Workload Process Research Articles

Justifying Diffusion Approximations for Stochastic Processing Networks Under a Moment Condition

Stochastic processing networks (SPN) are, in general, difficult objects to study analytically. The diffusion approximation refers to using the stationary distribution of the diffusion limit as an approximation of the diffusion-scaled process (say, the workload) in the original SPN. To validate such an approximation amounts to justifying the interchange of two limits, t→∞ and k→∞, with t being the time index and k, the scaling parameter. Here, we show this interchange of limits is justified for a broad class of SPN under a p*-th moment condition on the primitive data, interarrival and service times; and we provide an explicit characterization of the required order (p*), which depends naturally on the desired order of convergence of the workload process. To illustrate the generality of this moment condition, we first use it to establish the justification for resource sharing networks, where, to be processed each job needs to concurrently occupy multiple resources (servers), whereas each resource is shared among different job classes following a so-called proportional fair allocation scheme. We then show the same approach applies to most multi-class queueing networks that are known to have diffusion limits.

M/E<sub>n</sub>/1 대기모형에서 얼랑분포의 성질을 이용한 오버슛의 분포에 대한 근사

본 논문은 <TEX>$M/E_n/1$</TEX> 대기모형에서 작업부하량과정의 오버슛의 분포함수에 대한 근사식을 제안한다. 오버슛이란 작업부하량과정이 미리 정해진 한계점을 처음으로 초과할 때 초과하는 양을 말하는데 정확한 분포함수는 수학적인 표현으로만 얻어졌을 뿐 분포함수를 실제로 계산하는 것은 거의 불가능하다. 그래서 기존 연구에서는 오버슛에 관한 몇가지 성질을 이용하여 오버슛의 분포함수에 대한 근사식이 구해졌다. 본 논문은 고객의 서비스시간의 분포가 얼랑분포라는 점을 활용하여 기존에 얻어진 근사식보다 더 정확한 근사식을 제안한다. 그리고 제안한 근사식이 얼마나 참값에 가까운지 판단하기 위하여 시뮬레이션을 통하여 얻어진 오버슛의 분포함수와 비교한다. We consider an <TEX>$M/E_n/1$</TEX> queueing model where customers arrive at a facility with a single server according to a Poisson process with customer service times assumed to be independent and identically distributed with Erlang distribution. We concentrate on the overshoot of the workload process in the queue. The overshoot means the excess over a threshold at the moment where the workload process exceeds the threshold. The approximation of the distribution of the overshoot was proposed by Bae et al. (2011); however, but the accuracy of the approximation was unsatisfactory. We derive an advanced approximation using the property of the Erlang distribution. Finally the newly proposed approximation is compared with the results of the previous study.

A correlated overflow model with a view towards applications in credit risk

In this paper we present and explicitly solve a specific multi-dimensional, correlated overflow problem. Due to the ordering of the underlying components in our model, explicit results are obtained for the probabilities under consideration; importantly, the results are not in terms of Laplace transforms. In our setting, each component behaves as a compound Poisson process with unit-sized upward jumps, decreased by a linear drift. The approach relies on a Benes (1963) type argumentation, that is, the idea of partitioning the overflow event with respect to the last 'exceedance epoch'. This type of problems arises naturally in various branches of applied probability, and therefore has several applications. In this paper we point out two specific application areas: one in mathematical finance, and one in queueing. In the former, several 'obligors' (whose 'distances-to-default' are correlated, as is the case in practice) are considered, and it is quantified how likely it is that there are defaults before a given time T. In the latter, one is often interested in the workload or storage process; the results of this paper provide expressions for overflow probabilities in the context of specific coupled queuing systems.

AN EXTENSION OF THE MATRIX-ANALYTIC METHOD FOR M/G/1-TYPE MARKOV PROCESSES

We consider a bivariate Markov process f (U(t);S(t)); t � 0g , where U(t) (t � 0) takes values in (0; 1 ) and S(t) (t � 0) takes values in a nite set. We assume that U(t) (t � 0) is skip-free to the left, and therefore we call it the M/G/1-type Markov process. The M/G/1-type Markov process was rst introduced as a generalization of the workload process in the MAP/G/1 queue and its stationary distribution was analyzed under a strong assumption that the conditional innitesimal generator of the underlying Markov chain S(t) given U(t) > 0 is irreducible. In this paper, we extend known results for the stationary distribution to the case that the conditional innitesimal generator of the underlying Markov chain given U(t) > 0 is reducible. With this extension, those results become applicable to the analysis of a certain class of queueing models.

