Steady-state Delay Research Articles

Polling systems with periodic server routeing in heavy traffic: distribution of the delay

We consider polling systems with mixtures of exhaustive and gated service in which the server visits the queues periodically according to a general polling table. We derive exact expressions for the steady-state delay incurred at each of the queues under standard heavy-traffic scalings. The expressions require the solution of a set of only M—N linear equations, where M is the length of the polling table and N is the number of queues, but are otherwise explicit. The equations can even be expressed in closed form for several routeing schemes commonly used in practice, such as the star and elevator visit order, in a general parameter setting. The results reveal a number of asymptotic properties of the behavior of polling systems. In addition, the results lead to simple and fast approximations for the distributions and the moments of the delay in stable polling systems with periodic server routeing. Numerical results demonstrate that the approximations are highly accurate for medium and heavily loaded systems.

Comparison of Delay on Arterials by Simulation and Highway Capacity Manual Computations

A comparison of delay on arterials was made using simulation and Highway Capacity Manual (HCM) computations. An extensive set of cases was defined over a range of flow rates, free-flow speeds, and signals per mile. Delay was calculated on the basis of the HCM computation for the specified signal progressions. The same cases were run using a specially revised version of the NETSIM microscopic simulation model. Detailed analysis showed that the discrepancies were traceable to the progression factor (PF). Corrections can be made (and are outlined) but must be done with care. There are a number of complex mechanisms at work. The primary result of this study may be clarifying the insufficiency in the current PF algorithm. The expectation that a single multiplicative factor could provide the needed correction to a long-link random-arrival steady-state delay model (that is, the existing HCM delay model), so that it could be a network-based model for delay estimation, appears to be simply unrealistic. For these and other reasons, research is focusing on the creation of a time-varying intersection delay model, building from basic principles.

On the temperature dependence of hysteresis effect in floating-body partially depleted SOI CMOS circuits

This paper presents a detailed study on the temperature dependence of the hysteresis effect in static CMOS circuits and pass-transistor-based circuits with floating-body partially depleted (PD) silicon-on-insulator (SOI) CMOS devices. Basic physical mechanisms underlying the temperature dependence of hysteretic delay variations are examined. It is shown that, depending on the initial state of the circuit, the initial circuit delays have distinct temperature dependence. For steady-state circuit delays, the temperature dependence is dictated solely by the various charge injection/removing mechanisms into/from the body. The use of the cross-coupled dual-rail configuration in pass-transistor-based circuits is shown to be effective in compensating and reducing the disparity in the temperature dependence of the delays.

Internet-Type Queues with Power-Tailed Interarrival Times and Computational Methods for Their Analysis

Internet traffic flows have often been characterized as having power-tailed (long-tailed, fat-tailed, heavy-tailed) packet interarrival times or service requirements. In this work, we focus on the development of complete and computationally efficient steady-state solutions of queues with power-tailed interarrival times when the service times are assumed exponential. The classical method for obtaining the steady-state probabilities and delay-time distributions for the G/M/1 (G/M/c) queue requires solving a root-finding problem involving the Laplace-Stieltjes transform of the interarrival-time distribution function. Then the exponential service assumption is combined with the derived geometric arrival-point probabilities to get both the limiting general-time state and delay distributions. However, in situations where there is a power tail, the interarrival transform is typically quite complicated and never analytically tractable. In addition, it is possible that there is only a degenerate steady-state system-size probability distribution. Thus, an alternative approach to obtaining a steady-state solution is typically needed when power-tailed interarrivals are present. We exploit the exponentiality of the steady-state delay distributions for the G/M/1 and G/M/c queues, using level-crossings and a transform approximation method, to develop an alternative root-finding problem when there are power-tailed interarrival times. Extensive computational results are given.

Polling Systems with Switch-over Times under Heavy Load: Moments of the Delay

Consider an asymmetric cyclic polling system with general service-time and switch-over time distributions, and with general mixtures of exhaustive and gated service, in heavy traffic. We obtain explicit expressions for all moments of the steady-state delay at each of the queues, under heavy-traffic scalings. The expressions are strikingly simple: they depend on only a few system parameters, and moreover, can be expressed as finite products of simple known terms. The exact results provide new and useful insights into the behavior of polling systems in heavy traffic. In addition, the results suggest simple and fast approximations for the moments of the delay in stable polling systems. Numerical experiments demonstrate the usefulness of the approximations for moderately and heavily loaded systems.

