Heavy Traffic Regime Research Articles

On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case

Two of the most popular approximations for the distribution of the steady-state waiting time, $W_{\infty}$, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, $\rho$, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of $W_{\infty}$ is roughly exponential at scale $O((1-\rho)^{-1})$, while the heavy tailed asymptotic describes power law decay in the tail of the distribution of $W_{\infty}$ for a fixed traffic intensity. In this paper, we assume a regularly varying processing time distribution and obtain a sharp threshold in terms of the tail value, or equivalently in terms of $(1-\rho)$, that describes the point at which the tail behavior transitions from the heavy-traffic regime to the heavy-tailed asymptotic. We also provide new approximations that are either uniform in the traffic intensity, or uniform on the positive axis, that avoid the need to use different expressions on the two regions defined by the threshold.

Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

In this paper we derive a technique of obtaining limit theorems for suprema of L\'evy processes from their random walk counterparts. For each $a>0$, let $\{Y^{(a)}_n:n\ge 1\}$ be a sequence of independent and identically distributed random variables and $\{X^{(a)}_t:t\ge 0\}$ be a L\'evy processes such that $X_1^{(a)}\stackrel{d}{=} Y_1^{(a)}$, $\mathbb E X_1^{(a)}<0$ and $\mathbb E X_1^{(a)}\uparrow0$ as $a\downarrow0$. Let $S^{(a)}_n=\sum_{k=1}^n Y^{(a)}_k$. Then, under some mild assumptions, $\Delta(a)\max_{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iff\Delta(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R$, for some random variable $R$ and some function $\Delta(\cdot)$. We utilize this result to present a number of limit theorems for suprema of L\'evy processes in the heavy-traffic regime.

State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic

Recently Gamarnik and Zeevi (Ann. Appl. Probab. 16:56---90, 2006) and Budhiraja and Lee (Math. Oper. Res. 34:45---56, 2009) established that, under suitable conditions, a sequence of the stationary scaled queue lengths in a generalized Jackson queueing network converges to the stationary distribution of multidimensional reflected Brownian motion in the heavy-traffic regime. In this work we study the corresponding problem in multiclass queueing networks (MQNs). In the first part of this work we consider the MQNs for which the fluid stability is valid, state-space collapse is exhibited under suitable initial conditions and a heavy traffic limit theorem holds. For such MQNs we establish that, under the assumption of the tightness of a sequence of stationary scaled workloads, the sequence converges to the stationary distribution of semimartingale reflecting Brownian motion in the heavy-traffic regime. The key to the proof is to show that state-space collapse occurs in the heavy-traffic regime in stationarity under the assumption of tightness. In the second part, using the result obtained, it is shown that such a convergence of stationary workload holds for a multiclass single-server queue with feedback routing, where the tightness is proved by the Lyapunov function method developed in (Gamarnik and Zeevi in Ann. Appl. Probab. 16:56---90, 2006).

Asymptotically optimal parallel resource assignment with interference

Motivated by scheduling in cellular wireless networks and resource allocation in computer systems, we study a service facility with two classes of users having heterogeneous service requirement distributions. The aggregate service capacity is assumed to be largest when both classes are served in parallel, but giving preferential treatment to one of the classes may be advantageous when aiming at minimization of the number of users, or when classes have different economic values, for example. We set out to determine the allocation policies that minimize the total number of users in the system. For some particular cases we can determine the optimal policy exactly, but in general this is not analytically feasible. We then study the optimal policies in the fluid regime, which prove to be close to optimal in the original stochastic model. These policies can be characterized by either linear or exponential switching curves. We numerically compare our results with existing approximations based on optimization in the heavy-traffic regime. By simulations we show that, in general, our simple computable switching-curve strategies based on the fluid analysis perform well.

Delay Analysis for Wireless Networks With Single Hop Traffic and General Interference Constraints

We consider a class of wireless networks with general interference constraints on the set of links that can be served simultaneously at any given time. We restrict the traffic to be single-hop, but allow for simultaneous transmissions as long as they satisfy the underlying interference constraints. We begin by proving a lower bound on the delay performance of any scheduling scheme for this system. We then analyze a large class of throughput optimal policies which have been studied extensively in the literature. The delay analysis of these systems has been limited to asymptotic behavior in the heavy traffic regime and order results. We obtain a tighter upper bound on the delay performance for these systems. We use the insights gained by the upper and lower bound analysis to develop an estimate for the expected delay of wireless networks with mutually independent arrival streams operating under the well-known maximum weighted matching (MWM) scheduling policy. We show via simulations that the delay performance of the MWM policy is often close to the lower bound, which means that it is not only throughput optimal, but also provides excellent delay performance.

