Tail Asymptotics for HOL Priority Queues Handling a Large Number of Independent Stationary Sources

In this paper we study the asymptotics of the tail of the buffer occupancy distribution in buffers accessed by a large number of stationary independent sources and which are served according to a strict HOL priority rule. As in the case of single buffers, the results are valid for a very general class of sources which include long-range dependent sources with bounded instantaneous rates. We first consider the case of two buffers with one of them having strict priority over the other and we obtain asymptotic upper bound for the buffer tail probability for lower priority buffer. We discuss the conditions to have asymptotic equivalents. The asymptotics are studied in terms of a scaling parameter which reflects the server speed, buffer level and the number of sources in such a way that the ratios remain constant. The results are then generalized to the case of M buffers which leads to the source pooling idea. We conclude with numerical validation of our formulae against simulations which show that the asymptotic bounds are tight. We also show that the commonly suggested reduced service rate approximation can give extremely low estimates.     

The MAP/(PH/PH)/1 queue with self-generation of priorities and non-preemptive service

Customers arriving according to a Markovian arrival process are served at a single server facility. Waiting customers generate priority at a constant rate γ$\gamma$; such a customer waits in a waiting space of capacity 1 if this waiting space is not already occupied by a priority generated customer; else it leaves the system. A customer in service will be completely served before the priority generated customer is taken for service (non-preemptive service discipline). Only one priority generated customer can wait at a time and a customer generating into priority at that time will have to leave the system in search of emergency service elsewhere. The service times of ordinary and priority generated customers follow PH-distributions. The matrix analytic method is used to compute the steady state distribution. Performance measures such as the probability of n consecutive services of priority generated customers, the probability of the same for ordinary customers, and the mean waiting time of a tagged customer are found by approximating them by their corresponding values in a truncated system. All these results are supported numerically.     

Delay analysis of multiclass queues with correlated train arrivals and a hybrid priority/FIFO scheduling discipline

We analyze the delay experienced in a discrete-time priority queue with a train-arrival process. An infinite user population is considered. Each user occasionally sends packets in the form of trains: a variable number of fixed-length packets is generated and these packets arrive to the queue at the rate of one packet per slot. This is an adequate arrival process model for network traffic. Previous studies assumed two traffic classes, with one class getting priority over the other. We extend these studies to cope with a general number M of traffic classes that can be partitioned in an arbitrary number N of priority classes (1 ≤ N ≤ M). The lengths of the trains are traffic-class-dependent and generally distributed. To cope with the resulting general model, an (M × )∞-sized Markovian state vector is introduced. By using probability generating functions, moments and tail probabilities of the steady-state packet delays of all traffic classes are calculated. Since this study can be useful in deciding how to partition traffic classes in priority classes, we demonstrate the impact of this partitioning for some specific cases.     

MAP 1, MAP2 /M/c retrial queue with guard channels and its application to cellular networks

In this paper, we investigate the impact of retrial phenomenon on loss probabilities and compare loss probabilities of several channel allocation schemes giving higher priority to hand-off calls in the cellular mobile wireless network. In general, two channel allocation schemes giving higher priority to hand-off calls are known; one is the scheme with the guard channels for hand-off calls and the other is the scheme with the priority queue for hand-off calls. For mathematical unified model for both schemes, we consider theMAP ₁,MAP ₂ /M/c/b, ∞ retrial queue with infinite retrial group, geometric loss, guard channels and finite priority queue for hand-off class. We approximate the joint distribution of two queue lengths by Neuts' method and also obtain waiting time distribution for hand-off calls. From these results, we obtain the loss probabilities, the mean waiting time and the mean queue lengths. We give numerical examples to show the impact of the repeated attempt and to compare loss probabilities of channel allocation schemes.     

M/G/1 retrial queueing systems with two types of calls and finite capacity

We consider anM/G/1 priority retrial queueing system with two types of calls which models a telephone switching system and a cellular mobile communication system. In the case that arriving calls are blocked due to the server being busy, type I calls are queued in a priority queue of finite capacityK whereas type II calls enter the retrial group in order to try service again after a random amount of time. In this paper we find the joint generating function of the numbers of calls in the priority queue and the retrial group in closed form. When ₁=0, it is shown that our results are consistent with the known results for a classical retrial queueing system.     

Discrete-Time Geo X /G/1 Queue with Preemptive Repeat Different Priority

This paper considers a discrete-time Geo ^X /G/1 queue accepting two classes of messages with preemptive repeat different priority. Service times of messages of each priority class are i.i.d. according to a general discrete distribution function that may differ between two classes. The completion time and the stability condition for our system are investigated. By using the supplementary variable method and the generating function technique, we derive the joint system contents distributions at various observation instants and also compute the probability distribution for the unfinished work.     

