Workload Process Research Articles

Justifying diffusion approximations for multiclass queueing networks under a moment condition

Multiclass queueing networks (MQN) 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 MQN. To validate such an approximation amounts to justifying the interchange of two limits, $t\to\infty$ and $k\to\infty$, with $t$ being the time index and $k$, the scaling parameter. Here, we show this interchange of limits is justified under a $p^{*}$th moment condition on the primitive data, the interarrival and service times; and we provide an explicit characterization of the required order ($p^{*}$), which depends naturally on the desired order of moment of the workload process.

Platform modelling and scheduling game with multiple intelligent cloud-computing pools for big data

ABSTRACTWe develop a generic game platform that can be used to model various real-world systems with multiple intelligent cloud-computing pools and parallel-queues for resources-competing users. Inside the platform, the software structure is modelled as Blockchain. All the users are associated with Big Data arrival streams whose random dynamics is modelled by triply stochastic renewal reward processes (TSRRPs). Each user may be served simultaneously by multiple pools while each pool with parallel-servers may also serve multi-users at the same time via smart policies in the Blockchain, e.g. a Nash equilibrium point myopically at each fixed time to a game-theoretic scheduling problem. To illustrate the effectiveness of our game platform, we model the performance measures of its internal data flow dynamics (queue length and workload processes) as reflecting diffusion with regime-switchings (RDRSs) under our scheduling policies. By RDRS models, we can prove our myopic game-theoretic policy to be an asymptotic Pareto minimal-dual-cost Nash equilibrium one globally over the whole time horizon to a randomly evolving dynamic game problem. Iterative schemes for simulating our multi-dimensional RDRS models are also developed with the support of numerical comparisons.

Conditional sojourn time distributions in M/G/1 and G/M/1 queues under PλM-service policy

PMλ-service policy is a workload dependent hysteretic policy. The policy has two service states comprised of the ordinary stage and the fast stage. An ordinary service stage is initiated by the arrival of a customer in an idle state. When the workload of the server surpasses threshold λ, the ordinary service stage changes to the fast service state, and it continues until the system is empty. These service stages alternate in this manner. When the cost of changing service stages is high, the hysteretic policy is more efficient than the threshold policy, where a service stage changes immediately into the other service stage at either case of the workload’s surpassing or crossing down a threshold. PMλ-service policy is a modification of PMλ-policy proposed to control finite dams, and also an extension of the well-known D-policy. The distributions of the stationary workload of PMλ-service policy and its variants are studied well. However, there is no known result on the sojourn time distribution. We prove that there is a relation between the sojourn time of a customer and the first up-crossing time of the workload process over the threshold λ after the arrival of the customer. Using the relation and the duality of M/G/1 and G/M/1 queues, we obtain conditional sojourn time distributions in M/G/1 and G/M/1 queues under the policy.

The Overshoot in M/G/1 Queueing Model with Weibull Service Time

본 논문에서는 고객의 서비스시간이 와이블분포를 따르는 M/G/1 대기모형에서 작업부하량과정을 고려한다. 작업부하량은 어떤 특정한 시각에 대기 중인 고객들의 서비스시간과 서비스 받는 중인 고객의 잔여서비스 시간의 합으로 정의되는데 시각에 따라 변하는 확률변수이므로 확률과정으로서 작업부하량과정을 생각한다. 작업부하량이 0에서 출발하여 어느 정도 시간이 흐른 후 임계치를 넘어서는 순간 그 임계치를 초과한 양을 오버슛이라 부른다. 오버슛의 분포는 대기모형의 최적화 문제를 해결하기 위해서 반드시 계산해야만 하는 것이다. 고객의 서비스시간이 지수분포를 따를 때는 오버슛의 분포 역시 지수분포를 따르지만, 서비스시간이 지수분포가 아닌 다른 분포를 따를 때는 오버슛의 분포가 이론적으로만 구하여졌을 뿐, 실제로 계산하기는 거의 어려운 식으로 표현된다. 본 연구에서는 M/G/1 대기모형에서 고객의 서비스시간의 분포가 와이블분포인 경우에 오버슛의 분포를 근사적으로 계산할 수 있는 식을 제안한다. 근사식을 구하기 위하여 사전에 알려진 오버슛에 대한 성질을 이용할 것이다. 그리고 시뮬레이션을 통하여 제안한 근사식의 정확도를 평가한다.It is considered the workload process in the M/G/1 queueing model with Weibull service time. The workload means the sum of the service times of waiting customers in queue and the remaining service time of customer in service. The overshoot is the exceeding amount of workload process at the moment which the process crosses over a threshold. The distribution of the overshoot needs to be calculated for solving various optimization problem of queueing model. However, the formula for distribution of the overshoot cannot be used to manipulate the distribution function or moments of the overshoot. In this study, the approximation of the distribution function of the overshoot is proposed which is useful to solve the optimization problem of the model. To approximate the distribution function of overshoot, the properties of the Weibull distribution and some known results on the distribution of the overshoot in references are used. The simulation study is also carried to test the accuracy of the approximation. The approximated distribution function is compared with the estimated distribution function which is obtained by the simulation.