Characterizing the impact of the workload on the value of dynamic resizing in data centers

Energy consumption imposes a significant cost for data centers; yet much of that energy is used to maintain excess service capacity during periods of predictably low load. Resultantly, there has recently been interest in developing designs that allow the service capacity to be dynamically resized to match the current workload. However, there is still much debate about the value of such approaches in real settings. In this paper, we show that the value of dynamic resizing is highly dependent on statistics of the workload process. In particular, both slow time-scale non-stationarities of the workload (e.g., the peak-to-mean ratio) and the fast time-scale stochasticity (e.g., the burstiness of arrivals) play key roles. To illustrate the impact of these factors, we combine optimization-based modeling of the slow time-scale with stochastic modeling of the fast time-scale. Within this framework, we provide both analytic and numerical results characterizing when dynamic resizing does (and does not) provide benefits.

Two-node fluid network with a heavy-tailed random input: the strong stability case

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

Two-node fluid network with a heavy-tailed random input: the strong stability case

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case ofstrong stabilitywhere both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

Queues and Risk Models with Simultaneous Arrivals

We focus on a particular connection between queueing and risk models in a multidimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue), we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline of how the two-dimensional risk model with a general two-dimensional claim size distribution (hence, without ordering of claim sizes) is related to a known Riemann boundary-value problem.

Queues and Risk Models with Simultaneous Arrivals

We focus on a particular connection between queueing and risk models in a multidimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue), we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline of how the two-dimensional risk model with a general two-dimensional claim size distribution (hence, without ordering of claim sizes) is related to a known Riemann boundary-value problem.

Detection of variations in cognitive workload using multi-modality physiological sensors and a large margin unbiased regression machine.

Physiological sensor based workload estimation technology provides a real-time means for assessing cognitive workload and has a broad range of applications in cognitive ergonomics, mental health monitoring, etc. In this paper we report a study on detecting changes in workload using multi-modality physiological sensors and a novel feature extraction and classification algorithm. We conducted a cognitive workload experiment involving multiple subjects and collected an extensive data set of EEG, ECG and GSR signals. We show that the GSR signal is consistent with the variations of cognitive workload in 75% of the samples. To explore cardiac patterns in ECG that are potentially correlated with the cognitive workload process, we computed various heart-rate-variability features. To extract neuronal activity patterns in EEG related to cognitive workload, we introduced a filter bank common spatial pattern filtering technique. As there can be large variations in e.g. individual responses to the cognitive workload, we propose a large margin unbiased recursive feature extraction and regression method. Our leave-one-subject-out cross validation test shows that, using the proposed method, EEG can provide significantly better prediction of the cognitive workload variation than ECG, with 87.5% vs 62.5% in accuracy rate.

Accelerated Consistent Estimation of a High Load Probability in M/G/1 and GI/G/1 Queues*

A special speed-up simulation technique based on a combination of splitting and regenerative approach is developed to estimate a rare event probability. A consistent estimation of the probability that the workload process in M/G/1 and GI/G/1 queues exceeds a high level within a regeneration cycle is considered. A similar problem is studied for the queue-size process in an M/G/1 queue. The key element of the approach is to consider an embedded Markov chain instead of the original non-Markovian process. A comparison with the known analytical results and with the crude Monte Carlo (CMC) simulation shows a high efficiency of the approach.

Stability Analysis of a General State-Dependent Multiserver Queue

The paper studies a general multiserver queue in which the service time of an arriving customer and the next interarrival period may depend on both the current waiting time and the server assigned to the arriving customer. Stability of the system is proved under general assumptions on the predetermined distributions describing the model. The proof exploits a combination of the Markov property of the workload process with a regenerative property of the process. The key idea leading to stability is a characterization of the limit behavior of the forward renewal process generated by regenerations. Extensions of the basic model are also studied. This model generalizes a number of the known state-dependent systems and describes some new systems as particular cases.

