We present a comparison of the service disciplines in real-time queueing systems (the customers have a deadline before which they should enter the service booth). We state that giving priority to customers having an early deadline minimizes the average stationary lateness. We show this result by comparing adequate random vectors with the Schur-Convex majorization ordering.
The shortest-remaining-processing-time (SRPT) scheduling policy has been extensively studied, for more than 50 years, in single-server queues with infinitely patient jobs. Yet, much less is known about its performance in multiserver queues. In this paper, we present the first theoretical analysis of SRPT in multiserver queues with abandonment. In particular, we consider the M/GI/s+GI queue and demonstrate that, in the many-sever overloaded regime, performance in the SRPT queue is equivalent, asymptotically in steady state, to a preemptive two-class priority queue where customers with short ser…
Queueing systems with batch Markovian arrival process (BMAP) have paramount applications in the domain of wireless communication. The BMAP has been used to model the superposition of video sources and to approximate the super-position of data, voice and video traffic. This paper analyzes an infinite-buffer generally distributed batch-service queue with BMAP, general bulk service ($a,b$) rule and batch-size-dependent service time. In this proposed analysis, we mainly focus on deriving the bivariate vector generating function of queue and server content distribution together at departure epoch u…
The mathematics of the finite single server queue with Poisson input and semi-Markov service times($M/SM/1/b$) is similar to that used for $BMAP/G/1/b$ systems. This observation results in new analytical formulas for a queue size in the $M/SM/1/b$ system. Both stationary and the transient solutions are considered
We study a single server queue under a processor-sharing type of scheduling policy, where the weights for determining the sharing are given by functions of each job's remaining service(processing) amount, and obtain a fluid limit for the scaled measure-valued system descriptors.
加入最短队列（Join the Shortest queue，JSQ）是指网络中的输入作业被分配到系统中随机选择的队列子集中的最短队列。在队列中完成服务后，作业离开网络。我们表明，对于所有的非空闲服务学科和一般的到达间隔和服务时间分布，这样的网络是稳定的，当他们是亚临界的。然后，我们得到了均匀的边界上的家庭的平衡点的边缘分布的尾部，这些界限是用来显示相对紧凑的边缘分布。我们还提出了一个家庭的亚临界JSQ网络，其工作量在平衡是远远大于相应的网络，其中每个传入的工作被随机分配到一个队列。这项工作的一部分推广了Foss和Chernova [12]的结果，该结果应用了流体极限来研究具有FIFO规则的网络。在这里，我们应用适当的李雅普诺夫函数。
The main target of this paper is to present the Markov chain C that, not giving explicitly the queue lengths stationary probabilities, has the necessary information to its determination for open networks of queues with several classes of customers and exponential service times, allowing to overcome ingeniously this problem. The situation for closed networks, in the same conditions, much easier is also presented.
Two networks of queues models, presented initially by Jackson, in the open case, and Gordon and Newell, in the closed case, stochastic processes are presented and studied in some of their details and problems. The service times are exponentially distributed and there is only one class of customers.
An infinite buffer batch service vacation queue has been studied where service rate of the batch is dependent on the size of the batch and vacation rate is dependent on the queue size at vacation initiation epoch. The arrivals follow the Markovian arrival process (MAP). For service rule, general bulk service (GBS) rule is considered. The service time and vacation time both are considered to be generally distributed. Several joint distributions of interest are obtained using the bivariate vector generating function method and the supplementary variable technique (SVT). Numerical results are pre…
We investigate the transient and stationary queue-length distributions of a class of service systems with correlated service times. The classical $M^X/G/1$ queue with semi-Markov service times is the most prominent example in this class and serves as a vehicle to display our results. The sequence of service times is governed by a modulating process $J(t)$. The state of $J(\cdot)$ at a service initiation time determines the joint distribution of the subsequent service duration and the state of $J(\cdot)$ at the next service initiation. Several earlier works have imposed technical conditions, on…
The queue system,with Poisson arrivals,constant service time and infinite servers, busy period distribution is intensively studied because, due to its probability density function quite easy interpretation, it may serve as a clue to interpret the queue systems, with Poisson arrivals, infinite servers, and other service time distributions, busy period distributions. In this work the queue system, with Poisson arrivals,constant service time and infinite servers, busy period distribution parameters are exactly computed and its probability density function is interpreted and explained. The only pr…
We consider a two-node tandem queueing network in which the upstream queue is GI/GI/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. To investigate the evolution of workload in the second queue, we introduce and study a process M, called the plateau process, which encodes most of the information in the workload process. We focus on the case of infinite-variance service times and show that under appropriate scaling, workload in the first queue converges, and although the workload in the second queue do…
The busy period length distribution function knowledge is important for any queue system, and for the MGINF queue. But the mathematical expressions are in general very complicated, with a few exceptions, involving usually infinite sums and multiple convolutions. So, in this work are deduced some bounds for the MMINF system busy period length distribution function, meaning the second M exponential service time, which analytic expressions are simpler than the exact one. As a consequence, also some bounds for the MMINF system busy cycle length distribution function are presented.