MAP/(PH/PH)/c Queue with Self-Generation of Priorities and Non-Preemptive Service

Abstract

Customers arriving according to a Markovian arrival process are served at a c server facility. Waiting customers generate into priority while waiting in the system (self-generation of priorities), at a constant rate γ; such a customer is immediately taken for service, if at least one of the servers is free. Else it waits at a waiting space of capacity c exclusively for priority generated customers, provided there is vacancy. A customer in service is not preempted to accommodate a priority generated customer. The service times of ordinary and priority generated customers follow distinct PH-distributions. It is proved that the system is always stable. We provide a numerical procedure to compute the optimal number of servers to be employed to minimize the loss to the system. Several performance measures are evaluated.     

Performance analysis of a multiplexer with the threshold based overload control in ATM networks

In this paper we analyze the performance of a statistical ATM multiplexer with bursty input traffic and two thresholds in the buffer by using queueing model. Two priority levels are considered for source traffic, which is modeled by Markov Modulated Poisson Process to represent the bursty characteristics. Service time distributions of two priority sources are assumed to be same and deterministic for ATM environment. The partial buffer sharing scheme with one threshold may experience a sensitive state change around the threshold. But the proposed multiplexer with two thresholds avoids this hysterical phenominon to improve the system operation.     

Delay analysis of a discrete-time multiclass slot-bound priority system

This paper introduces a new priority mechanism in discrete-time queueing systems that compromises between first-come-first-served (FCFS) and head-of-line priority. In this scheduling discipline—which we dubbed slot-bound priority—customers of different priority classes entering the system during the same time-slot are served in order of their respective priority class. Customers entering during different slots are served on a FCFS basis. In this paper we study the delay in an N-class discrete-time queueing system under slot-bound priority. General independent arrivals and class-specific general service time distributions are assumed. Expressions for the probability generating function of the delay of a random type-j customer are derived, from which the respective moments are easily obtained. The tail behaviour of these distributions is analyzed as well, and some numerical examples show the effect slot-bound priority can have on the performance measures.     

An Iterative Algorithm for a Multiple Finite-source Queueing Model with Dynamic Priority Scheduling

A multiple finite source queueing model with a single server and dynamic, non-preemptive priority service discipline is studied in this paper. The times the customers spend at the corresponding sources are exponentially distributed. The service times of the customers can follow exponential, Erlang or hyperexponential probability density function. By using results published earlier and an extension of mean value analysis, an iterative algorithm was developed to obtain approximate values of the mean waiting times in queues for the priority classes. The mean number of waiting customers and the server utilization of each class are obtained using the result of this algorithm and Little's formula. The algorithm is preferable to the earlier method, because it does not increase in complexity as the number of customer classes increases.     

��ǿռ��������ȨM/M/n/m�Ŷ�ϵͳ

Concerning the problem that network congestion risk of computer network service system for some data frames having a full priority of transmission, a method about nonpreemptive limited-priority M/M/n/m queuing system model was proposed. Firstly, as the parameter r of limited-priority was introduced into the model, the data frame with full priority was converted to the one with limited priority. Secondly, in order to lower the risk of computer network service system and stabilize the network system further, the fairness among different priorities was studied in the model. Moreover, by making use of Total Probability Theorem, three results of the models, the average waiting time, the average dwelling time and the average queue length were obtained.     

On distorted probabilities and m-separable fuzzy measures

Fuzzy measures are used in conjunction with fuzzy integrals for aggregation. Their role in the aggregation is to permit the user to express the importance of the information sources (either criteria or experts). Due to the fact that fuzzy measures are set functions, the definition of such measures requires the definition of 2ⁿ parameters, where n is the number of information sources. To make the definition easier, several families of fuzzy measures have been defined in the literature.In this paper m-separable fuzzy measures are introduced. We present some results on this type of measures and we relate them to some of the previous existing ones. We study generating functions for m-separable fuzzy measures and some properties related to these generating functions.     

A production system with two job classes,changeover times and revisitation

This paper, motivated by the need to predict performance of production systems with random arrivals, setup times and revisitation, presents an imbedded Markov chain analysis of the underlyingM/G/1 queue with two customer classes, changeover times and instantaneous Bernoulli feedback. It is assumed that jobs are scheduled according to the exhaustive alternative priority queue discipline. Expressions for the mean waiting time and the nonsaturation condition are derived under two different priority assignments to the repeat customers. Sojourn times under these priority assignments are shown to possess a convex ordering. Results of the study are also applicable to data communication networks that operate under cyclic switching mechanisms. Research supported in part by Natural Sciences and Engineering Research Council of Canada.     

Relative priority policies for minimizing the cost of queueing systems with service discrimination

This paper considers several single-server two-class queueing systems with different cost functions. Customers in the two classes are discriminated by service rates and relative priorities. Most attention is focused on the ones with general quadratic bivariable and exponential cost functions that are usually applied in the relatively complicated systems. To the best of the authors’ knowledge, there is no literature analyzing these two kinds of cost functions on the subject of relative priority. We explicitly present the conditions under which relative priority outperforms absolute priority for reducing system cost and further provide the method to find the optimal DPS policy. Moreover, we also discuss variations where service rates of the two classes are decision variables under service equalization and service discrimination disciplines, respectively.     

