General Service Disciplines Research Articles

Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse.

Randomized pushing-out in the priority queueing with a finite buffer size

Consider the nonpreemptive priority queueing system with two classes of packets. Class-1 packets have priority over class-2 packets. The packets of class 1 (2) arrive into the buffer according to the Poisson process with rate λ1 (λ2), respectively. The service time has the exponential distribution with the same rate μ for each class. The service times are independent of the arrival processes. The buffer has a finite size N and it is shared by both types of customers. If the buffer is full, a new incoming customer of class 1 can push out of the buffer a customer of class 2 with probability α. Note that if α = 1, we retrieve the standard nonrandomized push-out mechanism. The infinite buffer priority queueing has been thoroughly studied in [4, 7, 8]. The case of finite buffer priority queueing has received considerably less attention. The M | M | C | K type finite-buffer nonpreemptive priority queueing with nonrandomized push-out mechanism is analyzed in [5, 6]. Bondi, in [1], considers the M | M | 1 | K type preemptive and nonpreemptive priority queueing with the following buffer management schemes: complete partitioning, complete sharing, and sharing with minimum allocation. Wagner and Krieger, in [9], analyze the M | M | 1 | K type nonpreemptive priority queueing with complete sharing buffer management scheme and with class-dependent service rates. In [2], Cheng and Akyildiz consider the priority queueing with general service-time distributions and a general service discipline function. Most of the above works use recursive relations to solve steady-state Kolmogorov equations. We use a generating function approach, which only requires the solution of a linear system ofN equations. As far as we know, the randomized push-out mechanism is analyzed for the first time.

Smoothed Perturbation Analysis for Single-Server Queues by Some General Service Disciplines

We consider some single-server queues with general service disciplines, where the family of the queueing processes are parameterized by the service time distributions. Through the smoothed perturbation analysis (SPA) technique, we present under some mild conditions a unified approach to give the strongly consistent estimator for the gradient of the steady-state mean sojourn time with respect to the parameter of service time distributions, provided that it exists. Although the implementation of the SPA requires the additional sub-paths in general, the derived estimator is given as suitable for single-run computation. Simulation results are presented for queues with non-preemptive and preemptive-resume priority disciplines which demonstrate the performance of our estimators.

Smoothed Perturbation Analysis for Single-Server Queues by Some General Service Disciplines

We consider some single-server queues with general service disciplines, where the family of the queueing processes are parameterized by the service time distributions. Through the smoothed perturbation analysis (SPA) technique, we present under some mild conditions a unified approach to give the strongly consistent estimator for the gradient of the steady-state mean sojourn time with respect to the parameter of service time distributions, provided that it exists. Although the implementation of the SPA requires the additional sub-paths in general, the derived estimator is given as suitable for single-run computation. Simulation results are presented for queues with non-preemptive and preemptive-resume priority disciplines which demonstrate the performance of our estimators.

On the mean sojourn time of jobs in queues by general service disciplines

Existence and finiteness of the sample-mean limit of sojourn times of jobs in a queueing system are investigated. The queueing system operates under rather general multiprocessor disciplines allowing job classes and priorities. The input stream of jobs consisting of job classes and interarrival and processing times is stationary and ergodic and may contain batch arrivals. Existence of the sample-mean limit is proved by means of the superadditive ergodic theorem, and its finiteness is controlled by uniform mixing of the input stream.

On the mean sojourn time of jobs in queues by general service disciplines

Existence and finiteness of the sample-mean limit of sojourn times of jobs in a queueing system are investigated. The queueing system operates under rather general multiprocessor disciplines allowing job classes and priorities. The input stream of jobs consisting of job classes and interarrival and processing times is stationary and ergodic and may contain batch arrivals. Existence of the sample-mean limit is proved by means of the superadditive ergodic theorem, and its finiteness is controlled by uniform mixing of the input stream.

Analysis of a finite buffer queue with different scheduling and push-out schemes

A queueing system M 1, M 2/ G 1, G 2/1/ N with different scheduling and push-out scheme is analyzed in this paper. This work is motivated by the study of the performance of an output link of ATM switches with traffic of two classes with different priorities. However, the queueing model developed in this paper is more general than that of the output link of ATM switches with two-class priority traffic. General service time distributions are allowed for classes 1 and 2 and a general service discipline function, α 1( i, j), is introduced where α 1( i, j) is the probability that a class 1 packet will be served, given that there are i class 1 and j class 2 packets waiting for service. An exact solution is obtained for the loss probabilities for classes 1 and 2, the queue length distribution and the mean waiting time for class 1. The queue length distribution and the mean waiting time for class 2 are calculated approximately. It is shown that the approximation is an upper bound and the error due to the approximation is very small when the loss probability of class 2 is small (e.g., less than 0.01).

Stability, monotonicity and invariant quantities in general polling systems

Consider a polling system withK≥1 queues and a single server that visits the queues in a cyclic order. The polling discipline in each queue is of general gated-type or exhaustive-type. We assume that in each queue the arrival times form a Poisson process, and that the service times, the walking times, as well as the set-up times form sequences of independent and identically distributed random variables. For such a system, we provide a sufficient condition under which the vector of queue lengths is stable. We treat several criteria for stability: the ergodicity of the process, the geometric ergodicity, and the geometric rate of convergence of the first moment. The ergodicity implies the weak convergence of station times, intervisit times and cycle times. Next, we show that the queue lengths, station times, intervisit times and cycle times are stochastically increasing in arrival rates, in service times, in walking times and in setup times. The stability conditions and the stochastic monotonicity results are extended to the polling systems with additional customer routing between the queues, as well as bulk and correlated arrivals. Finally, we prove that the mean cycle time, the mean intervisit time and the mean station times are invariant under general service disciplines and general stationary arrival and service processes.