PTAT

As the memory access footprints of applications in areas like data analytics increase, the latency overhead of translation lookaside buffer (TLB) misses increases. Thus, the efficiency of TLB becomes increasingly critical for overall system performance. Analyzing TLB miss traces is useful for hardware architecture design and software application optimization. Utilizing cycle-accurate simulators or instrumentation tools is very time-consuming and/or inaccurate for tracing and profiling TLB misses. In this article, we propose an efficient and precise tool to collect and profile last-level TLB misses. This tool utilizes a novel software method called Page Table Access Tracing (PTAT), storing last-level page table entries of certain workload processes into a reserved uncached memory region. Therefore, each last-level TLB miss incurred by user process corresponds to one uncached page table access to main memory, which can be captured and recorded by a hardware memory bus monitor. The detected information is then dumped into offline storage. In this manner, full TLB miss traces are collected and can be analyzed flexibly. Compared to previous software-based methods, this method achieves higher performance. Experiments show that, compared with a state-of-the-art kernel instrumentation method (BadgerTrap), which lacks complete dumping trace function, the speedup is still up to 3.88-fold for memory-intensive benchmarks. Due to the improved efficiency and completeness of tracing, case studies validate that more flexible profiling can be conducted, which is of great significance for TLB performance optimization. The accuracy of PTAT is verified by both dedicated sequence and performance counters.

Occupation times for the finite buffer fluid queue with phase-type ON-times

In this short communication we study a fluid queue with a finite buffer. The performance measure we are interested in is the occupation time over a finite time period, i.e., the fraction of time the workload process is below some fixed target level. Using an alternating renewal sequence, we determine the double transform of the occupation time; the occupation time for the finite buffer M/G/1 queue with phase-type jumps follows as a limiting case.

Rare events of transitory queues

AbstractWe study the rare-event behavior of the workload process in a transitory queue, where the arrival epochs (or 'points') of a finite number of jobs are assumed to be the ordered statistics of independent and identically distributed (i.i.d.) random variables. The service times (or 'marks') of the jobs are assumed to be i.i.d. random variables with a general distribution, that are jointly independent of the arrival epochs. Under the assumption that the service times are strictly positive, we derive the large deviations principle (LDP) satisfied by the workload process. The analysis leverages the connection between ordered statistics and self-normalized sums of exponential random variables to establish the LDP. In this paper we present the first analysis of rare events in transitory queueing models, supplementing prior work that has focused on fluid and diffusion approximations.

