16 research outputs found
Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime. Asymptotics of the stationary distribution
We consider a heterogeneous queueing system consisting of one large pool of
identical servers, where is the scaling parameter. The
arriving customers belong to one of several classes which determines the
service times in the distributional sense. The system is heavily loaded in the
Halfin-Whitt sense, namely the nominal utilization is where
is the spare capacity parameter. Our goal is to obtain bounds on the
steady state performance metrics such as the number of customers waiting in the
queue . While there is a rich literature on deriving process level
(transient) scaling limits for such systems, the results for steady state are
primarily limited to the single class case.
This paper is the first one to address the case of heterogeneity in the
steady state regime. Moreover, our results hold for any service policy which
does not admit server idling when there are customers waiting in the queue. We
assume that the interarrival and service times have exponential distribution,
and that customers of each class may abandon while waiting in the queue at a
certain rate (which may be zero). We obtain upper bounds of the form
on both and the number of idle servers. The bounds
are uniform w.r.t. parameter and the service policy. In particular, we show
that . Therefore, the
sequence is tight and has a uniform exponential tail
bound. We further consider the system with strictly positive abandonment rates,
and show that in this case every weak limit of
has a sub-Gaussian tail. Namely .Comment: 21 page
Holistic assessment of call centre performance
In modern call centres 60–70% of the operational costs come in the form of the human agents who take the calls. Ensuring that the call centre operates at lowest cost and maximum efficiency involves a trade‐off of the cost of agents against lost revenue and increased customer dissatisfaction due to lost calls. Modelling the performance characteristics of a call centre in terms of the agent queue alone misses key performance influencers, specifically the interaction between channel availability at the media gateway and the time a call is queued. A blocking probability at the media gateway, as low as 0.45%, has a significant impact on the degree of queuing observed and therefore the cost and performance of the call centre. Our analysis also shows how abandonment impacts queuing delay. However, the call centre manager has less control over this than the level of contention at the media gateway. Our commercial assessment provides an evaluation of the balance between abandonment and contention, and shows that the difference in cost between the best and worst strategy is £130K per annum, however this must be balanced against a possible additional £2.98 m exposure in lost calls if abandonment alone is used
A computational approach to steady-state convergence of fluid limits for Coxian queuing networks with abandonment
Many-server queuing networks with general service and abandonment times have proven to be a realistic model for scenarios such as call centers and health-care systems. The presence of abandonment makes analytical treatment difficult for general topologies. Hence, such networks are usually studied by means of fluid limits. The current state of the art, however, suffers from two drawbacks. First, convergence to a fluid limit has been established only for the transient, but not for the steady state regime. Second, in the case of general distributed service and abandonment times, convergence to a fluid limit has been either established only for a single queue, or has been given by means of a system of coupled integral equations which does not allow for a numerical solution. By making the mild assumption of Coxian-distributed service and abandonment times, in this paper we address both drawbacks by establishing convergence in probability to a system of coupled ordinary differential equations (ODEs) using the theory of Kurtz. The presence of abandonments leads in many cases to ODE systems with a global attractor, which is known to be a sufficient condition for the fluid and the stochastic steady state to coincide in the limiting regime. The fact that our ODE systems are piecewise affine enables a computational method for establishing the presence of a global attractor, based on a solution of a system of linear matrix inequalities
Diffusion Models for Double-ended Queues with Renewal Arrival Processes
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
Fluid Models of Many-server Queues with Abandonment
We study many-server queues with abandonment in which customers have general
service and patience time distributions. The dynamics of the system are modeled
using measure- valued processes, to keep track of the residual service and
patience times of each customer. Deterministic fluid models are established to
provide first-order approximation for this model. The fluid model solution,
which is proved to uniquely exists, serves as the fluid limit of the
many-server queue, as the number of servers becomes large. Based on the fluid
model solution, first-order approximations for various performance quantities
are proposed
Many-server queues with customer abandonment: numerical analysis of their diffusion models
We use multidimensional diffusion processes to approximate the dynamics of a
queue served by many parallel servers. The queue is served in the
first-in-first-out (FIFO) order and the customers waiting in queue may abandon
the system without service. Two diffusion models are proposed in this paper.
They differ in how the patience time distribution is built into them. The first
diffusion model uses the patience time density at zero and the second one uses
the entire patience time distribution. To analyze these diffusion models, we
develop a numerical algorithm for computing the stationary distribution of such
a diffusion process. A crucial part of the algorithm is to choose an
appropriate reference density. Using a conjecture on the tail behavior of a
limit queue length process, we propose a systematic approach to constructing a
reference density. With the proposed reference density, the algorithm is shown
to converge quickly in numerical experiments. These experiments also show that
the diffusion models are good approximations for many-server queues, sometimes
for queues with as few as twenty servers