TY - JOUR
T1 - Convergence of a queueing system in heavy traffic with general patience-time distributions
AU - Lee, Chihoon
AU - Weerasinghe, Ananda
PY - 2011/11
Y1 - 2011/11
N2 - We analyze a sequence of single-server queueing systems with impatient customers in heavy traffic. Our state process is the offered waiting time, and the customer arrival process has a state dependent intensity. Service times and customer patient-times are independent; i.i.d. with general distributions subject to mild constraints. We establish the heavy traffic approximation for the scaled offered waiting time process and obtain a diffusion process as the heavy traffic limit. The drift coefficient of this limiting diffusion is influenced by the sequence of patience-time distributions in a non-linear fashion. We also establish an asymptotic relationship between the scaled version of offered waiting time and queue-length. As a consequence, we obtain the heavy traffic limit of the scaled queue-length. We introduce an infinite-horizon discounted cost functional whose running cost depends on the offered waiting time and server idle time processes. Under mild assumptions, we show that the expected value of this cost functional for the n-th system converges to that of the limiting diffusion process as n tends to infinity.
AB - We analyze a sequence of single-server queueing systems with impatient customers in heavy traffic. Our state process is the offered waiting time, and the customer arrival process has a state dependent intensity. Service times and customer patient-times are independent; i.i.d. with general distributions subject to mild constraints. We establish the heavy traffic approximation for the scaled offered waiting time process and obtain a diffusion process as the heavy traffic limit. The drift coefficient of this limiting diffusion is influenced by the sequence of patience-time distributions in a non-linear fashion. We also establish an asymptotic relationship between the scaled version of offered waiting time and queue-length. As a consequence, we obtain the heavy traffic limit of the scaled queue-length. We introduce an infinite-horizon discounted cost functional whose running cost depends on the offered waiting time and server idle time processes. Under mild assumptions, we show that the expected value of this cost functional for the n-th system converges to that of the limiting diffusion process as n tends to infinity.
KW - Controlled queueing systems
KW - Customer abandonment
KW - Customer impatience
KW - Diffusion approximations
KW - Heavy traffic theory
KW - Reneging
KW - Stochastic control
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U2 - 10.1016/j.spa.2011.07.003
DO - 10.1016/j.spa.2011.07.003
M3 - Article
AN - SCOPUS:80052505462
SN - 0304-4149
VL - 121
SP - 2507
EP - 2552
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
IS - 11
ER -