Stationary Arrival Process Research Articles

Precise large deviations in a non stationary risk model with arbitrary dependence between subexponential claim sizes and waiting times

Consider a risk model in which the claims follow a non stationary arrival process that satisfies a large deviation principle. Supposing that the claim sizes form a sequence of i.i.d. random variables with subexponential tail, precise large deviations for the aggregate claims are obtained, by allowing the claim-sizes and claim inter-arrival (waiting) times to be arbitrarily dependent.

A busy period approach to some queueing games

The traditional approach when looking for a symmetric equilibrium behavior in queueing models with strategic customers who arrive according to some stationary arrival process is to look for a strategy which, if used by all, is also a best response of an individual customer under the resulting stochastic steady-state conditions. This description lacks a key component: a proper definition of the set of players. Hence, many of these models cannot be defined as proper noncooperative games. This limitation raises two main concerns. First, it is hard to formulate these models in a canonical way, and hence, results are typically limited to a specific model or to a narrow class of models. Second, game theoretic results cannot be applied directly in the analysis of such models and call for ad hoc adaptations. We suggest a different approach, one that is based on the stationarity of the underlying stochastic processes. In particular, instead of considering a system that is functioning from time immemorial and has already reached stochastic steady-state conditions, we look at the process as a series of isolated, a priori identical, and independent strategic situations with a random, yet finite, set of players. Each of the isolated games can be analyzed using existing tools from the classical game theoretic literature. Moreover, this approach suggests a canonic definition of strategic queueing models as properly defined mathematical objects. A significant advantage of our approach is its compatibility with models in the existing literature. This is exemplified in detail for the famous model of the unobservable “to queue or not to queue” problem and other related models.

Consistent Sampling of Churn Under Periodic Non-Stationary Arrivals in Distributed Systems

Characterizing user churn has become an important research area of networks and distributed systems, both in theoretical analysis and system design. A realistic churn model, often measured using periodic observation, should replicate two key properties of deployed systems -- (1) the arrival process and (2) the lifetime distribution of participating agents. Because users can be sampled only by sending packets to them and eliciting responses, there is an inherent tradeoff between overhead (i.e., bandwidth needed to perform the measurement) and accuracy of obtained results. Furthermore, all observations are censored, i.e., rounded up or down to a multiple of Δ, where Δ is the minimum delay between repeat visits to the same user. Assuming a stationary arrival process, previous work shows that consistent (i.e., asymptotically accurate) estimation of the lifetime distribution is possible; however, the problem remains open for non-stationary cases. Questions include what distributions these methods sample when the assumptions on the arrival process are violated, under what conditions consistency is possible with existing techniques, and what avenues exist for improving their accuracy and overhead. To investigate these issues, we first use random-measure theory to develop a novel churn model that allows rather general non-stationary scenarios and even synchronized joins (e.g., flash crowds). We not only dispose with common assumptions, such as existence of arrival rate and ergodicity, but also show that this model can produce all metrics of interest (e.g., sampled lifetime distributions, bandwidth overhead) using simple expressions. We apply these results to study the accuracy of prior techniques and discover that they are biased unless user lifetimes are exponential or the arrival measure is stationary. To overcome these limitations, we then create a new lifetime-sampling technique that remains asymptotically robust under all periodic arrival measures and provide a methodology for undoing the bias in the sampled arrival rate created by missed users. We demonstrate that the proposed approach exhibits accuracy advantages and 1-2 orders of magnitude less bandwidth consumption compared to the alternatives. We finish by implementing the proposed framework and applying it to experimental data from massive crawls of Gnutella.

Precise large deviations of aggregate claims in a risk model with size dependence and non stationary arrivals

ABSTRACTConsider a risk model with claims of heavy tails for non stationary arrival processes that satisfy a large-deviation principle. Assume that the claim sizes and interarrival times form a sequence of random pairs, with each pair obeying a dependence structure via the conditional distribution of the interarrival time given the subsequent claim size being large, and then a precise large-deviation formula of the aggregate amount of claims is obtained.

