76 research outputs found

    Convexity Properties and Comparative Statics for M/M/S Queues with Balking and Reneging

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    We use sample path arguments to derive convexity properties of an M/M/S queue with impatient customers that balk and renege. First, assuming that the balking probability and reneging rate are increasing and concave in the total number of customers in the system (head-count), we prove that the expected head-count is convex decreasing in the capacity (service rate). Second, with linear reneging and balking, we show that the expected lost sales rate is convex decreasing in the capacity. Finally, we employ a sample-path sub-modularity approach to comparative statics. That is, we employ sample path arguments to show how the optimal capacity changes as we vary the parameters of customer demand and impatience. We find that the optimal capacity increases in the demand rate and decreases with the balking probability, but is not monotone in the reneging rate. This means, surprisingly, that failure to account for customersâ reneging may result in over-investment in capacity. Finally, we show that a seemingly minor change in system structure, customer commitment during service, produces qualitatively different convexity properties and comparative statics.Operations Management Working Papers Serie

    A note on the central limit theorem for a one-sided reflected Ornstein-Uhlenbeck process

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    In this short communication we present a (functional) central limit theorem for the idle process of a one-sided reflected Ornstein-Uhlenbeck proces

    Diffusion Models for Double-ended Queues with Renewal Arrival Processes

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    We study a double-ended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be non-zero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an ordinary differential equation, and the diffusion limit is a time-inhomogeneous asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis is also developed, and the diffusion limit in the stronger heavy traffic regime is a time-homogeneous asymmetric O-U process. The limiting distributions of both diffusion limits are obtained. We also show the interchange of the heavy traffic and steady state limits

    BRAVO for many-server QED systems with finite buffers

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    This paper demonstrates the occurrence of the feature called BRAVO (Balancing Reduces Asymptotic Variance of Output) for the departure process of a finite-buffer Markovian many-server system in the QED (Quality and Efficiency-Driven) heavy-traffic regime. The results are based on evaluating the limit of a formula for the asymptotic variance of death counts in finite birth--death processes

    Engineering Solution of a Basic Call-Center Model

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    Convexity Properties and Comparative Statics for M/M/S Queues with Balking and Reneging

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    We use sample path arguments to derive convexity properties of an M/M/S queue with impatient customers that balk and renege. First, assuming that the balking probability and reneging rate are increasing and concave in the total number of customers in the system (head-count), we prove that the expected head-count is convex decreasing in the capacity (service rate). Second, with linear reneging and balking, we show that the expected lost sales rate is convex decreasing in the capacity. Finally, we employ a sample-path sub-modularity approach to comparative statics. That is, we employ sample path arguments to show how the optimal capacity changes as we vary the parameters of customer demand and impatience. We find that the optimal capacity increases in the demand rate and decreases with the balking probability, but is not monotone in the reneging rate. This means, surprisingly, that failure to account for customersâ reneging may result in over-investment in capacity. Finally, we show that a seemingly minor change in system structure, customer commitment during service, produces qualitatively different convexity properties and comparative statics.Operations Management Working Papers Serie

    On the generalized drift Skorokhod problem in one dimension

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    We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/N queue in a many-server heavy traffic regime
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