The single-server queue with independent GI/G and M/G input streams

A single-server queueing system is studied, the input into which consists of the sum of two independent stochastic processes. One of these is an ‘M/G' type input process, the other a much more general process which need not be Markov. There are two types of busy period, depending on which arrival process started the busy period. Stochastic monotonicity results are derived and it is found that under a stationarity-like condition the probability of being in a busy period which started with an ‘M/G' arrival is independent of time and is the same it would be with the ‘M/G' process as only input process. Also, distributional results are obtained for the virtual waiting-time process, and these results are used to reduce the study of a single-server queueing system with as input the sum of independent ‘M/G' and ‘GI/G' input streams to the study of a related GI/G/1 queueing system.The purpose of this paper is to pave the way for a study of an M/G/1 queueing system with periodic arrivals of additional work, and for optimal scheduling of maintenance processes in certain real-time computer systems.

Single server queueing models are important in the study of a wide variety of stochastic systems. A particularly important example is the study of task schedules for computer systems with real-time applications. In this paper, we present a class of approximations for the waiting time distribution in single server queueing systems with general independent (renewal) input and general (independent) service time distributions. These approximations allow the analyst to use as much (or as little) of the structure of the input and service processes as desired. Moreover, they also allow him to concentrate on specific quantities associated with the delay distribution that may represent the most relevant performance criteria. Examples include probability of delay, mean delay, and tails of the delay distribution. The methods given in this paper have been used to analyze the performance of task schedules in a variety of processor-based systems.

Single server queues with batch arrivals are useful in the modeling of computer and communication systems amongst others. In this paper we give for a wide class of single server queues with batch arrivals a stable recursive method to compute state probabilities both at arrival and at arbitrary epochs. This class includes finite capacity models with state dependent Markovian input in which batches could be totally or partially rejected. The customers of a batch are served individually and the service time has a general distribution function. In particular, we discuss as application the computation of operating characteristics, such as rejection probability and probability of server idleness, for a finite capacity M/G/1 queue for which arriving customers are rejected if their waiting plus service times exceed the capacity of the system.

Sleep modes are important for user equipment to reduce power consumption or to reduce battery reloading. To keep the user equipment connected, it has to be activated for uplink or downlink data transfer which causes overhead and additional delay. Wireless or mobile equipment has to be monitored repeatedly for location updates. These classes of problems can be modeled by queuing systems with modified busy and idle periods. In this paper, we consider a generalized infinite-buffer, single-server queuing systems with activation overhead and under various types of sleep modes during inactive periods. For the special assumption of Markovian arrival processes of data units and generally distributed activation times and vacation (sleep) periods, the models can be analyzed exactly by a mean value analysis method which is based on regenerative stochastic processes, renewal theory and classical queuing theorems. The method allows also for dynamic sleep periods which are dependent on each other as in the case of the standard IEEE 802.16m for wireless equipment. Explicit results are derived for average values of data unit delays, resource utilization and power- saving margins dependent on given parameters for activation times and wakeup signaling frequencies. The models include the special cases of M/G/1 queuing systems with a modified service time of the first arrival initiating a busy period and M/G/1 queuing systems with vacation periods, respectively, which are well known from literature providing general solutions for the state and delay distributions. Finally, all models are extended to the exact analysis of Batch Poisson arrival processes.

The busy period for a queue is cast as the area swept under the random walk until it first returns to zero, $B$. Encompassing non-i.i.d. increments, the large-deviations asymptotics of $B$ is addressed, under the assumption that the increments satisfy standard conditions, including a negative drift. The main conclusions provide insight on the probability of a large busy period, and the manner in which this occurs: I) The scaled probability of a large busy period has the asymptote, for any $b>0$, \lim_{n\to\infty} \frac{1}{\sqrt{n}} \log P(B\geq bn) = -K\sqrt{b}, \hbox{where} \quad K = 2 \sqrt{-\int_0^{\lambda^*} \Lambda(\theta) d\theta}, \quad \hbox{with $\lambda^*=\sup\{\theta:\Lambda(\theta)\leq0\}$,} and with $\Lambda$ denoting the scaled cumulant generating function of the increments process. II) The most likely path to a large swept area is found to be a simple rescaling of the path on $[0,1]$ given by, [\psi^*(t) = -\Lambda(\lambda^*(1-t))/\lambda^*.] In contrast to the piecewise linear most likely path leading the random walk to hit a high level, this is strictly concave in general. While these two most likely paths have very different forms, their derivatives coincide at the start of their trajectories, and at their first return to zero. These results partially answer an open problem of Kulick and Palmowski regarding the tail of the work done during a busy period at a single server queue. The paper concludes with applications of these results to the estimation of the busy period statistics $(\lambda^*, K)$ based on observations of the increments, offering the possibility of estimating the likelihood of a large busy period in advance of observing one.

