Heavy-traffic analysis of k-limited polling systems

In this paper we study a two-queue polling model with zero switch-over times and $k$-limited service (serve at most $k_i$ customers during one visit period to queue $i$, $i=1,2$) in each queue. The arrival processes at the two queues are Poisson, and the service times are exponentially distributed. By increasing the arrival intensities until one of the queues becomes critically loaded, we derive exact heavy-traffic limits for the joint queue-length distribution using a singular-perturbation technique. It turns out that the number of customers in the stable queue has the same distribution as the number of customers in a vacation system with Erlang-$k_2$ distributed vacations. The queue-length distribution of the critically loaded queue, after applying an appropriate scaling, is exponentially distributed. Finally, we show that the two queue-length processes are independent in heavy traffic

Queueing models for appointment-driven systems.

Many service systems are appointment-driven. In such systems, customers make an appointment and join an external queue(also referred to as the “waiting list”). At the appointed date, the customer arrives at the service facility, joins an internal queue and receives service during a service session. After service, the customer leaves the system. Important measures of interest include the size of the waiting list, the waiting time at the service facility and server overtime. These performance measures may support strategic decisionmaking concerning server capacity (e.g. how often, when and for how long should a server be online). We develop an ew model to assess these performance measures. The model is a combination of a vacation queueing system and an appointment system.Queueing system; Appointment system; Vacation model; Overtime; Waiting list;

Queueing models for token and slotted ring networks

Currently the end-to-end delay characteristics of very high speed local area networks are not well understood. The transmission speed of computer networks is increasing, and local area networks especially are finding increasing use in real time systems. Ring networks operation is generally well understood for both token rings and slotted rings. There is, however, a severe lack of queueing models for high layer operation. There are several factors which contribute to the processing delay of a packet, as opposed to the transmission delay, e.g., packet priority, its length, the user load, the processor load, the use of priority preemption, the use of preemption at packet reception, the number of processors, the number of protocol processing layers, the speed of each processor, and queue length limitations. Currently existing medium access queueing models are extended by adding modeling techniques which will handle exhaustive limited service both with and without priority traffic, and modeling capabilities are extended into the upper layers of the OSI model. Some of the model are parameterized solution methods, since it is shown that certain models do not exist as parameterized solutions, but rather as solution methods

Iterative approximation of k-limited polling systems

The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) k-limited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowledge of the queue length distributions is needed to operate the system properly. The multi-queue polling system is decomposed into single-queue vacation systems with k-limited service and state-dependent vacations, for which the vacation distributions are computed in an iterative approximate manner. These vacation models are analyzed via matrix-analytic techniques. The accuracy of the approximation scheme is verified by means of an extensive simulation study. The developed approximation turns out be accurate, robust and computationally efficient

Two queues with random time-limited polling

In this paper, we analyse a single server polling model with two queues. Customers arrive at the two queues according to two independent Poisson processes. There is a single server that serves both queues with generally distributed service times. The server spends an exponentially distributed amount of time in each queue. After the completion of this residing time, the server instantaneously switches to the other queue, i.e., there is no switch-over time. For this polling model we derive the steady-state marginal workload distribution, as well as heavy traffic and heavy tail asymptotic results. Furthermore, we also calculate the joint queue length distribution for the special case of exponentially distributed service times using singular perturbation analysis

Queueing system with vacations after a random amount of work

This paper considers an M/G/1 queue with the following vacation discipline. The server takes a vacation as soon as it has served a certain amount of work since the end of the previous vacation. If the system becomes empty before the server has completed this amount of work, then it stays idle until the next customer arrival and then becomes active again. Such a vacation discipline arises, for example, in the maintenance of production systems, where machines or equipment mainly degrade while being operational. We derive an explicit expression for the distribution of the time it takes until the prespecified amount of work has been served. For the case the total amount of work till vacation is exponentially distributed, we derive the transforms of the steady-state workload at various epochs, busy period, waiting time, sojourn time, and queue length distributions