Idle Period Distribution Research Articles

On the Discrete-TimeGeoX/G/1Queues underN-Policy with Single and Multiple Vacations

We consider the discrete-timeGeoX/G/1queue underN-policy with single and multiple vacations. In this queueing system, the server takes multiple vacations and a single vacation whenever the system becomes empty and begins to serve customers only if the queue length is at least a predetermined threshold valueN. Using the well-known property of stochastic decomposition, we derive the stationary queue-length distributions for both vacation models in a simple and unified manner. In addition, we derive their busy as well as idle-period distributions. Some classical vacation models are considered as special cases.

On the exact expression for single-stage CSMA/CA’s idle period distribution

The presence of CSMA/CA transmissions in a wired or wireless medium, as well as the number of active transmitters may be deducible via profiling energy level measurements in its band of operation. A distinct characteristic that may be useful in this application is its idle period distribution. In this paper, we propose an exact analytical model for this distribution for CSMA/CA with a single backoff stage, and verify its accuracy with simulation results alongside two approximation models.

Deriving the Suspended Backoff Counter Value Distribution in CSMA/CA

of today's wireless LANs and PANs employ the use of the CSMA/CA protocol. In CSMA/CA, a randomly initialised counter is used to reduce the probability of nodes accessing the channel at the same time. In most implementations, this counter is suspended when the channel is idle. In this paper, we derive and verify through simulation the exact analytical expression for the distribution of this counter's value. For further validation, we show how it can be used to obtain the protocol's idle period distribution under saturation loads. For this latter distribution, we provide simulation results to show the accuracy of the formula compared to one derived using the de+facto first+order Markov channel state model.

Delay Performance of Threshold Policies for Dynamic Spectrum Access

In this paper, we analyze the delay performance of a secondary user (SU) under dynamic spectrum access. We design simple time-threshold policies for the SU to minimize the average delay while satisfying the collision probability constraint of the primary user (PU). Such policies perform closely to an optimized policy found by a Markov Decision Process (MDP) formulation, while facilitating analytical analysis of the delay and collision probability. For general PU busy and idle period distributions, we analyze the performance of threshold policies through a one-dimensional Markov chain, and develop analytical expressions to approximate the delay and collision probability. The accuracy of the Markov chain analysis and the analytical approximations is examined under various busy and idle distributions. We investigate the impact of busy and idle distributions on system performance. We find that while the idle distribution determines the time capacity of SU access, the busy distribution significantly affects the delay performance of the threshold policies. The effect of imperfect sensing is also studied.

Microarchitecture-level leakage reduction with data retention

In this paper, we study microarchitecture-level leakage energy reduction by power gating. We consider the virtual power/ground rails clamp (VRC) and multithreshold CMOS (MTCMOS) techniques and apply VRC to memory-based units for data retention and MTCMOS to the other units. We propose a systematic methodology for leakage reduction at the microarchitecture level, in which profiling of idle period distribution and ideal power gating analysis are used to select a target component for realistic power gating. We show that the ideal leakage energy reduction can be up to 30% of the total energy for the modern high-performance very long instruction word processors we study and that the secondary level (L2) cache contributes most to the reduction. We further improve the existing adaptive cache decay method for leakage reduction by using VRC for data retention and name it VRC decay . Applied to L2 cache, the VRC decay, on average, increases performance by 5.6% and reduces system energy by 24.1%, compared to the adaptive cache decay without data retention.

Idle period approximations and bounds for the GI/G/1 queue

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

Idle period approximations and bounds for the GI/G/1 queue

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

GI/M/1/1 queue with finite retrials and finite orbits

In this paper we consider the GI/M/1/1 retrial queue with finite number of retrials to each orbital customer and a finite number of orbits. The interarrival times from outside the system have a general distribution. The sojourn time of a unit in an orbit until its retrial for service and its service time are exponentially distributed with parameters depending on the orbit number. The maximum number of retrials any unit is permitted to take is restricted to k. There are a finite number, say m, of orbits with at most one customer in each orbit. At the time of an arrival if the server is busy and all orbits are occupied, then the customer is lost to the system. If the server is idle at an arrival epoch, the unit directly goes for service. If the server is busy and at least one of the orbits is empty then the arriving customer occupies the first empty orbit. A unit in the orbit retries for service which returns to the same orbit (if the server is busy) with probability P and leaves the system with probability 1−P, 0<P<1. We compute the system state distribution. The distribution of the number of customers reneging and that of the number of units served during an interarrival time are computed. Also waiting time distribution and the idle period distribution are computed. Optimal values of k and m are investigated.

Characterizing the Idle-Period Distribution of a &lt;i&gt;GI/G/&lt;/i&gt;1 Queue with Vacation

Characterizing the Idle-Period Distribution of a &lt;i&gt;GI/G/&lt;/i&gt;1 Queue with Vacation

An Approximation Method for the Analysis of GI/G/1 Queues

We study in this paper an approximation method for the calculation of various performance measures of a GI/G/1 queue. Instead of solving the waiting time directly, we analyze the idle-period distribution as the starting point. The result is then taken as input to many known results to get other performance measures. We show that the distribution of the GI/G/1 idle period satisfies a nonlinear integral equation. This equation directly leads to an accurate approximate solution of the idle-period distribution of the GI/G/1 queue where the interarrival times have a generalized hyperexponential distribution (GH). Since all distribution functions can be approximated by a GH distribution at any given accuracy (Botta and Harris [Botta, R. F., C. M. Harris. 1986. Approximation with generalized hyperexponential distributions: Weak convergence results. Queueing Systems 2 169–190.]), the solution method developed in this paper serves as a unified basis for the analysis of GI/G/1 queues.

