Heavy-Traffic Limits for Queues with Many Exponential Servers

In this paper, we study a number of closely related paradoxes of queuing theory, each of which is based on the intuitive notion that the level of congestion in a queuing system should be directly related to the stochastic variability of the arrival process and the service times. In contrast to such an expectation, it has previously been shown that, in all Hk/G/1 queues, PW (the steady-state probability that a customer has to wait for service) decreases when the service-time becomes more variable. An analagous result has also been proved for ploss (the steady-state probability that a customer is lost) in all Hk/G/1 loss systems. Such theoretical results can be seen, in this paper, to be part of a much broader scheme of paradoxical behaviour which covers a wide range of queuing systems. The main aim of this paper is to provide a unifying explanation for these kinds of behaviour. Using an analysis based on a simple, approximate model, we show that, for an arbitrary set of nGI/Gk/1 loss systems (k=1,..., n), if the interarrival-time distribution is fixed and ‘does not differ too greatly’ from the exponential distribution, and if the n systems are ordered in terms of their ploss values, then the order that results whenever cA<1 is the exact reverse of the order that results whenever cA>1, where cA is the coefficient of variation of the interarrival time. An important part of the analysis is the insensitivity of the ploss value in an M/G/1 loss system to the choice of service-time distribution, for a given traffic intensity. The analysis is easily generalised to other queuing systems for which similar insensitivity results hold. Numerical results are presented for paradoxical behaviour of the following quantities in the steady state: ploss in the GI/G/1 loss system; PW and Wq (the expected queuing time of customers) in the GI/G/1 queue; and pK (the probability that all K machines are in the failed state) in the GI/G/r machine interference model. Two of these examples of paradoxical behaviour have not previously been reported in the literature. Additional cases are also discussed.

Sensitivity analysis for output performance measures in long-range dependent queueing system

A queueing system with stepwise randomly available additional space

On the M/G/1 queueing system with multiclass customers and fixed feedback

We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general time-varying arrival process (the Gt), a general service-time distribution (the first GI), a time-dependent number of servers (the st) and allows abandonment from queue according to a general abandonment-time distribution (the +GI). This fluid model approximates the associated queueing system when the arrival rate and number of servers are both large. We also show that the system dynamics greatly simplifies in two special cases: (i) when the service time distribution is exponential (M) and (ii) when the service time distribution is deterministic (D) and the model is stationary. We develop an efficient algorithm to compute all standard performance functions in both cases. In case (i), we establish an asymptotic loss of memory (ALOM) property, i.e., asymptotic independence from the initial conditions as time evolves. We show that the difference in the performance functions with different initial conditions dissipates over time exponentially fast, under regularity conditions. In contrast, in case (ii) we show that ALOM fails dramatically. Instead, although all model parameters are constants, we show that the performance rapidly approaches a periodic steady state (PSS) with a period equal to the service time, whenever the system does not start with the unique stationary distribution. Moreover, the form of the PSS depends on the initial condition. Simulation and a heavy-traffic limit confirm that this anomalous behavior also occurs in the large-scale queueing model.

In this paper we analyze a single removable and unreliable server in the N policy M/G/1 queueing system in which the server breaks down according to a Poisson process and the repair time obeys an arbitrary distribution. The method of maximum entropy is used to develop the approximate steady-state probability distributions of the queue length in the M/G(G)/1 queueing system, where the second and the third symbols denote service time and repair time distributions, respectively. A study of the derived approximate results, compared to the exact results for the M/M(M)/1, M/E2(E3)/1, M/H2(H3)/1 and M/D(D)/1 queueing systems, suggest that the maximum entropy principle provides a useful method for solving complex queueing systems. Based on the simulation results, we demonstrate that the N policy M/G(G)/1 queueing model is sufficiently robust to the variations of service time and repair time distributions.

We iteratively derive the product‐form solutions of stationary distributions for a type of preemptive priority multiclass queueing networks with multiserver stations. This type of queueing systems can typically be used to model the stochastic dynamics of some large scale backbone networks with multiprocessor shared‐memory switches or local (edge) cloud computing centers with parallel‐server pools. The queueing networks are Markovian with exponential interarrival and service time distributions. The obtained iterative solutions can be used to conduct performance analysis or as comparison criteria for approximation and simulation studies. Numerical comparisons with existing Brownian approximating model (BAM) related to general interarrival and service times are provided to show the effectiveness of our current designed algorithm and our previous derived BAM. Furthermore, based on the iterative solutions, we can also give some analysis concerning network stability for some cases of these queueing systems, which provides some insight for more general study.

To analyze networks of queues, it is important to be able to analyze departure processes from single queues. For the M/M/s and M/G/∞ models, the stationary departure process is simple (Poisson), but in general the stationary departure process is quite complicated. As a basis for approximations, this paper shows that the stationary departure process is approximately Poisson when there are many busy slow servers in a large class of stationary G/GI/s congestion models having s servers, infinite waiting room, the first-come first-served discipline, and mutually independent and identically distributed service times that are independent of a stationary arrival process. Limit theorems are proved for the departure process in a G/GI/s system in which the number of servers and the offered load (arrival rate divided by the service rate) both increase. The asymptotic behavior of the departure process depends on the way the arrival rate changes. If the arrival rate is held fixed, so that the offered load increases by slowing down the service rate, then the departure process converges to a Poisson process. For this result, the service-time distribution is assumed to be phase-type. Other limiting behavior occurs if the arrival rate approaches zero or infinity. Convergence is established in each case by applying previous heavy-traffic limit theorems.

