Modeling circulation systems in buildings using state dependent queueing models

Circulation systems within buildings are analyzed using M/G/C/C queueing models. Congestion aspects of the traffic flow are represented by introducing state dependent service rates as a function of the number of occupants in each region of the circulation system. Analytical models for unidirectional and multi-source/single sink flows are presented. Finally, use of the queueing models to analytically determine the optimal size and capacity of the links of the circulation systems is incorporated into a series of software programs available from the authors.This material is based upon work supported by the National Science Foundation under grant #MSM-8417942 and #MSM-8715152.     

Validating state-dependent queueing models for uninterrupted traffic flows using simulation

In this paper, the effectiveness of state-dependent queueing models for analyzing traffic flows is tested by comparing the speeds generated by the queueing models with the ones obtained by simulation. Simulation is thus used to evaluate speeds generated by the different queueing models. Different state-dependency functions are described and their performance is assessed. An M/G/1 queueing model with Gaussian state-dependency outperforms all other state-dependent queueing models. Different test results and insights are provided. Received: July 2004, Revised: September 2005 AMS classification: 60K25, 60K30     

Empirical validation of a queueing approach to uninterrupted traffic flows

In this paper, the use of queueing theory for modeling uninterrupted traffic flows is evaluated. Empirical data on speeds and flows are used to evaluate speeds generated by the different queueing models. Using the Theil inequality coefficient as evaluation criterion, the speeds generated by the queueing models are compared to the empirical speeds. Queueing models that best fit the observed speeds are obtained. It appears that traffic flow on a highway during non-congested hours is best described using a M/G/1 queueing model. During the congested hours however, the state dependent queueing GI/G/z models are more realistic. Because the queueing models describe the empirical data well, they can also be used to evaluate potential improvements in existing traffic conditions. Received: April 2005 / Revised version: June 2005 AMS classification: 60K30, 68M20     

Heavy traffic approximations of large deviations of feedforward queueing networks

We consider a multi-class feedforward queueing network with first come first serve queueing discipline and deterministic services and routing. The large deviation asymptotics of tail probabilities of the distribution of the workload vector can be characterized by convex path space minimization problems with non-linear constraints. So far there exists no numerical algorithm which could solve such minimization problems in a general way. When the queueing network is heavily loaded it can be approximated by a reflected Brownian motion. The large deviation asymptotics of tail probabilities of the distribution of this heavy traffic limit can again be characterized by convex path space minimization problems with non-linear constraints. However, due to their less complicated structure there exist algorithms to solve such minimization problems. In this paper we show that, as the network tends to a heavy traffic limit, the solution of the large deviation minimization problems of the network approaches the solution of the corresponding minimization problems of a reflected Brownian motion. Stated otherwise, we show that large deviation and heavy traffic asymptotics can be interchanged. We prove this result in the case when the network is initially empty. Without proof we state the corresponding result in the stationary case. We present an illuminating example with two queues. This revised version was published online in June 2006 with corrections to the Cover Date.     

An empirical extension of the M/G/1 heavy traffic approximation

Numerical evaluation of waiting time distributions for M/G/1 systems is somewhat difficult. This paper examines a simple variation of the heavy traffic formula which may be useful at modest levels of traffic intensity. One can justify the heavy traffic approximation by expressing the Laplace transform of the service time distribution as a Maclaurin series and then truncating to three terms. The spectrum factorization and inversion leads in a straightforward fashion to the heavy traffic approximation. If one carries two additional terms from the Maclaurin series, the characteristic equation is a cubic with exactly one real negative root. This root provides an easy way to extend the heavy traffic formula to cases where the traffic is not so heavy. This paper studies the quality of this approximation and includes some numerical evaluation based on data actually encountered.     

State space collapse with application to heavy traffic limits for multiclass queueing networks

Heavy traffic limits for multiclass queueing networks are a topic of continuing interest. Presently, the class of networks for which these limits have been rigorously derived is restricted. An important ingredient in such work is the demonstration of state space collapse. Here, we demonstrate state space collapse for two families of networks, first-in first-out (FIFO) queueing networks of Kelly type and head-of-the-line proportional processor sharing (HLPPS) queueing networks. We then apply our techniques to more general networks. To demonstrate state space collapse for FIFO networks of Kelly type and HLPPS networks, we employ law of large number estimates to show a form of compactness for appropriately scaled solutions. The limits of these solutions are next shown to satisfy fluid model equations corresponding to the above queueing networks. Results from Bramson [4,5] on the asymptotic behavior of these limits then imply state space collapse. The desired heavy traffic limits for FIFO networks of Kelly type and HLPPS networks follow from this and the general criteria set forth in the companion paper Williams [41]. State space collapse and the ensuing heavy traffic limits also hold for more general queueing networks, provided the solutions of their fluid model equations converge. Partial results are given for such networks, which include the static priority disciplines. This revised version was published online in June 2006 with corrections to the Cover Date.     