On the tail asymptotics of the area swept under the Brownian storage graph

In this paper, the area swept under the workload graph is analyzed: with {Q(t): t≥0} denoting the stationary workload process, the asymptotic behavior of πT(u)(u):=P(∫T(u)0Q(r)dr>u) is analyzed. Focusing on regulated Brownian motion, first the exact asymptotics of πT(u)(u) are given for the case that T(u) grows slower than u√, and then logarithmic asymptotics for (i) T(u)=Tu√ (relying on sample-path large deviations), and (ii) u√=o(T(u)) but T(u)=o(u). Finally, the Laplace transform of the residual busy period are given in terms of the Airy function.

Validity of Heavy-Traffic Steady-State Approximations in Multiclass Queueing Networks: The Case of Queue-Ratio Disciplines

A class of stochastic processes known as semi-martingale reflecting Brownian motions (SRBMs) is often used to approximate the dynamics of heavily loaded queueing networks. In two influential papers, Bramson [Bramson M (1998) State space collapse with applications to heavy-traffic limits for multiclass queueing networks. Queueing Systems 30:89–148] and Williams [Williams RJ (1998b) Diffusion approximations for open multiclass queueuing networks: Sufficient conditions involving state space collapse. Queueing Systems 30:27–88] laid out a general and structured approach for proving the validity of such heavy-traffic approximations, in which an SRBM is obtained as a diffusion limit from a sequence of suitably normalized workload processes. However, for multiclass networks it is still not known in general whether the steady-state distribution of the SRBM provides a valid approximation for the steady-state distribution of the original network. In this paper we study the case of queue-ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative. In addition to standard assumptions made in the literature towards the stability of the pre- and post-limit processes and the existence of diffusion limits, we add a requirement that solutions to the fluid model are attracted to the invariant manifold at a linear rate. For the special case of static-priority networks such linear attraction is known to hold under certain conditions on the network primitives. The analysis elucidates interesting connections between stability of the pre- and post-limit processes, their respective fluid models and state-space collapse, and identifies the respective roles played by all of the above in establishing validity of heavy-traffic steady-state approximations.

Two Coupled Lévy Queues with Independent Input

We consider a pair of coupled queues driven by independent spectrally-positive Lévy processes. With respect to the bi-variate workload process this framework includes both the coupled processor model and the two-server fluid network with independent Lévy inputs. We identify the joint transform of the stationary workload distribution in terms of Wiener-Hopf factors corresponding to two auxiliary Lévy processes with explicit Laplace exponents. We reinterpret and extend the ideas of Cohen and Boxma (1983) to provide a general and uniform result with a neat transform expression.

Optimal Rate Scheduling via Utility-Maximization for J-User MIMO Markov Fading Wireless Channels with Cooperation

We design a dynamic rate scheduling policy of Markov type by using the solution (a social optimal Nash equilibrium point) to a utility-maximization problem over a randomly evolving capacity set for a stochastic system of generalized processor-sharing queues in a random environment whose job arrivals to each queue follow a doubly stochastic renewal process (DSRP). Both the random environment and the random arrival rate of each DSRP are driven by a finite state continuous time Markov chain. The scheduling policy optimizes in a greedy fashion with respect to each queue and environmental state. Since the closed-form solution for the performance of such a queuing system under the policy is difficult to obtain, we establish a reflecting diffusion with regime-switching model for its measures of performance. Furthermore, we justify its asymptotic optimality by deriving the stochastic fluid and diffusion limits for the corresponding system under heavy traffic. In addition, we identify a cost function related to the utility function, which is minimized by minimizing the workload process in the diffusion limit. More importantly, our queuing model includes typical systems in the future wireless networks, such as the J-user multi-input multioutput multiple access channel and the broadcast channel under Markov fading with cooperation and admission control as special cases.

Predicting the continuation of a function with applications to call center data

We show how to construct the best linear unbiased predictor (BLUP) for the continuation of a curve, and apply the proposed estimator to real-world call center data. Using the BLUP, we demonstrate prediction of the workload process, both directly and based on prediction of the arrival counts. The Matlab code and all data sets in the presented examples are available in the supplementary material.

Asymptotics of the area under the graph of a Lévy-driven workload process

Let (Qt)t∈R be the stationary workload process of a Lévy-driven queue, where the driving Lévy process is light-tailed. For various functions T(u), we analyze P(∫0T(u)Qsds>u) for u large. For T(u)=o(u) the asymptotics resemble those of the steady-state workload being larger than u/T(u). If T(u) is proportional to u they look like e−αu for some α>0. Interestingly, the asymptotics are still valid when u=o(T(u)), T(u)=o(u), and T(u)=βu for β suitably small.

A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).