A statistical Markov chain approximation of transient hospital inpatient inventory

Inventory levels are critical to the operations, management, and capacity decisions of inventory systems but can be difficult to model in heterogeneous, non-stationary throughput systems. The inpatient hospital is a complicated throughput system and, like most inventory systems, hospitals dynamically make managerial decisions based on short term subjective demand predictions. Specifically, short term hospital staffing, resource capacity, and finance decisions are made according to hospital inpatient inventory predictions. Inpatient inventory systems have non-stationary patient arrival and service processes. Previously developed models present poor inventory predictions due to model subjectivity, high model complexity, solely expected value predictions, and assumed stationary arrival and service processes. Also, no models present statistical testing for model significance and quality-of-fit. This paper presents a Markov chain probability model that uses maximum likelihood regression to predict the expectations and discrete distributions of transient inpatient inventories. The approach has a foundation in throughput theory, has low model complexity, and provides statistical significance and quality-of-fit tests unique to this Markov chain. The Markov chain is shown to have superior predictability over Seasonal ARIMA models.

Estimating network link characteristics using packet-pair dispersion: A discrete-time queueing theoretic analysis

We present a queueing theoretic analysis of packet-dispersion based probing. The links are modeled as independent discrete-time queues with i.i.d. arrivals. We first derive an algorithm to obtain the (joint) distribution of the separation between the probes at the destination(s) for a given distribution of the spacing at the input. The parameter estimates of the arrival processes are obtained as the minimizer of a cost function between the empirical and calculated distributions. We also carry out extensive simulations and numerical experiments on the model to study the performance of the estimation algorithm for some non stationary arrival process. We find that the estimations work fairly well for two queues in series and for multicast. We also identify issues related to packet-dispersion based parameter estimation when there are multiple queues.

A cross-layer optimized adaptive modulation and coding scheme for transmission of streaming media over wireless links

In this paper, a novel cross-layer Adaptive Modulation and Coding scheme that optimizes the overall packet loss (by both transmission errors and excessive delays) probability under a given arrival process is developed. To this end, an improved Large Deviations approximation for the fraction of packets that suffer from excessive queuing delay is proposed. This approximation is valid for G/G/1 queues with infinite buffers that are driven by stationary arrival and service processes which satisfy certain conditions. Such models can capture the time correlations in the amount of traffic generated by streaming media sources and the time varying service capacity of a wireless link. Through numerical examples, the proposed AMC policy is shown to achieve a significant reduction in the overall packet loss rate compared to previously proposed schemes. This algorithmic performance gain can be translated into a sizeable decrease in the required transmit power or an analogous increase in the rate of the arrival process, subject to a given maximum packet loss rate Quality of Service constraint. Furthermore, the proposed AMC policy can be combined with ARQ in order to achieve an even lower overall packet loss probability.

Fluid Models for Multiserver Queues with Abandonments

Deterministic fluid models are developed to provide simple first-order performance descriptions for multiserver queues with abandonment under heavy loads. Motivated by telephone call centers, the focus is on multiserver queues with a large number of servers and nonexponential service-time and time-to-abandon distributions. The first fluid model serves as an approximation for the G/GI/s+GI queueing model, which has a general stationary arrival process with arrival rate λ, independent and identically distributed (IID) service times with a general distribution, s servers and IID abandon times with a general distribution. The fluid model is useful in the overloaded regime, where λ > s, which is often realistic because only a small amount of abandonment can keep the system stable. Numerical experiments, using simulation for M/GI/s+GI models and exact numerical algorithms for M/M/s+M models, show that the fluid model provides useful approximations for steady-state performance measures when the system is heavily loaded. The fluid model accurately shows that steady-state performance depends strongly upon the time-to-abandon distribution beyond its mean, but not upon the service-time distribution beyond its mean. The second fluid model is a discrete-time fluid model, which serves as an approximation for the Gt (n)/GI/s+GI queueing model, having a state-dependent and time-dependent arrival process. The discrete-time framework is exploited to prove that properly scaled queueing processes in the queueing model converge to fluid functions as s → ∞. The discrete-time framework is also convenient for calculating the time-dependent fluid performance descriptions.