Light traffic heuristic for anM/G/1 queue with limited inventory

Motivated by solving a stylized location problem, we develop a light traffic heuristic for anM/G/1 queue with limited inventory that gives rise to a closed form expression for average delay in terms of basic system parameters. Simulation experiments show that the heuristic works well. The inventory operates as follows: the inventory level drops by one unit after each service completion and whenever it drops to a pre-specified levelu, an order is placed with replenishment time exp(). Upon replenishment the inventory is restocked to a pre-specified levels and any arrivals when there is no inventory are placed in queue. Suggestions are given to cover the more general case of a New Better than Used (NBU) replenishment time distribution. Applications to inventory management problems are also discussed.The research was supported in part by ONR Contract N00014-90-J-1649, NSF Contract DDM-8922712 and the Center for Telecommunications Research under NSF grant CDR 84-21402.     

Bounding blocking probabilities and throughput in queueing networks with buffer capacity constraints

We propose a new technique for upper and lower bounding of the throughput and blocking probabilities in queueing networks with buffer capacity constraints, i.e., some buffers in the network have finite capacity. By studying the evolution of multinomials of the state of the system in its assumed steady state, we obtain constraints on the possible behavior of the system. Using these constraints, we obtain linear programs whose values upper and lower bound the performance measures of interest, namely throughputs or blocking probabilities. The main advantages of this new technique are that the computational complexity does not increase with the size of the finite buffers and that the technique is applicable to systems in which some buffers have infinite capacity. The technique is demonstrated on examples taken from both manufacturing systems and communication networks. As a special case, for the M/M/s/s queue, we establish the asymptotic exactness of the bounds, i.e., that the bounds on the blocking probability asymptotically approach the exact value as the degree of the multinomials considered is increased to infinity. This revised version was published online in June 2006 with corrections to the Cover Date.     

Structure-reversibility and departure functions of queueing networks with batch movements and state dependent routing

We consider characterizations of departure functions in Markovian queueing networks with batch movements and state-dependent routing in discrete-time and in continuous-time. For this purpose, the notion of structure-reversibility is introduced, which means that the time-reversed dynamics of a queueing network corresponds with the same type of queueing network. The notion is useful to derive a traffic equation. We also introduce a multi-source model, which means that there are different types of outside sources, to capture a wider range of applications. Characterizations of the departure functions are obtained for any routing mechanism of customers satisfying a recurrent condition. These results give a unified view to queueing network models with linear traffic equations. Furthermore, they enable us to consider new examples as well as show limited usages of this kind of queueing networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

Branching/queueing networks: Their introduction and near-decomposability asymptotics

A new class of models, which combines closed queueing networks with branching processes, is introduced. The motivation comes from MIMD computers and other service systems in which the arrival of new work is always triggered by the completion of former work, and the amount of arriving work is variable. In the variant of branching/queueing networks studied here, a customer branches into a random and state-independent number of offspring upon completing its service. The process regenerates whenever the population becomes extinct. Implications for less rudimentary variants are discussed. The ergodicity of the network and several other aspects are related to the expected total number of progeny of an associated multitype Galton-Watson process. We give a formula for that expected number of progeny. The objects of main interest are the stationary state distribution and the throughputs. Closed-form solutions are available for the multi-server single-node model, and for homogeneous networks of infinite-servers. Generally, branching/queueing networks do not seem to have a product-form state distribution. We propose a conditional product-form approximation, and show that it is approached as a limit by branching/queueing networks with a slowly varying population size. The proof demonstrates an application of the nearly complete decomposability paradigm to an infinite state space. This revised version was published online in June 2006 with corrections to the Cover Date.     

Heavy traffic analysis of M/G/1 type queueing systems with Markov-modulated arrivals

M/G/1 type queueing systems whose arrival rate is a function of an independent continuous time Markov chain are considered. We suggest a simple analytical approach which allows rigorous mathematical analysis of the stationary characteristics under heavy traffic. Their asymptotic behaviour is described in terms of characteristics of the modulating process (defined as a solution of a set of linear algebraic equations). The analysis is based on certain “semi-explicit” formulas for the performance characteristics. This research was supported by INTAS under grant No. 96-0828.     

Brownian models of multiclass queueing networks: Current status and open problems

This paper is concerned with Brownian system models that arise as heavy traffic approximations for open queueing networks. The focus is on model formulation, and more specifically, on the formulation of Brownian models for networks with complex routing. We survey the current state of knowledge in this dynamic area of research, including important open problems. Brownian approximations culminate in estimates of complete distributions; we present numerical examples for which complete sojourn time distributions are estimated, and those estimates are compared against simulation.     