Distribution of the delay in polling systems in heavy traffic

We consider asymmetric cyclic polling systems with an arbitrary number of queues, with general mixtures of exhaustive and gated service and with generally distributed service-times and switch-over times, in heavy traffic. We derive closed-form expressions for the Laplace–Stieltjes transform (LST) of the steady-state delay incurred at each of the queues, under standard heavy-traffic scalings. The expressions give an explicit characterization of the complete (scaled) waiting-time distributions at each of the queues. The results are strikingly simple and provide a variety of new insights into the behavior of heavily loaded polling systems. In addition, the results lead to simple and fast-to-evaluate approximations for the waiting-time distributions in stable polling systems that are close to saturation. Numerical results demonstrate that the approximations are highly accurate in many practical heavy-traffic scenarios.

Statistical analysis of delay bound violations at an earliest deadline first (EDF) scheduler

In this paper, we develop an analytical framework for providing statistical delay guarantees in an Earliest Deadline First (EDF) scheduler which multiplexes traffic from multiple markovian sources with heterogeneous delay requirements. Our framework permits the computation of steady-state delay bound violation probabilities (i.e., the fraction of traffic that does not meet its delay bounds) at the EDF scheduler, and can be therefore used to characterize the schedulable region of EDF in a statistical setting. Our method employs results from the theory of large deviations and the theory of effective bandwidths, and demonstrates that effective bandwidths at both infinite and finite time scales have to be considered in the analysis of delays at the EDF scheduler (this is in contrast to the analysis of packet losses at a multiplexor, where only the effective bandwidth at infinite time scales is relevant). Our framework is of general use, and suitable to handle a broad range of markovian sources. As illustrating examples, we apply our method to two simple models, poisson and markovian on–off fluid traffic, and compare the analytical results with simulations, showing that the analysis is quite accurate. The framework presented in this paper can serve as the basis for the design of a Call Admission Control (CAC) mechanism which provides statistical guarantees on traffic transfer delays. Such a statistical CAC approach can offer dramatic advantages in network utilization over CAC frameworks based on deterministic delay bounds.

Polling systems in heavy traffic: Higher moments of the delay

We study an asymmetric cyclic polling model with general mixtures of exhaustive and gated service, and with zero switch-over times, in heavy traffic. We derive closed-form expressions for all moments of the steady-state delay at each of the queues, under standard heavy-traffic scalings. The expressions obtained provide new and useful insights into the behavior of polling systems under heavy-load conditions.

Delay moments for FIFO GI/GI/s queues

For stable FIFO GI/GI/s queues, s \geq 2, we show that finite (k+1)st moment of service time, S, is not in general necessary for finite kth moment of steady-state customer delay, D, thus weakening some classical conditions of Kiefer and Wolfowitz (1956). Further, we demonstrate that the conditions required for \mathrm{E}[D^k]<\infty are closely related to the magnitude of traffic intensity \rho (defined to be the ratio of the expected service time to the expected interarrival time). In particular, if \rho is less than the integer part of s/2, then \mathrm{E}[D] <\infty if \mathrm{E}[S^{3/2}]<\infty, and \mathrm{E}[D^k]<\infty if \mathrm{E}[S^k]<\infty,\ k\geq 2. On the other hand, if s-1 < \rho < s, then \mathrm{E}[D^k]<\infty if and only if \mathrm{E}[S^{k+1}]<\infty,\ k\geq 1. Our method of proof involves three key elements: a novel recursion for delay which reduces the problem to that of a reflected random walk with dependent increments, a new theorem for proving the existence of finite moments of the steady-state distribution of reflected random walks with stationary increments, and use of the classic Kiefer and Wolfowitz conditions.

Ballot theorems applied to the transient analysis of nD/D/1 queues

The problem of transporting constant-bit-rate (CBR) traffic through a packet network is analyzed. In the system considered, CBR traffic is packetized and packets from several similar sources are multiplexed on a transmission link. The bit streams are recreated at the receiving end by demultiplexing the packets and then playing out the packets of each CBR stream. Traffic fluctuations may cause gaps to appear in the playout process. Their frequency can be reduced by adding a smoothing delay to each stream. The queueing system analyzed has periodic arrivals and deterministic service times. A method of analysis, based on the ballot theorems of Takacs (1967), is presented to provide steady-state delay distributions as well as a transient analysis of the system to predict the statistics of the time for a gap to develop in the CBR stream as a function of the smoothing delay. >

Queue Size and Delay Analysis for a Communication System Subject to Traffic Activity Mode Changes

We consider a discrete single-server queueing/communication system, in which the arrival distribution is changing stochastically with time. The distribution of the number of messages to arrive within a time slot depends on a traffic activity process of modes. This process of modes is a two-state Markov chain: mode 0 and mode 1, which represent light and heavy traffic conditions, respectively. The transition probabilities of this Markov chain [herein called mode duration parameters (MDP's)] determine the distribution of the duration of time the process stays in each mode. Expressions for the z -transform, expectation, and variance are obtained for the steady-state queue size and delay. A special consideration is given to the effect of the MDP's on the system performance, and the limiting cases (where the MDP's approach zero) are studied.

