Work-conserving Service Disciplines Research Articles

On the Power-of-d-choices with Least Loaded Server Selection

Motivated by distributed schedulers that combine the power-of-d-choices with late binding and systems that use replication with cancellation-on-start, we study the performance of the LL(d) policy which assigns a job to a server that currently has the least workload among d randomly selected servers in large-scale homogeneous clusters. We consider general job size distributions and propose a partial integro-differential equation to describe the evolution of the system. This equation relies on the earlier proven ansatz for LL(d) which asserts that the workload distribution of any finite set of queues becomes independent of one another as the number of servers tends to infinity. Based on this equation we propose a fixed point iteration for the limiting workload distribution and study its convergence. For exponential job sizes we present a simple closed form expression for the limiting workload distribution that is valid for any work-conserving service discipline as well as for the limiting response time distribution in case of first-come-first-served scheduling. We further show that for phase-type distributed job sizes the limiting workload and response time distribution can be expressed via the unique solution of a simple set of ordinary differential equations. Numerical and analytical results that compare response time of the classic power-of-d-choices algorithm and the LL(d) policy are also presented and the accuracy of the limiting response time distribution for finite systems is illustrated using simulations.

Perfect and nearly perfect sampling of work-conserving queues

In this paper, we explore algorithms for perfect and nearly perfect sampling from the stationary distribution of the waiting times in various Poisson arrival multi-class and multi-server queues with non-preemptive work-conserving service disciplines. The service duration distributions of these classes may be identical or may vary from class to class. The algorithms follow the idea of dominated coupling from the past (Kendall, Adv Appl Probab 32:844–865 2000) and are variations on an algorithm of Sigman (J Appl Prob 48A:37–43, 2011). A coupled first come first serve queue is constructed for each work-conserving queue. When the service duration distributions do not vary, we achieve perfect simulation by finding times when the system is known to be totally idle. When the distributions differ, the totally idle times may be impossible to determine exactly, but we can achieve simulations with a specified error limit \(\epsilon > 0\).

Heavy traffic limit theorems for a queue with Poisson ON/OFF long-range dependent sources and general service time distribution

In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions (FBMs) or certain type of longrange dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little’s law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.

Lyapunov method for the stability of fluid networks

One of the primary tools in establishing the stability of a fluid network is to construct a Lyapunov function. In this paper, we establish the sufficiency in the use of a Lyapunov function. Specifically, we show that a necessary and sufficient condition for the stability of a generic fluid network is the existence of a Lyapunov function for its fluid level process. Then by applying this result to various specific fluid networks, including a fluid network under all work-conserving service disciplines, a fluid network under a priority service discipline, and a fluid network under a first-in-first-out service discipline, we establish the existence of a Lyapunov function for their fluid level processes is a necessary and sufficient condition for their stabilities. The result is also applied to various fluid limit models and a linear Skorohod problem.

Classification of Markov Processes of Matrix M/G/l type with a Tree Structure and its Applications to the MMAP[K]/G[K]/1 Queues

This paper studies the classification problem of discrete time and continuous time Markov processes of matrix M/G/l type with a tree structure. It is shown that the Perron-Frobenius eigenvalue of a nonnegative matrix provides information for a complete classification of the Markov process of interest. A computational method is developed to find whether a Markov process of matrix M/G/l type with a tree structure is positive recurrent, null recurrent, or transient. The method is then used to study the impact of the last-come-first-served general preemptive resume (LCFS-GPR) service discipline on the stability of a MAP/PHA queue. Two sufficient conditions are identified for the positive recurrence and transience of the Markov processes of interest, respectively. As an example, the results are used to show that the discrete time (and continuous time) MMAP[K]/G[K]/1queue with a work conserving service discipline is stable if and only if its traffic intensity is less than one, unstable if its traffic intensity is larger than one

Numerical Solution of Piecewise-Stationary Mt/Gt/1 Queues

We develop an algorithm for computing the (exact) cumulative distribution function of the time-dependent workload in a piecewise-stationary Mt/Gt/1 queue with a work-conserving service discipline and general service-time distributions, where service times are determined at arrival instants. The t subscripts indicate that the arrival rate and the general service-time distribution may change with time, but we allow changes only at finitely many time points. The algorithm is based on numerical transform inversion, using the classical Takács double-transform of the transient workload in an M/G/1 queue recursively over the successive stationary intervals. In particular, we apply our recently developed Fourier-series-based inversion algorithms for two-dimensional transforms and nested one-dimensional transforms. We also do additional work to greatly speed up the computation while tightly controlling the error. As a consequence, the computation time grows only quadratically with the number of intervals. The algorithm is effective for ten or fewer intervals, where the intervals may have unlimited and possibly unequal lengths, typically running in at most a few minutes and maintaining high accuracy. We have also demonstrated that the algorithm can solve a 21-interval example with 7-to-10-digit accuracy in about half an hour. Models with only a few intervals are useful to study overload control strategies.