具有两类失效模式的D-策略M/G/1可修排队系统分析

研究具有两类失效模式的D策略M/G/1可修排队系统,其中第一类失效是服务台在服务顾客期间发生的失效,第二类失效是服务台在空闲期间发生的失效,且两类失效模式的失效率不同.使用全概率分解技术和利用拉普拉斯变换与母函数等工具,从任意初始状态出发,讨论了系统队长的瞬时分布和稳态分布,获得了系统稳态队长分布的递推表达式与稳态队长的随机分解结果.进一步,在建立费用模型的基础上,通过数值计算实例讨论了使得系统在长期单位时间内达到最小值的最优控制策略D^*,并在同一组参数取值下与服务台不发生故障时的最优控制策略进行了比较.     

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.     

Busy period analysis of the state dependent M/M/1/K queue

In this paper, we study the transient behavior of a state dependent M/M/1/K queue during the busy period. We derive in closed-form the joint transform of the length of the busy period, the number of customers served during the busy period, and the number of losses during the busy period. For two special cases called the threshold policy and the static policy we determine simple expressions for their joint transform.     

A Markov-modulated M/G/1 queue I: Stationary distribution

We consider an M/G/1 queueing system in which the arrival rate and service time density are functions of a two-state stochastic process. We describe the system by the total unfinished work present and allow the arrival and service rate processes to depend on the current value of the unfinished work. We employ singular perturbation methods to compute asymptotic approximations to the stationary distribution of unfinished work and in particular, compute the stationary probability of an empty queue.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

Co‐existence of Poisson and non‐Poisson processes in ordered parallel multilane pedestrian traffic

We show that non‐Poisson and Poisson processes can coexist in ordered parallel multilane pedestrian traffic, in the presence of lane switching which asymmetrically benefits the switchers and nonswitchers. Pedestrians join at the tail end of a queue and transact at the opposite front end. Their aim is to complete a transaction within the shortest possible time, and they can transfer to a shorter queue with probability p_s. Traffic is described by the utilization parameter U = λ〈t_s〉/N, where λ is the average rate of pedestrians entering the system, 〈t_s〉 is the average transaction time, and N is the number of lanes. Using an agent‐based model, we investigate the dependence of the average completion time 〈t_c〉 with variable K = 1 + (1 ? U)^?1 for different N and 〈t_s〉 values. In the absence of switching (p_s = 0), we found that 〈t_c〉 ∝ K^τ, where τ ≈ 1 regardless of N and 〈t_s〉. Lane switching (p_s = 1) reduces 〈t_c〉 for a given K, but its characteristic dependence with K differs for nonswitchers and switchers in the same traffic system. For the nonswitchers, 〈t_c〉 ∝ K^τ, where τ < 1. At low K values, switchers have a larger 〈t_c〉 that also increases more rapidly with K. At large K, the increase rates become equal for both. For nonswitchers, the possible t_c values obey an exponentially decaying probability density function p(t_c). The switchers on the other hand, are described by a fat‐tailed p(t_c) implying that a few are penalized with t_c values that are considerably longer than any of those experienced by nonswitchers. © 2006 Wiley Periodicals, Inc. Complexity 11: 35–42, 2006     

Vacation policies in an M/G/1 type queueing system with finite capacity

This paper deals with a queueing system with finite capacity in which the server passes from the active state to the inactive state each time a service terminates withv customers left in the system. During the active (inactive) phases, the arrival process is Poisson with parameter (₀). Denoting byu _n the duration of thenth inactive phase and byx _n the number of customers present at the end of thenth inactive phase, we assume that the bivariate random vectors {(v _n ,x _n ),n 1} are i.i.d. withx _n v+l a.s. The stationary queue length distributions immediately after a departure and at an arbitrary instant are related to the corresponding distributions in the classical model.     

M/M~B/1排队模型的时间依赖解的存在唯一性

应用线性算子的积分半群理论证明 M/MB/1排队模型的时间依赖解的存在唯一性 ,其次推出 M/M/1排队模型的时间依赖解的存在唯一性 .     

The M/G/c queue in light traffic

Under light traffic, we investigate the quality of a well‐known approximation for first‐moment performance measures for an M/G/c queue, and, in particular, conditions under which the approximation is either an upper or a lower bound. The approach is to combine known relationships between quantities such as average delay and time‐average work in system with direct sample‐path comparisons of system operation under two modes of operation: conventional FIFO and a version of preemptive LIFO. We then use light traffic limit theorems to show an inequality between time‐average work of the M/G/c queue and that of the approximation. In the process, we obtain new and improved approximations. This revised version was published online in June 2006 with corrections to the Cover Date.