Analysis of a Two‐Phase Queueing System with Vacations and Bernoulli Feedback

Abstract

We consider a two‐phase queueing system with server vacations and Bernoulli feedback. Customers arrive at the system according to a Poisson process and receive batch service in the first phase followed by individual services in the second phase. Each customer who completes the individual service returns to the tail of the second phase service queue with probability 1 ? σ. If the system becomes empty at the moment of the completion of the second phase services, the server takes vacations until he finds customers. This type of queueing problem can be easily found in computer and telecommunication systems. By deriving a relationship between the generating functions for the system size at various embedded epochs, we obtain the system size distribution at an arbitrary time. The exhaustive and gated cases for the batch service are considered.     

Study on Transient Queue-Size Distribution in the Finite-Buffer Model with Batch Arrivals and Multiple Vacation Policy

The transient behavior of the finite-buffer queueing model with batch arrivals and generally distributed repeated vacations is analyzed. Such a system has potential applications in modeling the functioning of production systems, computer and telecommunication networks with energy saving mechanism based on cyclic monitoring the queue state (Internet of Things, wireless sensors networks, etc.). Identifying renewal moments in the evolution of the system and applying continuous total probability law, a system of Volterra-type integral equations for the time-dependent queue-size distribution, conditioned by the initial buffer state, is derived. A compact-form solution for the corresponding system written for Laplace transforms is obtained using an algebraic approach based on Korolyuk’s potential method. An illustrative numerical example presenting the impact of the service rate, arrival rate, initial buffer state and single vacation duration on the queue-size distribution is attached as well.     

Perturbation Theory for the Asymptotic Decay Rates in the Queues with Markovian Arrival Process and/or Markovian Service Process

Three kinds of queues with Markovian arrival process and/or Markovian service process, are considered in this paper. In great generality, their basic steady-state distributions have asymptotically exponential tails. We investigate the sensitivity of these asymptotic decay rates to the small entrywise perturbations in the parameter matrices of the Markovian arrival process.     

Stability and queueing time analysis of a reader-writer queue with alternating exhaustive priorities

This paper considers a reader-writer queue with alternating exhaustive priorities. The system can process an unlimited number of readers simultaneously. However, writers have to be processed one at a time. Both readers and writers arrive according to Poisson processes. Writer and reader service times are general iid random variables. There is infinite waiting room for both. The alternating exhaustive priority policy operates as follows. Assume the system is initially idle. The first arriving customer initiates service for the class (readers or writers) to which it belongs. Once processing begins for a given class of customers, this class is served exhaustively, i.e. until no members of that class are left in the system. At this point, if customers of the other class are in the queue, priority switches to this class, and it is served exhaustively. This system is analyzed to produce a stability condition and Laplace-Stieltjes transforms (LSTs) for the steady state queueing times of readers and writers. An example is also given.     

Lattice path counting andM/M/c queueing systems

We apply the lattice path counting method to the analysis of the transientM/M/c queueing system. A closed-form solution is obtained for the probability of exactlyi arrivals andj departures within a time interval of lengtht in anM/M/c queueing system that is empty at the initial time. The derivation of the probability is based on the counting of paths from the origin to(i,j) on thexy-plane, that have exactly r_d x-steps whose depth from the liney=x isd (d=0,1,...,c–1). The closed-form solution has an expression useful for numerical calculation.     

Performance analysis of a GI/M/1 queue with single working vacation

Consider a GI/M/1 queue with single working vacation. During the vacation period, the server works at a lower rate rather than stopping completely, and only takes one vacation each time. Using the matrix analytic approach, the steady-state distributions of the number of customers in the system at both arrival and arbitrary epochs are obtained. Then the closed property of the conditional probability of gamma distribution is proved and using it the waiting time of an arbitrary customer is analyzed. Finally, Some numerical results and effect of critical model parameters on performance measures have been presented.     

Connection admission control for constant bit rate traffic at a multi‐buffer multiplexer using the oldest‐cell‐first discipline

Consider an ATM multiplexer where M input links contend for time slots on an output link which transmits C cells per second. Each input link has its own queue of size B cells. The traffic is delay sensitive so B is small (e.g., B=20). We assume that each of the M input links carries Constant Bit Rate (CBR) traffic from a large number of independent Virtual Connections (VCs) which are subject to jitter. The fluctuations of the aggregate traffic arriving at queue i, i=1,...,M, is modeled by a Poisson process with rate λ_i. The Quality of Service (QoS) of one connection is determined in part by the queueing delay across the multiplexer and the Cell Loss Ratio (CLR) or proportion of cells from this connection lost because the buffer is full. The Oldest‐Customer(Cell)‐First (OCF) discipline is a good compromise between competing protocols like round‐robin queueing or serving the longest queue. The OCF discipline minimizes the total cell delay among all cells arriving at the contending queues. Moreover, the CLR is similar to that obtained by serving the longest queue. We develop QoS formulae for this protocol that can be calculated on‐line for Connection Admission Control (CAC). These formulae follow from a simple new expression for the exact asymptotics of a M/D/1 queue. This revised version was published online in June 2006 with corrections to the Cover Date.     

Telecommunication traffic,queueing models,and subexponential distributions

This article reviews various models within the queueing framework which have been suggested for teletraffic data. Such models aim to capture certain stylised features of the data, such as variability of arrival rates, heavy-tailedness of on- and off-periods and long-range dependence in teletraffic transmission. Subexponential distributions constitute a large class of heavy-tailed distributions, and we investigate their (sometimes disastrous) influence within teletraffic models. We demonstrate some of the above effects in an explorative data analysis of Munich Universities’ intranet data. This revised version was published online in June 2006 with corrections to the Cover Date.     

空竭服务Geom/G/1休假模型

本文用嵌入Markov链的方法证明了Geom／G／1的边界状态变体模型的随机分解定理，同时又证明两种具体模型的结果．