Stability of Data Networks: Stationary and Bursty Models

This paper studies stability of network models that capture macroscopic features of data communication networks, including the Internet. The network model consists of a set of links and a set of possible routes that are fixed subsets of links. A connection is dynamically established along one of the routes to transmit data as requested and is terminated after the transmission is over. The transmission bandwidth of a link is dynamically allocated, according to specific bandwidth allocation policy, to ongoing connections that traverse the link. A network model is said to be stable under a given bandwidth allocation policy if, roughly, the number of ongoing connections in the network will not blow up over time. We consider a stationary and a bursty network model; the former assumes stochastically stationary arrival processes of connections as did many theoretical studies, while the latter allows more realistic bursty and correlated arrival processes. For both models under a necessary stability condition (i.e., the average offered transmission load on each link is within its bandwidth capacity), we show that the proportionally fair, the minimum potential delay, the max-min fair, and a class of utility-maximizing bandwidth allocation policies ensure network model stability, while some priority-oriented and maximum throughput policies do not. Interestingly, the bandwidth allocation policy that maximizes the arctan(·) utility ensures the stability of the stationary model but not the bursty model. This raises a serious concern about the current practice in the Internet protocol design, since such a policy is thought of as a good approximation of one of the most widely used TCP in the Internet.

Entropy and the timing capacity of discrete queues

Queueing systems which map Poisson input processes to Poisson output processes have been well-studied in classical queueing theory. This paper considers two discrete-time queues whose analogs in continuous-time possess the Poisson-in-Poisson-out property. It is shown that when packets arriving according to an arbitrary ergodic stationary arrival process are passed through these queueing systems, the corresponding departure process has an entropy rate no less (some times strictly more) than the entropy rate of the arrival process. Some useful by-products are discrete-time versions of: (i) a proof of the celebrated Burke's (1956) theorem, (ii) a proof of the uniqueness, amongst renewal inputs, of the Poisson process as a fixed point for exponential server queues proposed by Anantharam (1993), and (iii) connections with the timing capacity of queues described by Anantharam and Verdu (1996).

The Stationary Arrival Process of Independent Diffusers from a Continuum to an Absorbing Boundary Is Poissonian

We consider the arrival process of infinitely many identical independent diffusion processes from an infinite bath to an absorbing boundary. Previous results on this problem were confined to independent Brownian particles arriving at an absorbing sphere. The present paper extends these results to general diffusion processes, without any symmetries and without resorting to explicit expressions for solutions to the relevant equations. It is shown that for general absorbing boundaries and force fields, the steady stream of arrivals is Poissonian with rate equal to the total flux on the absorbing boundary, as calculated from the continuum theory of diffusion with transport. The considered arrival problem arises in the theory of Langevin simulations of ions in electrolytic solutions. In a Langevin simulation ions enter and exit the simulation region, and it is necessary to compute the probability laws for their entrance times into the simulation. While the simulated ions inside the small simulation region interact with each other and with the far field of the surrounding bath and the applied voltage, the physical chemistry continuum description of the surrounding bath implies independent diffusion in a mean field. Under these conditions the result of this paper applies to the stream of new ions that arrive from the continuum bath into the discrete simulation region. The recirculation problem, of ions that have already visited and exited the simulation region, as well as the integration of these results into a simulation of interacting ions will be studied in separate papers.