Short-term fairness and long-term QoS in the Internet

We consider connection-level models of resource allocation in a simple symmetric three-link network, where files arrive into the network according to a Poisson process and the size of each file is exponentially distributed. For the simple network under consideration, we derive an optimal resource allocation policy which minimizes the steady-state expected number of files in the network. Using this result, we show that, in a heavy traffic regime, the mean file-transfer delay under the proportionally fair policy is at most 1.5 times the delay under the optimal policy. Simulation results indicate that the gap is even smaller.

Simplified Control Problems for Multiclass Many-Server Queueing Systems

We consider scheduling and routing control problems for queueing models with I customer classes and J server pools, each consisting of many statistically identical, exponential servers. Customers require a single service that can be performed by a server from one of the pools; the service rate is μij ≥ 0, which depends on the customer's class i and the server's pool j, and customers can abandon the system while waiting to be served. In the heavy traffic regime of Halfin and Whitt, these problems are formally equivalent to I-dimensional diffusion control problems. We analyze the diffusion control problems is two special cases. First, when the service rates depend only on the pool (μij = μj), the diffusion control problem is shown to be similar to (but distinct from) the diffusion control problem for a single class model, which greatly reduces the complexity of the problem. Second, when the service rates depend only on the class (μij = μi), the diffusion control problem is shown to be equivalent to a diffusion control problem for a single pool model, a problem that has previously been studied. In the first case, we also establish a rigorous relation between the queueing control problem and the diffusion control problem, showing that a policy for the queueing model, based on an ordinary differential equation of Hamilton-Jacobi-Bellman type, is asymptotically optimal.

State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy

We consider a connection-level model of Internet congestion control, introduced by Massouli\'{e} and Roberts [Telecommunication Systems 15 (2000) 185--201], that represents the randomly varying number of flows present in a network. Here, bandwidth is shared fairly among elastic document transfers according to a weighted $\alpha$-fair bandwidth sharing policy introduced by Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556--567] [$\alpha\in (0,\infty)$]. Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [Ann. Appl. Probab. 14 (2004) 1055--1083] by two of the authors. Here, we use the long-time behavior of the solutions of the fluid model established in that paper to derive a property called multiplicative state space collapse, which, loosely speaking, shows that in diffusion scale, the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process.

Monotonicity Properties for Multi-Class Queueing Systems

We study multi-dimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted ?-fair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter ? and the relative weights is established. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments. In addition, we study a single-server queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the single-server queue with an arbitrary number of classes as well.

Performance analysis of wireless opportunistic schedulers using stochastic Petri nets

In this paper, performance of wireless opportunistic schedulers in multiuser systems is studied under a dynamic data arrival setting. Different from the previous studies which mostly focus on the network stability and the worst case scenarios, we emphasize on the average performance of wireless opportunistic schedulers. We first develop a framework based on Markov queueing model and then analyze it by applying decomposition and iteration techniques in the stochastic Petri nets (SPN). Since the size of the state space in our analytical model is small, the proposed framework shows an improved efficiency in computational complexity. Based on the established analytical model, performance of both opportunistic and non-opportunistic schedulers are studied and compared in terms of average queue length, mean throughput, average delay and dropping probability. Analytical results demonstrate that the multiuser diversity effect as observed in the infinite backlog scenario is only valid in the heavy traffic regime. The performance of the opportunistic schedulers in the light traffic regime is worse than that of the non-opportunistic round-robin scheduler, and becomes worse especially with the increase of the number of users. Simulations are also performed to verify the accuracy of the analytical results.

Efficient Routing in Heavy Traffic Under Partial Sampling of Service Times

We consider a queue with renewal arrivals and n exponential servers in the Halfin-Whitt heavy traffic regime, where n and the arrival rate increase without bound, so that a critical loading condition holds. Server k serves at rate μk, and the empirical distribution of {μk}k = 1,…,n is assumed to converge weakly. We show that very little information on the service rates is required for a routing mechanism to perform well. More precisely, we construct a routing mechanism that has access to a single sample from the service time distribution of each of n1/2 + ε randomly selected servers (ε &gt; 0), but not to the actual values of the service rates, the performance of which is asymptotically as good as the best among mechanisms that have the complete information {μk}k = 1,…,n.

Heavy traffic analysis of maximum pressure policies for stochastic processing networks with multiple bottlenecks

A class of open processing networks operating under a maximum pressure policy is considered in the heavy traffic regime. We prove that the diffusion-scaled workload process for a network with several bottleneck resources converges to a semimartingale reflecting Brownian motion (SRBM) living in a polyhedral cone. We also establish a state space collapse result that the queue length process can be lifted from the lower-dimensional workload process.