Steady-state and transient delay analysis of ARQ protocols

A technique for the performance evaluation of ARQ protocols is presented. General distributions for the packet interarrival times at the transmitter are assumed. Two specific ARQ protocols, the stop-and-wait ARQ and the continuous ARQ are examined in particular. Both the transient and steady-state behaviour are obtained. A simple iterative scheme is proposed for the computation of the delay experienced by the packets and for the packet interdeparture time distribution. The latter result can be applied to the analysis of networks using an ARQ protocol in their datalink control layer. This method is demonstrated for the special case of a network consisting of N nodes connected in tandem. Analytical results are compared with simulation runs and excellent agreement is observed.

Minimizing Delays in the GI/G/1 Queue

What service-time distribution minimizes delays in a GI/G/1 queue with given renewal arrival process and given mean service time? It is natural to conjecture that the deterministic service-time distribution with unit mass on the mean is optimal for all objective functions of the form Ef(W) where f is a nondecreasing function of the steady-state delay W. However, we show that this conjecture is false. In fact, for hyperexponential interarrival-time distributions (mixtures of two exponential distributions), the service-time distribution minimizing the average delay maximizes the proportion of customers delayed.

On Stochastic Bounds for the Delay Distribution in the GI/G/s Queue

A counterexample is constructed to show that the steady-state delay distribution in the GI/G/s queue with the FIFO discipline need not be stochastically less (in the sense of first-order stochastic dominance) than the steady-state delay distribution in the same system with the cyclic queue discipline. Such an ordering was conjectured by R. W. Wolff. However, it is shown that the stochastic dominance does hold for many GI/M/s queues. When it does, this bound on the delay distribution is often better than the bound for GI/M/s systems obtained by S. L. Brumelle, but neither bound appears to be very good, as is illustrated here for the M/M/s case.

The light-response rhythm and seasonal entrainment of the endogenous circadian locomotor rhythm ofTalitrus saltator(Crustacea: Amphipoda)

Talitrus saltator, a supralittoral, sand-beach amphipod, exhibits a nocturnal, circadian, locomotor rhythm entrained by the natural photoperiod (nLD). A phase-response curve (PRC) (light pulse: 2h/400 lx) of the free-running rhythm exhibits a conventional shape; steady-state delay shifts of approximately 3·5 h occur in response to light pulses around subjective dusk, with similar advance shifts when exposed to light pulses around dawn. The seasonal variation of the overt expression of the rhythm indicates that entrainment is predominantly non-parametric, while the constant phase-angle difference between the peak of locomotor activity and dawn confirms the latter as the effective field signal for entrainment. The qualitative relationship between the time scale of the PRC and nLD, together with the absence of any seasonal variation in the periodicity of the free-running rhythm (τ), suggests that entrainment occurs through the interaction of phase shifts defined by the PRC, essentially in response to the systematically changing photoperiod.

An Approximate Method for Calculating Delays for a Family of Cyclic-Type Queues

A study of the marker-register dial-tone delay problem in No. 5 crossbar switching machines led to a special type of cyclic queuing model. In this paper, we present a method for calculating approximately the steady-state delays of an arriving customer. When applied to the marker-register problem, the model emphasizes the order in which markers are assigned to waiting calls and the fact that part of the markers' time is unproductive when an “all registers busy” condition occurs. Some numerical results are presented, which agree with the observed phenomenon that, for a constant marker load, the delays of Touch-Tone® calls are influenced by the load on the dial-pulse originating registers, and vice versa. The results are compared to those of a simulation of the same problem. The numerical results compare favorably in the range of loads that produce a dial-tone speed of between 0.05 and 0.15.

Regenerative processes in the theory of queues, with applications to the alternating-priority queue

Using some well-known and some recently proved asymptotic properties of regenerative processes, we present a new proof in a general regenerative setting of the equivalence of the limiting distributions of a stochastic process at an arbitrary point in time and at the time of an event from an associated Poisson process. From the same asymptotic properties, several conservation equations are derived that hold for a wide class of GI/G/1 priority queues. Finally, focussing our attention on the alternating-priority queue with Poisson arrivals, we use both types of result to give a new, simple derivation of the expected steady-state delay in the queue in each class.

Regenerative processes in the theory of queues, with applications to the alternating-priority queue

Using some well-known and some recently proved asymptotic properties of regenerative processes, we present a new proof in a general regenerative setting of the equivalence of the limiting distributions of a stochastic process at an arbitrary point in time and at the time of an event from an associated Poisson process. From the same asymptotic properties, several conservation equations are derived that hold for a wide class of GI/G/1 priority queues. Finally, focussing our attention on the alternating-priority queue with Poisson arrivals, we use both types of result to give a new, simple derivation of the expected steady-state delay in the queue in each class.