Open-loop routeing to M parallel servers with no buffers

In this paper we study the assignment of packets to M parallel heterogeneous servers with no buffers. The controller does not have knowledge on the state of the servers and bases all decisions on the number of time slots ago that packets have been sent to the different servers. The objective of the controller is to minimize the expected average cost. We consider a general stationary arrival process, with no independence assumptions. We show that the problem can be transformed into a Markov Decision Process (MDP). We further show under quite general conditions on cost functions that only a finite number of states have to be considered in this MDP, which then implies the optimality of periodic policies. For the case of two servers we obtain a more detailed structure of the cost and optimal policies. In particular we show that the average cost function is multimodular, and we obtain expressions for the cost for MMPP and MAP processes. Finally we present an application to optimal robot scheduling for Web search engines.

Open-loop routeing to M parallel servers with no buffers

In this paper we study the assignment of packets to M parallel heterogeneous servers with no buffers. The controller does not have knowledge on the state of the servers and bases all decisions on the number of time slots ago that packets have been sent to the different servers. The objective of the controller is to minimize the expected average cost. We consider a general stationary arrival process, with no independence assumptions. We show that the problem can be transformed into a Markov Decision Process (MDP). We further show under quite general conditions on cost functions that only a finite number of states have to be considered in this MDP, which then implies the optimality of periodic policies. For the case of two servers we obtain a more detailed structure of the cost and optimal policies. In particular we show that the average cost function is multimodular, and we obtain expressions for the cost for MMPP and MAP processes. Finally we present an application to optimal robot scheduling for Web search engines.

On variations of queue response for inputs with the same mean and autocorrelation function

This paper explores the variations in mean queue length for stationary arrival processes with the same mean and autocorrelation functions, or equivalently, the same mean and power spectrum. Three types of processes, namely, two-state Markov-modulated Poisson processes, periodic-sequence modulated Poisson processes and processes generated by randomly filtering a white noise process, are investigated. Results show that the mean queue length can vary substantially for the first type of process, and can vary moderately for the second type of process, as the parameters of the processes are varied, subject to a specified mean and autocorrelation function. However, the mean queue lengths for the third type of arrival processes are determined by the input mean and autocorrelation functions. The results suggest that the queueing performance can be hard to predict from spectral data alone when the power in low frequencies is large.

Stochastic Bounds for Queueing Systems with Multiple On–Off Sources

Consider a queueing system where the input traffic consists of background traffic, modeled by a Markov Arrival Process, and foreground traffic modeled by N ≥ 1 homogeneous on–off sources. The queueing system has an increasing and concave service rate, which includes as a particular case multiserver queueing systems. Both the infinite-capacity and the finite-capacity buffer cases are analyzed. We show that the queue length in the infinite-capacity buffer system (respectively, the number of losses in the finite-capacity buffer system) is larger in the increasing convex order sense (respectively, the strong stochastic order sense) than the queue length (respectively, the number of losses) of the queueing system with the same background traffic and M N homogeneous on–off sources of the same total intensity as the foreground traffic, where M is an arbitrary integer. As a consequence, the queue length and the loss with a foreground traffic of multiple homogeneous on–off sources is upper bounded by that with a single on–off source and lower bounded by a Poisson source, where the bounds are obtained in the increasing convex order (respectively, the strong stochastic order). We also compare N ≥ 1 homogeneous arbitrary two-state Markov Modulated Poisson Process sources. We prove the monotonicity of the queue length in the transition rates and its convexity in the arrival rates. Standard techniques could not be used due to the different state spaces that we compare. We propose a new approach for the stochastic comparison of queues using dynamic programming which involves initially stationary arrival processes.