Central limit theorem for a many-server queue with random service rates

Given a random variable $N$ with values in ${\mathbb{N}}$, and $N$ i.i.d. positive random variables $\{\mu_k\}$, we consider a queue with renewal arrivals and $N$ exponential servers, where server $k$ serves at rate $\mu_k$, under two work conserving routing schemes. In the first, the service rates $\{\mu_k\}$ need not be known to the router, and each customer to arrive at a time when some servers are idle is routed to the server that has been idle for the longest time (or otherwise it is queued). In the second, the service rates are known to the router, and a customer that arrives to find idle servers is routed to the one whose service rate is greatest. In the many-server heavy traffic regime of Halfin and Whitt, the process that represents the number of customers in the system is shown to converge to a one-dimensional diffusion with a random drift coefficient, where the law of the drift depends on the routing scheme. A related result is also provided for nonrandom environments.

Approximating the GI/GI/1+GI Queue with a Nonlinear Drift Diffusion: Hazard Rate Scaling in Heavy Traffic

We study a single-server queue, operating under the first-in-first-out (FIFO) service discipline, in which each customer independently abandons the queue if his service has not begun within a generally distributed amount of time. Under some mild conditions on the abandonment distribution, we identify a limiting heavy-traffic regime in which the resulting diffusion approximation for both the offered waiting time process (the process that tracks the amount of time an infinitely patient arriving customer would wait for service) and the queue-length process contain the entire abandonment distribution. To use a continuous mapping approach to establish our weak convergence results, we additionally develop existence, uniqueness, and continuity results for nonlinear generalized regulator mappings that are of independent interest. We further perform a simulation study to evaluate the quality of the proposed approximations for the steady-state mean queue length and the steady-state probability of abandonment suggested by the limiting diffusion process.

State space collapse for asymptotically critical multi-class fluid networks

We consider a class of fluid queueing networks with multiple fluid classes and feedback allowed, which are fed by N heavy tailed ON/OFF sources. We study the asymptotic behavior when N?? of these queueing systems in a heavy traffic regime (that is, when they are asymptotically critical). As performance processes we consider the workload W N (the amount of time needed for each server to complete processing of all the fluid in queue), and the fluid queue Z N (the quantity of each fluid class in the system). We show the convergence of $\sqrt{N}W^{N}$ and $\sqrt{N}Z^{N}$ (to $\hat{W}$ and $\hat{Z}$ ) in heavy traffic if state space collapse (SSC) holds. (SSC) is a condition that establishes a relationship between those components of $\hat{Z}$ that correspond to fluid classes processed by the same server, which implies that $\hat{Z}=\Delta\hat{W}$ for a deterministic lifting matrix Δ. Our main contribution is to prove that assuming that the other hypotheses are true, (SSC) is not only sufficient for this convergence, but necessary. Furthermore, we prove that processes $\hat{W}$ and $\hat{Z}$ , conveniently scaled in time, converge to W (a reflected fractional Brownian motion) and Z (=ΔW). We illustrate the application of our results with some examples including a tandem queue.

Optimal control of parallel server systems with many servers in heavy traffic

We consider a parallel server system that consists of several customer classes and server pools in parallel. We propose a simple robust control policy to minimize the total linear holding and reneging costs. We show that this policy is asymptotically optimal under the many-server heavy traffic regime for parallel server systems when the service times are only server pool dependent and exponentially distributed.

Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

We consider a multiserver queue in the Halfin-Whitt regime: as the number of servers n grows without a bound, the utilization approaches 1 from below at the rate Assuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

Product form stationary distributions for diffusion approximations to a flow-level model operating under a proportional fair sharing policy

We consider a flow-level model of Internet congestion control introduced by Massoulié and Roberts [2]. We assume that bandwidth is shared amongst elastic documents according to a weighted proportional fair bandwidth sharing policy. With Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. In [1], under a mild local traffic condition, we establish a diffusion approximation for the workload process (and hence for the flow count process) in this model. We first recall that result in this paper. We then state results showing that when all of the weights are equal (proportional fair sharing) the diffusion has a product form invariant distribution with a strikingly simple interpretation in terms of dual random variables, one for each of the resources of the network. This result can be extended to the case where document sizes are distributed as finite mixtures of exponentials, and to models that include multi-path routing (these extensions are not described here, but can be found in [1]).

Dynamic Routing in Large-Scale Service Systems with Heterogeneous Servers

Motivated by modern call centers, we consider large-scale service systems with multiple server pools and a single customer class. For such systems, we propose a simple routing rule which asymptotically minimizes the steady-state queue length and virtual waiting time. The proposed routing scheme is FSF which assigns customers to the Fastest Servers First. The asymptotic regime considered is the Halfin-Whitt many-server heavy-traffic regime, which we refer to as the Quality and Efficiency Driven (QED) regime; it achieves high levels of both service quality and system efficiency by carefully balancing between the two. Additionally, expressions are provided for system limiting performance measures based on diffusion approximations. Our analysis shows that in the QED regime this heterogeneous server system outperforms its homogeneous server counterpart.