Diffusion approximations for double-ended queues with reneging in heavy traffic

Abstract

We study a double-ended queue which consists of two classes of customers. Whenever there is a pair of customers from both classes, they are matched and leave the system immediately. The matching follows first-come-first-serve principle. If a customer from one class cannot be matched immediately, he/she will stay in a queue and wait for the upcoming arrivals from the other class. Thus there cannot be non-zero numbers of customers from both classes simultaneously in the system. We also assume that each customer can leave the queue without being matched because of impatience. The arrival processes are assumed to be independent renewal processes, and the patience times for both classes are generally distributed. Under suitable heavy traffic conditions, assuming that the diffusion-scaled queue length process is stochastically bounded, we establish a simple asymptotic relationship between the diffusion-scaled queue length process and the diffusion-scaled offered waiting time processes, and further show that the diffusion-scaled queue length process converges weakly to a diffusion process. We also provide a sufficient condition for the stochastic boundedness of the diffusion-scaled queue length process. At last, the explicit form of the stationary distribution of the limit diffusion process is derived.AMS 2010 subject classifications: Primary: 60F05, 60K25, 90B22; Secondary: 60K05.

Admission Control for Double-ended Queues

“…Without loss of generality, we assume that the initial number of sellers is given by X n (0−) ≥ 0. We refer to [26,25] and [11] for similar representations of the state process. Let the quantities G n s (t) and G n b (t) represent the number of sellers and buyers abandon the system during [0, t], respectively.…”

“…[8,12,33,41]). In particular, the diffusion approximations of the uncontrolled double-ended queues have been investigated in [26,25]. In [26], diffusion approximations for double-ended queues with renewal arrivals and exponential patience times are studied.…”

“…In [26], diffusion approximations for double-ended queues with renewal arrivals and exponential patience times are studied. This work is extended in [25] to a setting with a general abandonment structure. In addition, a linear relationship between the the diffusion-scaled queue length and offered waiting time (asymptotic Little's law) is established and the unique stationary distribution for the limiting diffusion process is also obtained in [25].…”

Optimal Control of a Time-Varying Double-Ended Production Queueing Model

“…In Liu et al (2015), rigorous fluid and diffusion models were developed for double-ended queues with renewal arrivals and exponential patience times. Later on, Liu (2019) established the heavy traffic asymptotics for the system with renewal arrivals and generally distributed patience times. Double-ended queues are the simplest matching queues.…”

Optimal Control of a Time-Varying Double-Ended Production Queueing Model

“…In [18], rigorous fluid and diffusion models were developed for double-ended queues with renewal arrivals and exponential patience times. Later on, [17] establishes the heavy traffic asymptotics for the system with renewal arrivals and generally distributed patience times.…”