11 km) is impatient, updates the belief on the human server’s rate of service based on the observed queue length, and balks in the case of a long queue a
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