This paper presents a unifying algorithmic analysis for a general class of single server queueing systems with a state dependent Markovian input process and a phase-type service time distribution including single server queues with random and quasirandom input. Using regenerative analysis we develop numerically stable and efficient recursion schemes to compute the state probabilities. The computation of the waiting times is based on the state probabilities.

In this article some approximations for the mean number in the single server queueing system are presented and compared. Based on numerical comparison a guide is provided to select the appropriate approximation(s) for different ranges, of the squared coefficient of variation of the inter-arrival and service time distributions. Two approximations for the distribution of the number in the system are given and the better is chosen.

Computability and realizability for interactive computations

We present a methodology for building timed models of real-time systems by adding time constraints to their application software. The applied constraints take into account execution times of atomic statements, the behavior of the system's external environment, and scheduling policies. The timed models of the application obtained in this manner can be analyzed by using time analysis techniques to check relevant real-time properties. We show an instance of the methodology developed in the TAXYS project for the modeling and analysis of real-time systems programmed in the Esterel language. This language has been extended to describe, by using pragmas, time constraints characterizing the execution platform and the external environment. An analyzable timed model of the real-time system is produced by composing instrumented C-code generated by the compiler. The latter has been re-engineered in order to take into account the pragmas. Finally, we report on applications of TAXYS to several nontrivial examples.

The five papers in this special issue are based on a selection of top theoretical papers from the ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, which took place at Austin, Texas, in June 2014. As the premier conference on the measurement and modeling of computer systems, the technical program featured papers that covered both theory and applications from a wide variety of areas, including algorithms, communication networks, dynamics and control, optimization, performance analysis, resource allocation and scheduling, and stochastic modeling, among others. The program consisted of 40 papers selected from more than 237 international submissions by a technical program committee of well-established researchers from all over the world. We seek with this special issue to highlight some of the recent theoretical work on the mathematical analysis and modeling of computer systems. The paper of Larranaga, Ayesta, and Verloop deals with a resource allocation problem in a multi-class server with convex holding costs and user impatience under the average cost criterion. Although the optimal policy has a complex dependency on all the input parameters and state information, the authors derive simple index policies obtained by solving a relaxed version of the optimal stochastic control problem and combining results from restless multi-armed bandits and queueing theory.

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.

We consider a queuing system of the type M θ/G/1/b in which the input stream is regulated by a certain threshold level. The asymptotic properties of the first busy period and the number of calls served during this period are studied.

The paper deals with problems of high level modeling of real-time systems in system level languages. Main advantages of transaction level modeling in SystemC are emphasized. The universal kernel for functionality, communication and timing issues modeling is presented. The problem of multithreading is exemplified on sorting procedure implemented in multi-core microprocessor system. Advantages of system level approach to modeling and design of real-time systems are outlined.

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

An approach to the modeling and verification of real-time systems is presented. The Specification PEARL description language is used for hardware/software modeling. It has its origins in the standard Multiprocessor PEARL language. Its usefulness has been enhanced for asymmetrical multiprocessor system modeling, and parameters for schedulability analysis have been added. It is meant to be an over-layer for PEARL programs and supports the corresponding program model. The detailed program models are represented internally by timed state transition automata. The model of the co-designed system is verified for feasibility with co-simulation. The resulting information may be used for changes in the current design to gain a feasible model of the system. After that, the program models may be enhanced to their full functionality. Schedulability analysis, which should provide the designer with more precise information, can then be done. The new information can be reused on the system model in order to discover bottlenecks or possible discrepancies with the estimated times. By utilising this methodology, the possibility of implementing a non-feasible system should be minimised.