Distraction or cognitive overload? Using modulations of the autonomic nervous system to discriminate the possible negative effects of advanced assistance system

The interaction with Advanced Driver Assistance Systems has several positive implications for road safety, but also some potential downsides such as mental workload and automation complacency. Malleable attentional resources allocation theory describes two possible processes that can generate workload in interaction with advanced assisting devices. The purpose of the present study is to determine if specific analysis of the different modalities of autonomic control of nervous system can be used to discriminate different potential workload processes generated during assisted-driving tasks and automation complacency situations. Thirty-five drivers were tested in a virtual scenario while using head-up advanced warning assistance system. Repeated MANOVA were used to examine changes in autonomic activity across a combination of different user interactions generated by the advanced assistance system: (1) expected take-over request without anticipatory warning; (2) expected take-over request with two-second anticipatory warning; (3) unexpected take-over request with misleading warning; (4) unexpected take-over request without warning. Results shows that analysis of autonomic modulations can discriminate two different resources allocation processes, related to different behavioral performances. The user's interaction that required divided attention under expected situations produced performance enhancement and reciprocally-coupled parasympathetic inhibition with sympathetic activity. At the same time, supervising interactions that generated automation complacency were described specifically by uncoupled sympathetic activation. Safety implications for automated assistance systems developments are considered.

Strong approximation method and the (functional) law of iterated logarithm for GI/G/1 queue

In this paper, a unified method based on the strong approximation (SA) of renewal process (RP) is developed for the law of the iterated logarithm (LIL) and the functional LIL (FLIL), which quantify the magnitude of the asymptotic rate of the increasing variability around the mean value of the RP in numerical and functional forms respectively. For the GI/G/1 queue, the method provides a complete analysis for both the LIL and the FLIL limits for four performance functions: The queue length, workload, busy time and idle time processes, covering three regimes divided by the traffic intensity.

Construction of Asymptotically Optimal Control for Crisscross Network from a Free Boundary Problem

An asymptotic framework for optimal control of multiclass stochastic processing networks, using formal diffusion approximations under suitable temporal and spatial scaling, by Brownian control problems (BCP) and their equivalent workload formulations (EWF), has been developed by Harrison (1988). This framework has been implemented in many works for constructing asymptotically optimal control policies for a broad range of stochastic network models. To date all asymptotic optimality results for such networks correspond to settings where the solution of the EWF is a reflected Brownian motion in ℝ+ or a wedge in [Formula: see text]. In this work we consider a well studied stochastic network which is perhaps the simplest example of a model with more than one dimensional workload process. In the regime considered here, the singular control problem corresponding to the EWF does not have a simple form explicit solution. However, by considering an associated free boundary problem one can give a representation for an optimal controlled process as a two dimensional reflected Brownian motion in a Lipschitz domain whose boundary is determined by the solution of the free boundary problem. Using the form of the optimal solution we propose a sequence of control policies, given in terms of suitable thresholds, for the scaled stochastic network control problems and prove that this sequence of policies is asymptotically optimal. As suggested by the solution of the EWF, the policy we propose requires a server to idle under certain conditions which are specified in terms of thresholds determined from the free boundary.

Heavy-traffic limits for a single-server queue leading up to a critical point

We establish heavy-traffic limits for the arrival and workload processes in a single-server queue with a time-varying arrival-rate function. We establish limits at and before a critical point, the onset of critical loading, where the arrival-rate function approaches its critical value from below. We extend results by Newell (1968) and Mandelbaum and Massey (1995) and present alternative views of the interesting scaling constants that arise in these limits.

Dynamic Scheduling for Parallel Server Systems in Heavy Traffic: Graphical Structure, Decoupled Workload Matrix and some Sufficient Conditions for Solvability of the Brownian Control Problem

We consider a dynamic scheduling problem for parallel server systems. J. M. Harrison has proposed a scheme for using diffusion control problems to approximately solve such control problems for heavily loaded systems. This approach has been very successfully used in the special case when the diffusion control problem can be reduced to an equivalent one for a one-dimensional workload process. However, it remains a challenging open problem to make substantial progress on using Harrison’s scheme when the workload process is more than one-dimensional. Here we present some new structural results concerning the diffusion control problem for parallel server systems with arbitrary workload dimension. Specifically, we prove that a certain server-buffer graph associated with a parallel server system is a forest of trees. We then exploit this graphical structure to prove that there exists a matrix, used to define the workload process, that has a block diagonal-like structure. An important feature of this matrix is that, except when the workload is one-dimensional, this matrix is frequently different from a choice of workload matrix proposed by Harrison. We demonstrate that our workload matrix simplifies the structure of the control problem for the workload process by proving that when the original diffusion control problem has linear holding costs, the equivalent workload formulation also has a linear cost function. We also use this simplification to give sufficient conditions for a certain least control process to be an optimal control for the diffusion control problem with linear holding costs. Under these conditions, we propose a continuous review threshold-type control policy for the original parallel server system that exploits pooling of servers within trees in the server-buffer graph and uses non-basic activities connecting different trees in a critical manner. We call this partial pooling. We conjecture that this threshold policy is asymptotically optimal in the heavy traffic limit. We illustrate the solution of the diffusion control problem and our proposed threshold control policy for a three-buffer, three-server example.