Discrete-time queueing system has widespread applications in packet switching networks, internet protocol, Broadband Integrated Services Digital Network (B-ISDN), circuit switched time-division multiple access etc. In this paper, we analyze an infinite-buffer discrete-time group-arrival queue with single and multiple vacation policy where processing/transmission/service times dynamically vary with batch-size under service. The bivariate probability generating function of queue length and server content distribution has been derived first, and from that we extract the probabilities in terms of …
A single server retrial queueing system with two-classes of orbiting customers, and general class dependent service times is considered. If an arriving customer finds the server unavailable, it enters a virtual queue, called the orbit, according to its type. The customers from the orbits retry independently to access the server according to the constant retrial policy. We derive the generating function of the stationary distribution of the number of orbiting customers at service completion epochs in terms of the solution of a Riemann boundary value problem. For the symmetrical system we also d…
In this paper we analyze a single server queue with batch arrivals and semi-Markovian service times. We also include the feature that the first service of each busy period might have a different distribution than subsequent service times. Our generating function based approach allows us to determine the heavy traffic limit of the scaled queue-length distribution. It turns out that this distribution converges to an exponential distribution. Nonsurprisingly, the exceptional first service does not influence this limiting distribution. We identify a sufficient and necessary condition under which t…
We consider many-server queueing systems with heterogeneous exponential servers and renewal arrivals. The service rate of each server is a random variable drawn from a given distribution. We develop a framework for analyzing the heavy traffic limit of these queues in random environment using probability measure-valued stochastic processes. We introduce the measure-valued fairness process which denotes the proportion of cumulative idleness experienced by servers whose rates fall in a Borel subset of the support of the service rates. It can be shown that these fairness processes do not converge …
We consider a discrete time parallel queue, which is two-queue network, where at each time-slot there is a the same batch arrival to both queues and at each queue there is a random service available. The service law at each time-slot for each queue is different. Let $(Q_n^1, Q_n^2)$ be the queue length at $n$th time-slot. We present several open questions related to the steady-state of this queue.
We study a general $k$ dimensional infinite server queues process with Markov switching, Poisson arrivals and where the service times are fat tailed with index $α\in (0,1)$. When the arrival rate is sped up by a factor $n^γ$, the transition probabilities of the underlying Markov chain are divided by $n^γ$ and the service times are divided by $n$, we identify two regimes (''fast arrivals'', when $γ>α$, and ''equilibrium'', when $γ=α$) in which we prove that a properly rescaled process converges pointwise in distribution to some limiting process. In a third ''slow arrivals'' regime, $γ<α$, we sh…
We consider a system of N queues with decentralized load balancing such as power-of-D strategies(where D may depend on N) and generic scheduling disciplines. To measure the dependence of the queues, we use the clan of ancestors, a technique coming from interacting particle systems. Relying in that analysis we prove quantitative estimates on the queues correlations implying propagation of chaos for systems with Markovian arrivals and general service time distribution. This solves the conjecture posed by Bramsom et. al. in [*] concerning the asymptotic independence of the servers in the case of …
Discrete-time queueing models find huge applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems, computer networks, etc. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain …
We consider three parallel service models in which customers of several types are served by several types of servers subject to a bipartite compatibility graph, and the service policy is first come first served. Two of the models have a fixed set of servers. The first is a queueing model in which arriving customers are assigned to the longest idling compatible server if available, or else queue up in a single queue, and servers that become available pick the longest waiting compatible customer, as studied by Adan and Weiss, 2014. The second is a redundancy service model where arriving customer…
This paper considers a multichannel preemptive-resume priority queueing system with a Poisson input and an arbitrary service time distribution depending on the priority of job. Jobs of the same priority are serviced according to the LIFO rule. If, at moment of job arrival, all servers are busy, and at least one server is busy with the service of a job of a not higher than the priority of arriving job, then the service of a job is preempted such that the priority of preempted job is lowest from the priorities of the jobs in service. The service of a preempted job is resumed later. The paper pro…
We characterize heavy-traffic process and steady-state limits for systems staffed according to the square-root safety rule, when the service requirements of the customers are perfectly correlated with their individual patience for waiting in queue. Under the usual many-server diffusion scaling, we show that the system is asymptotically equivalent to a system with no abandonment. In particular, the limit is the Halfin-Whitt diffusion for the $M/M/n$ queue when the traffic intensity approaches its critical value $1$ from below, and is otherwise a transient diffusion, despite the fact that the pr…
Solving a Riccati equation, induced by the study of the transient behaviour of the MGInf queue system, a collection of service times distributions is determined. For the MGInf queue, which service time distribution is a member of that collection, the busy period and busy cycle probabilistic studies are performed. In extra, the properties of that distributions collection are deduced and presented.