Convex ordering of the attained waiting times in single-server queues and related problems

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

The mean waiting time in a G/G/m/∞ queue with the LCFS-P service discipline

A single- or multiserver queue with work-conserving service discipline and a stationary and ergodic input stream with bounded service times and arbitrarily light traffic intensity may have infinite mean waiting time. We give an example of this paradox and we also give a mixing condition which, in the case of the preemptive-resume LCFS discipline, excludes this phenomenon. Furthermore, the same methods allow to estimate the durations of the first busy period and cycle and the number of customers served in the first busy cycle of a work-conserving queue.

The mean waiting time in a G/G/m/∞ queue with the LCFS-P service discipline

A single- or multiserver queue with work-conserving service discipline and a stationary and ergodic input stream with bounded service times and arbitrarily light traffic intensity may have infinite mean waiting time. We give an example of this paradox and we also give a mixing condition which, in the case of the preemptive-resume LCFS discipline, excludes this phenomenon. Furthermore, the same methods allow to estimate the durations of the first busy period and cycle and the number of customers served in the first busy cycle of a work-conserving queue.

Convex ordering of the attained waiting times in single-server queues and related problems

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

A conservation law for single-server queues and its applications

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

A conservation law for single-server queues and its applications

We establish a conservation law forG/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in aG/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in aGI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

On extremal service disciplines in single-stage queueing systems

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.

On extremal service disciplines in single-stage queueing systems

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.

Interrelationships between performance measures for the machine-repairman problem

The machine-repairman problem with heterogeneous machines and multiple repairmen is considered. It is assumed that machine up times and repair times have general distributions, and a work-conserving service discipline is employed. Under these general assumptions, steady-state relationships between performance measures are examined for the individual machines. Also, various performance measures are defined for the aggregate system and interrelationships between these measures are established.

On the relationship between work load and waiting time in single server queues with batch inputs

The relationship between work load and waiting time in single server queues with batch inputs is discussed under a work-conserving service discipline. Based on a result of Brumelle, the relationship is newly presented especially under the preemptive-resume discipline. This relationship is applied to analyze batch Poisson input models.

Convex ordering of sojourn times in single-server queues: extremal properties of FIFO and LIFO service disciplines

In this paper, the extremal properties of the ergodic sojourn times in G/G/1queues under various service disciplines are studied in terms of the convex ordering. It is shown that among work-conserving non-preemptive service disciplines that are service time independent, the FIFO and the LIFO service disciplines provide the minima and the maxima, respectively, of the ergodic sojourn times for any G/G/1 queue. For G/M/1 queues, this class of work-conserving service disciplines is extended to include preemptive/resume disciplines. In this case the FIFO and LIFO-P (preemptive/resume LIFO) service disciplines attain the minima and maxima, respectively. These extend results of Durr (1971), Kingman (1962) and a recent result of Ramaswami (1984). Further results are obtained for G/Em/1 and G/D/1 queues.

Convex ordering of sojourn times in single-server queues: extremal properties of FIFO and LIFO service disciplines

In this paper, the extremal properties of the ergodic sojourn times in G/G/1queues under various service disciplines are studied in terms of the convex ordering. It is shown that among work-conserving non-preemptive service disciplines that are service time independent, the FIFO and the LIFO service disciplines provide the minima and the maxima, respectively, of the ergodic sojourn times for any G/G/1 queue. For G/M/1 queues, this class of work-conserving service disciplines is extended to include preemptive/resume disciplines. In this case the FIFO and LIFO-P (preemptive/resume LIFO) service disciplines attain the minima and maxima, respectively. These extend results of Durr (1971), Kingman (1962) and a recent result of Ramaswami (1984). Further results are obtained for G/Em/1 and G/D/1 queues.