Transient analysis of one-sided Lévy-driven queues

ABSTRACTIn this article, we analyze the transient behavior of the workload process in a Lévy-driven queue. We are interested in the value of the workload process at a random epoch; this epoch is distributed as the sum of independent exponential random variables. We consider both cases of spectrally one-sided Lévy input processes, for which we succeed in deriving explicit results. As an application, we approximate the mean and the Laplace transform of the workload process after a deterministic time.

A two-queue polling model with priority on one queue and heavy-tailed On/Off sources: a heavy-traffic limit

We consider a single-server polling system consisting of two queues of fluid with arrival process generated by a big number of heavy-tailed On/Off sources, and application in road traffic and communication systems. Class-j fluid is assigned to queue j, $$j=1,2$$j=1,2. Server 2 visits both queues to process or let pass the corresponding fluid class. If there is class-2 fluid in the system, it is processed by server 2 until the queue is empty, and only then server 2 visits queue 1, revisiting queue 2 and restarting the cycle as soon as new class-2 fluid arrives, with zero switchover times. Server 1 is an "extra" server which continuously processes class-1 fluid (if there is any). During the visits of server 2 to queue 1, class-1 fluid is simultaneously processed by both servers (possibly at different speeds). We prove a heavy-traffic limit theorem for a suitable workload process associated with this model. Our limit process is a two-dimensional reflected fractional Brownian motion living in a convex polyhedron. A key ingredient in the proof is a version of the Invariance Principle of Semimartingale reflecting Brownian motions which, in turn, is also proved.

Dynamic scheduling for parallel server systems in heavy traffic: Graphical structure, decoupled workload matrix and some sufficient conditions for solvability of the Brownian control problem

We consider a dynamic scheduling problem for parallel server systems. J. M. Harrison has proposed a scheme for using diffusion control problems to approximately solve such control problems for heavily loaded systems. This approach has been very successfully used in the special case when the diffusion control problem can be reduced to an equivalent one for a one-dimensional workload process. However, it remains a challenging open problem to make substantial progress on using Harrison’s scheme when the workload process is more than one-dimensional. Here we present some new structural results concerning the diffusion control problem for parallel server systems with arbitrary workload dimension. Specifically, we prove that a certain server-buffer graph associated with a parallel server system is a forest of trees. We then exploit this graphical structure to prove that there exists a matrix, used to define the workload process, that has a block diagonal-like structure. An important feature of this matrix is that, except when the workload is one-dimensional, this matrix is frequently different from a choice of workload matrix proposed by Harrison. We demonstrate that our workload matrix simplifies the structure of the control problem for the workload process by proving that when the original diffusion control problem has linear holding costs, the equivalent workload formulation also has a linear cost function. We also use this simplification to give sufficient conditions for a certain least control process to be an optimal control for the diffusion control problem with linear holding costs. Under these conditions, we propose a continuous review threshold-type control policy for the original parallel server system that exploits pooling of servers within trees in the server-buffer graph and uses non-basic activities connecting different trees in a critical manner. We call this partial pooling. We conjecture that this threshold policy is asymptotically optimal in the heavy traffic limit. We illustrate the solution of the diffusion control problem and our proposed threshold control policy for a three-buffer, three-server example.