Approximating a Cumulative Distribution Function by Generalized Hyperexponential Distributions

Recent developments in stochastic modeling show that enormous analytical advantages can be gained if a general cumulative distribution function (c.d.f.) can be approximated by generalized hyperexponential distributions. In this paper, we introduce a procedure to explicitly construct such approximations of an arbitrary c.d.f. Although our approach can be used in different types of stochastic models, the main motivation comes from queueing theory in obtaining approximations of the idle-period distribution and other performance measures in GI/G/1 queues.

Characterizing the idle-period distribution of GI/G/1 queues

A variety of performance measures of a GI/G/1 queue are explicitly related to the idle-period distribution of the queue, suggesting that the system analysis can be accomplished by the analysis of the idle period. However, the ‘stand-alone' relationship for the idle-period distribution of the GI/G/1 queue (i.e. the counterpart of Lindley's equation) has not been found in the literature. In this paper we develop a non-linear integral equation for the idle period distribution of the GI/G/1 queue. We also show that this non-linear system defines a unique solution. This development makes possible the analysis of the GI/G/1 queue in a different perspective.

Characterizing the idle-period distribution of GI/G/1 queues

A variety of performance measures of aGI/G/1 queue are explicitly related to the idle-period distribution of the queue, suggesting that the system analysis can be accomplished by the analysis of the idle period. However, the ‘stand-alone' relationship for the idle-period distribution of theGI/G/1 queue (i.e. the counterpart of Lindley's equation) has not been found in the literature. In this paper we develop a non-linear integral equation for the idle period distribution of theGI/G/1 queue. We also show that this non-linear system defines a unique solution. This development makes possible the analysis of theGI/G/1 queue in a different perspective.

Discrete-time analysis of the busy and idle period distributions of a finite-capacity ATM multiplexer with periodic input

In this paper we present an exact solution for the distribution of the busy and idle periods of a finite-capacity ATM multiplexer. The multiplexer input lines and also the output line is considered as discretized into time slots of cell-duration length. The input traffic is the superposition of a number of independent and periodic sources. For different load situations we present recursive algorithms whose complexity is of polynomial order and which are efficient in terms of computing time and memory requirements. By special parameter choice we are able to compute the exact busy and idle period distributions also for the infinite-capacity multiplexer. Numerical examples show that the mean value and especially the coefficient of variation of the busy period depend strongly on the traffic intensity. By the numerical results we get some insight into the changes of the traffic characteristics when one ATM multiplexer stage is passed.

A Functional Approach to the Stationary Waiting Time and Idle Period Distributions of the GI/G/1 Queue

The GI/G/1 queueing model is regarded as a functional which maps the service and interarrival time distributions onto output quantities of interest, such as the stationary waiting time distribution. For the case where the input distributions have densities, techniques from infinite-dimensional analysis are used to obtain derivatives and Taylor series expansions for the functionals. These yield approximations to the output distributions which can be viewed as nonparametric alternatives to parametric approximations such as those provided by infinitesimal perturbation analysis or the phase method.

On the concavity of the waiting-time distribution in some GI/G/1 queues

In this paper the concavity property for the distribution of a geometric random sum (geometric compound) X, + · ·· + XN is established under the assumption that Xi are i.i.d. and have a DFR distribution. From this and the fact that the actual waiting time in GI/G/1 queues can be written as a geometric random sum, the concavity of the waiting-time distribution in GI/G/1 queues with a DFR service-time distribution is derived. The DFR property of the idle-period distribution in specialized GI/G/1 queues is also mentioned.

On the concavity of the waiting-time distribution in some GI/G/1 queues

In this paper the concavity property for the distribution of a geometric random sum (geometric compound) X, + · ·· + XN is established under the assumption that Xi are i.i.d. and have a DFR distribution. From this and the fact that the actual waiting time in GI/G/1 queues can be written as a geometric random sum, the concavity of the waiting-time distribution in GI/G/1 queues with a DFR service-time distribution is derived. The DFR property of the idle-period distribution in specialized GI/G/1 queues is also mentioned.

PH/M/S/K型待ち行列の過渡現象

We analyze the transient behavior of the PH/M/S/K queue. Emphasis is placed on the overflow process, busy period distribution, idle period distribution, accepted arrival process, departure process and the transition probabilities of the system states. All results are represented by the addition and multiplication of the matrices whose dimensions are at most equal to the number of phases for the inter-arrival time distribution.

On the eelationship of the transient behaviour of a general queueing model to its idle and busy period distributions

A general queueing model is studied. The main limitation is that bulk arrivals and bulk services are not allowed. Simple relations between the transient state probabilities and the busy and idle period distributions are obtained. The LAPLACE Transform of the idle period distribution always follows from the transient solution. The same is true for one busy period distribution, when intervals between arrivals have negative exponential distributions. On the other hand the transient state probabilities can always be obtained directly from an investigation of a “busy cycle process”.

Asymptotic behaviour of the Ek/G/1 queue with finite waiting room

The asymptotic behaviour of the Ek/G/1 queue with finite waiting room is studied. Using a combination of the supplementary variable and phase techniques, queue length and waiting time distributions are obtained. Also the idle period distribution and mean length of the idle and busy periods are found.