Stationary-Process Approximations for the Nonstationary Erlang Loss Model

In this paper we consider the Mt/G/s/0 model, which has s servers in parallel, no extra waiting space, and i.i.d. service times that are independent of a nonhomogeneous Poisson arrival process. Arrivals finding all servers busy are blocked (lost). We consider approximations for the average blocking probabilities over subintervals (e.g., an hour when the expected service time is five minutes) obtained by replacing the nonstationary arrival process over that subinterval by a stationary arrival process. The stationary-Poisson approximation, using a Poisson (M) process with the average rate, tends to significantly underestimate the blocking probability. We obtain much better approximations by using a non-Poisson stationary (G) arrival process with higher stochastic variability to capture the effect of the time-varying deterministic arrival rate. In particular, we propose a specific approximation based on the heavy-traffic peakedness formula, which is easy to apply with either known arrival-rate functions or data from system measurements. We compare these approximations to exact numerical results for the Mt/M/s/0 model with linear arrival rate.

Convergence of Departures in Tandem Networks of·/GI/ ∞ Queues

We consider an infinite series of independent and identical -/GI/∞ queues fed by an arbitrary stationary and ergodic arrival process, A1. Let Ai be the arrival process to the ith node, and let νi be the law of Ai. Denote by (·) the input–output map of the ·/GI/∞ node; that is, νi+1 = (νi). It is known that the Poisson process is a fixed point for . In this paper, we are interested in the asymptotic distribution of the departure process from the nth node, νn+1 = n(ν1), as n ∞. Using couplings for random walks, this limiting distribution is shown to be either a Poisson process or a stationary ν-Poisson process, depending on the joint distribution of A1 and the service process. This generalizes a result of Vere-Jones (1968, Journal of the Royal Statistical Society, Series B 30: 321–333) and is similar in flavor to Mountford and Prabhakar (1995, Annals of Applied Probability 5(1): 121–127), where Poisson convergence is established for departures from a series of exponential server queues using coupling methods.

Exponential Approximations for Tail Probabilities in Queues II: Sojourn Time and Workload

We continue to focus on simple exponential approximations for steady-state tail probabilities in queues based on asymptotics. For the G/GI/1 model with i.i.d. service times that are independent of an arbitrary stationary arrival process, we relate the asymptotics for the steady-state waiting time, sojourn time, and workload. We show that the three asymptotic decay rates coincide and that the three asymptotic constants are simply related. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact numerical results for BMAP/G/1 queues, which have batch Markovian arrival processes. Numerical examples show that the exponential approximations for the tail probabilities are remarkably accurate at the 90th percentile and beyond. Thus, these exponential approximations appear very promising for applications.

The input-output map of a monotone discrete-time quasireversible node (queueing theory)

A class of discrete-time quasi-reversible nodes called monotone, which includes discrete-time analogs of the ./M/ infinity and ./M/1 nodes, is considered. For stationary ergodic nonnegative integer valued arrival processes, the existence and uniqueness of stationary regimes are proven when a natural rate condition is met. Coupling is used to prove the contractiveness of the input-output map relative to a natural distance on the space of stationary arrival processes that is analogous to Ornstein's d distance. A consequence is that the only stationary ergodic fixed points of the input-output map are the processes of independent and identically distributed Poisson random variables meeting the rate condition. >

Stability, monotonicity and invariant quantities in general polling systems

Consider a polling system withK≥1 queues and a single server that visits the queues in a cyclic order. The polling discipline in each queue is of general gated-type or exhaustive-type. We assume that in each queue the arrival times form a Poisson process, and that the service times, the walking times, as well as the set-up times form sequences of independent and identically distributed random variables. For such a system, we provide a sufficient condition under which the vector of queue lengths is stable. We treat several criteria for stability: the ergodicity of the process, the geometric ergodicity, and the geometric rate of convergence of the first moment. The ergodicity implies the weak convergence of station times, intervisit times and cycle times. Next, we show that the queue lengths, station times, intervisit times and cycle times are stochastically increasing in arrival rates, in service times, in walking times and in setup times. The stability conditions and the stochastic monotonicity results are extended to the polling systems with additional customer routing between the queues, as well as bulk and correlated arrivals. Finally, we prove that the mean cycle time, the mean intervisit time and the mean station times are invariant under general service disciplines and general stationary arrival and service processes.