非强占优先权下的M/M/1排队

分析带有两个优先权的非强占M/M/1系统的性能,用补充变量法构造向量马尔可夫过程对此排队系统的状态转移方程进行分析,得到两类顾客在非强占优先权的队长联合分布的母函数,进一步讨论,得出了服务台被两类顾客占有和闲置的概率以及两类信元各自的平均队长.     

A modified HOL priority scheduling discipline: Performance analysis

In this paper, we introduce and analyze a modified HOL (head-of-the-line) priority scheduling discipline. The modification is incorporated to cope with the so-called starvation problem of regular HOL priority queues. We consider a discrete-time single-server queueing system with two priority queues of infinite capacity and with the introduced priority scheme. We show that the use of probability generating functions is suitable for analyzing the system contents and the packet delay. Some performance measures (such as means and variances) of these stochastic quantities will be derived. Furthermore, approximate expressions of the tail probabilities are obtained from the probability generating functions, by means of the dominant-singularity method. These expressions, together with their characteristics, constitute one of the main contributions of this paper. Finally, the impact and significance of the m-HOL (modified HOL) priority scheduling on these performance measures is illustrated by some numerical examples.     

Performance of service policies in a specialized service system with parallel servers

This paper considers a scheduling problem occurring in a specialized service system with parallel servers. In the system, customers are divided into the “ordinary” and “special” categories according to their service needs. Ordinary customers can be served by any server, while special customers can be served only by the flexible servers. We assume that the service time for any ordinary customer is the same and all special customers have another common service time. We analyze three classes of service policies used in practice, namely, policies with priority, policies without priority and mixed policies. The worst-case performance ratios are obtained for all of these service policies.     

Reservation and allocation policies for influenza vaccines

This research investigates the impact of alternative allocation mechanisms that can be employed in the context of vaccine inventory rationing. Available vaccine inventory can be allocated to arrivals from high priority (target groups such as healthcare professionals) and low priority (non-target groups) demand classes using Partitioned Allocation (PA), Standard Nesting (SN), and Theft Nesting (TN). In any one of the mechanisms, a part of the available inventory is reserved for the exclusive use of the high priority demand class. They differ, however, in how the unreserved portion of the inventory is utilized: Under PA, demand from the high (low) priority class consumes only the reserved (unreserved) quantity. Under SN, demand from the high priority class first consumes the reserved quantity; once and if this quantity is exhausted, high priority demand competes with low priority demand for the remaining inventory. Under TN the sequence of allocation is reversed: both demand classes first compete for the unreserved inventory. Once this portion of inventory is exhausted, high priority demand is fulfilled from the reserved inventory and low priority demand is rejected. We develop service level (probability of fulfilling the entire demand) and fill rate (fraction of demand fulfilled) expressions for all three allocation mechanisms. Based on these expressions, numerical analyses are conducted to illustrate which allocation mechanism a health planner should choose depending on the availability of vaccines, and how the health planner should set the reserved quantity for the high priority class. We observe that (1) there exist certain conditions under which one of the allocation mechanisms outperforms the others and (2) this effect is determined by the decision maker’s choice of the performance measure.     

Competitive queue policies for differentiated services

We consider the setting of a network providing differentiated services. As is often the case in differentiated services, we assume that the packets are tagged as either being a high priority packet or a low priority packet. Outgoing links in the network are serviced by a single FIFO queue.Our model gives a benefit of α1 to each high priority packet and a benefit of 1 to each low priority packet. A queue policy controls which of the arriving packets are dropped and which enter the queue. Once a packet enters the queue it is eventually sent. The aim of a queue policy is to maximize the sum of the benefits of all the packets it sends.We analyze and compare different queue policies for this problem using the competitive analysis approach, where the benefit of the online policy is compared to the benefit of an optimal offline policy. We derive both upper and lower bounds for the policies we consider. We believe that competitive analysis gives important insight to the performance of these queuing policies.     

A matrix-analytic solution for the DBMAP/PH/1 priority queue

Priority queueing models have been commonly used in telecommunication systems. The development of analytically tractable models to determine their performance is vitally important. The discrete time batch Markovian arrival process (DBMAP) has been widely used to model the source behavior of data traffic, while phase-type (PH) distribution has been extensively applied to model the service time. This paper focuses on the computation of the DBMAP/PH/1 queueing system with priorities, in which the arrival process is considered to be a DBMAP with two priority levels and the service time obeys a discrete PH distribution. Such a queueing model has potential in performance evaluation of computer networks such as video transmission over wireless networks and priority scheduling in ATM or TDMA networks. Based on matrix-analytic methods, we develop computation algorithms for obtaining the stationary distribution of the system numbers and further deriving the key performance indices of the DBMAP/PH/1 priority queue. AMS subject classifications: 60K25 · 90B22 · 68M20 The work was supported in part by grants from RGC under the contracts HKUST6104/04E, HKUST6275/04E and HKUST6165/05E, a grant from NSFC/RGC under the contract N_HKUST605/02, a grant from NSF China under the contract 60429202.