This research investigates the unexplored domain of negative k parameters within dynamic systems, focusing on their influence across stability, traffic intensity, and chaotic phases. Using an iterative computational framework, we examine the sigma function (σ) to characterize system responses under varying conditions of k and ρ. Visualizations and insights demonstrate transitions between phases for a first-time-ever exploration for negative values of the number of sets of phases, namely k, with novel findings extending foundational studies. This work establishes a baseline for further explorations of negative parameter spaces in complex systems. It is to be noted that the current work provides new contributions to Ismail’s Contemporary Pointwise Stationary Fluid Approximation Theory by offering a comprehensive computational framework for exploring dynamic system behaviors across varying parameters.

Motivated by dynamic scheduling control for queueing networks, Chen and Yao [8] developed a systematic method to generate dynamic scheduling control policies for a fluid network, a simple and highly aggregated model that approximates the queueing network. This study addresses the question of how good these fluid policies are as heuristic scheduling policies for queueing networks. Using simulation on some examples these heuristic policies are compared with traditional simple scheduling rules. The results show that the heuristic policies perform at least comparably to classical priority rules, regardless of the assumptions made about the traffic intensities and the arrival and service time distributions. However, they are certainly not always the best and, even when they are, the improvement is seldom dramatic. The comparative advantage of these policies may lie in their application to nonstationary situations such as might occur with unreliable machines or nonstationary demand patterns.

Previously established necessary and sufficient conditions for finite stationary moments in stable FIFO GI/GI/s queues exist only for the first component of the workload vector, the delay, and the final component, which behaves as the total work in the system. In this paper, we derive moment results for all the components of the stationary workload vector in stable FIFO GI/GI/s queues. As in the case of stationary delay, the moment conditions for workload components incorporate the interaction between service-time distribution, traffic intensity and the number of servers in the queue. If we denote a generic service-time random variable by S, a generic interarrival time by T, and define the traffic intensity as ?=ES/ET, then sufficient conditions for EW i ???, where for any real x, ?x? denotes the smallest integer greater than or equal to x. Explicitly, for i????, $\mbox {E}W_{i}^{\alpha} ???; these components have individual finite moment conditions: $\mbox {E}W_{i}^{\alpha}<\infty$ provided that $\mbox {E}S^{\beta_{2}(i)}<\infty$ , where β 2(i)=(s?i+?)/(s?i), for ??1. Finally, for S in a large class of service distributions, these conditions are also necessary.

Several queueing systems in heavy traffic regimes are shown to admit a diffusive approximation in terms of the Reflected Brownian Motion. The latter is defined by solving the Skorokhod reflection problem on the trajectories of a standard Brownian motion. In recent years, fractional queueing systems have been introduced to model a class of queueing systems with heavy-tailed interarrival and service times. In this paper, we consider a subdiffusive approximation for such processes in the heavy traffic regime. To do this, we introduce the Delayed Reflected Brownian Motion by either solving the Skorohod reflection problem on the trajectories of the delayed Brownian motion or by composing the Reflected Brownian Motion with an inverse stable subordinator. The heavy traffic limit is achieved via the continuous mapping theorem. As a further interesting consequence, we obtain a simulation algorithm for the Delayed Reflected Brownian Motion via a continuous-time random walk approximation.

In this paper, we are interested in solving queueing systems having Poisson batch arrivals, exponential servers and negative customers. Preconditioned Conjugate Gradient (PCG) method is applied to solving the steady-state probability distribution of the queueing system. Preconditioners are constructed by exploiting near-Toeplitz structure of the generator matrix and the Gohberg-Semumcul formula. We proved that the preconditioned system has singular values clustered around one. Therefore Conjugate Gradient (CG) methods when applied to solving the preconditioned system, we expect fast convergence rate. Numerical examples are given to demonstrate our claim.

Now the information system of the iron and steel industry is a typical sensor cloud computing system. The system has accumulated a lot of relevant manufacturing data. Using these data can well solve the slab storage problem in the production process for the iron and steel industry. In this paper, we investigate a queueing system where customers arrive according to a Markov Modulated Poisson Process (MMPP). MMPP can describe how the arrival rate changes with the environment, which is more realistic. We develop an MMPP(3)/M/C queueing model to solve the congestion problem in the iron and steel industry. In the actual production process, the slab arrival rates vary with states, therefore MMPP is used to model the arrival process in this paper. Based on explicit performance measures, we develop a nonlinear optimization model of queueing system, and convert the model into a convex optimization problem. Through the convex optimization method, the MMPP(3)/M/C model, resulted from the practical system, can be analyzed by the M/M/C model approximately.

Queueing systems with catastrophes and state-dependent arrival and service rates are considered. For two types of queueing systems namely, queues with discouraged arrivals and infinite server queue, explicit expressions for the transient probabilities of system size are obtained by using continued fractions technique. Some system performance measures and steady-state probabilities are studied. The effect of system parameters on system size probabilities are also illustrated numerically. It is observed that the steady-state probabilities differ when catastrophes are present, while they are identical in the absence of catastrophes.