The rate of convergence to stationarity for M/G/1 models with admission controls via coupling

We study the workload processes of two M/G/1 queueing systems with restricted capacity: in Model 1 any service requirement that would exceed a certain capacity threshold is truncated; in Model 2 new arrivals do not enter the system if they have to wait more than a fixed threshold time in line. For Model 1 we obtain several results concerning the rate of convergence to equilibrium. In particular, we derive uniform bounds for geometric ergodicity with respect to certain subclasses. For Model 2 geometric ergodicity follows from the finiteness of the moment-generating function of the service time distribution. We derive bounds for the convergence rates in special cases. The proofs use the coupling method.

Diffusion limits for shortest remaining processing time queues under nonstandard spatial scaling

We develop a heavy traffic diffusion limit theorem under nonstandard spatial scaling for the queue length process in a single server queue employing shortest remaining processing time (SRPT). For processing time distributions with unbounded support, it has been shown that standard diffusion scaling yields an identically zero limit. We specify an alternative spatial scaling that produces a nonzero limit. Our model allows for renewal arrivals and i.i.d. processing times satisfying a rapid variation condition. We add a corrective spatial scale factor to standard diffusion scaling, and specify conditions under which the sequence of unconventionally scaled queue length processes converges in distribution to the same nonzero reflected Brownian motion to which the sequence of conventionally scaled workload processes converges. Consequently, this corrective spatial scale factor characterizes the order of magnitude difference between the queue length and workload processes of SRPT queues in heavy traffic. It is determined by the processing time distribution such that the rate at which it tends to infinity depends on the rate at which the tail of the processing time distribution tends to zero. For Weibull processing time distributions, we restate this result in a manner that makes the resulting state space collapse more apparent.

Perfect simulation of M/G/c queues

In this paper we describe a perfect simulation algorithm for the stable M/G/c queue. Sigman (2011) showed how to build a dominated coupling-from-the-past algorithm for perfect simulation of the super-stable M/G/c queue operating under first-come-first-served discipline. Sigman's method used a dominating process provided by the corresponding M/G/1 queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service). The method exploited the fact that the workload process for the M/G/1 queue remains the same under different queueing disciplines, in particular under the processor sharing discipline, for which a dynamic reversibility property holds. We generalise Sigman's construction to the stable case by comparing the M/G/c queue to a copy run under random assignment. This allows us to produce a naïve perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled M/G/c queues started respectively from the empty state and the state of the M/G/c queue under random assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarise statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.

Perfect simulation of M/G/c queues

In this paper we describe a perfect simulation algorithm for the stable M/G/cqueue. Sigman (2011) showed how to build a dominated coupling-from-the-past algorithm for perfect simulation of the super-stable M/G/cqueue operating under first-come-first-served discipline. Sigman's method used a dominating process provided by the corresponding M/G/1 queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service). The method exploited the fact that the workload process for the M/G/1 queue remains the same under different queueing disciplines, in particular under the processor sharing discipline, for which a dynamic reversibility property holds. We generalise Sigman's construction to the stable case by comparing the M/G/cqueue to a copy run under random assignment. This allows us to produce a naïve perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled M/G/cqueues started respectively from the empty state and the state of the M/G/cqueue under random assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarise statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.

A packet-switched network with On/Off sources and a fair bandwidth sharing policy: state space collapse and heavy-traffic

We consider a flow-level model for packet-switched telecommunications networks handling elastic flows with concurrent occupancy of resources, in which digital objects are transferred at a rate determined by capacity allocation on each route. The capacity of each node is dynamically allocated to the routes passing by it through a weighted proportional fair sharing policy, and the arrival request for transfer on each route is generated by N heavy-tailed On/Off sources. Under heavy-traffic, we combine state space collapse (SSC) and an Invariance Principle to show that when $$N\rightarrow +\infty $$N?+? the conveniently scaled workload and flow count processes converge. SSC establishes a relationship between the corresponding limits by means of a deterministic operator. In Theorem 1 we prove that assuming the other hypotheses hold, SSC is not only sufficient for the convergence, but necessary. In Theorem 2 we prove that when $$r\rightarrow +\infty $$r?+?, r being a scale parameter, the workload limit process converges to a reflected fractional Brownian motion on a polyhedral cone.