Scheduling Policy: A Performability Study

13th IEEE International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems

This paper introduces an analytical method for approximating the performability of a firm realtime system modeled by a multi-server queue. The service discipline in the queue is earliestdeadline-first (EDF), which is an optimal scheduling algorithm. Real-time jobs with exponentially distributed relative deadlines arrive according to a Poisson process. All jobs have deadlines until the end of service and are served non-preemptively. An important performance measure to calculate is the loss probability. The performance of the system is approximated by a Markovian model in the long run. A key parameter, namely, the loss rate when there are n jobs in the system is used in the model, which is estimated by partitioning the system into two subsystems. The resulting model can then be solved analytically using standard Markovian solution techniques. The number of servers in the system may change due to failure or repair. The performability of the system is evaluated in the presence of such structural changes. The latter measure is approximated by a Markov reward model, considering the loss probability as the reward rate. Comparing numerical and simulation results, we find that the existing errors are relatively small.

Journal of Industrial and Systems Engineering, 2007

This paper introduces an analytical method for approximating the performability of a firm realtime system modeled by a multi-server queue. The service discipline in the queue is earliestdeadline-first (EDF), which is an optimal scheduling algorithm. Real-time jobs with exponentially distributed relative deadlines arrive according to a Poisson process. All jobs have deadlines until the end of service and are served non-preemptively. An important performance measure to calculate is the loss probability. The performance of the system is approximated by a Markovian model in the long run. A key parameter, namely, the loss rate when there are n jobs in the system is used in the model, which is estimated by partitioning the system into two subsystems. The resulting model can then be solved analytically using standard Markovian solution techniques. The number of servers in the system may change due to failure or repair. The performability of the system is evaluated in the presence of such structural changes. The latter measure is approximated by a Markov reward model, considering the loss probability as the reward rate. Comparing numerical and simulation results, we find that the existing errors are relatively small.

The Journal of Supercomputing, 2006

This paper introduces an analytical method to approximate the fraction of jobs missing their deadlines in a soft real-time system when the earliest-deadline-first (EDF) scheduling policy is used. In the system, jobs either all have deadlines until the beginning of service (DBS) and are non-preemptive, or have deadlines until the end of service (DES) and are preemptive. In the former case, the system is represented by an M/M/m/EDF+G model, i.e., a multi-sever queue with Poisson arrival, exponential service, and generally distributed relative deadlines. In the latter case, it is represented by an M/M/1/EDF+G model, i.e., a singleserver queue with the same specifications as before. EDF is known to be optimal in both of the above cases. The optimality property of EDF scheduling policy is used for the estimation of a key parameter, namely the loss rate when there are n jobs in the system. The estimation is possible by assuming an upper bound and a lower bound for this parameter and then linearly combining these two bounds together. The resulting Markov chains can then be easily solved numerically. Comparing numerical and simulation results, we find that the existing errors are relatively small.

Computers & Operations Research, 2007

In this paper, we consider a service system in which the server can process N + 1 different types of jobs. Jobs of type 0 are generated randomly according to a Poisson stream. Jobs of types 1 to N are non-queue types which may or may not be available for completion by the server. To optimally allocate the server's time to these jobs, we formulate a finite state semi-Markov decision process model to this environment. With this model, the optimal stationary policies can be numerically determined via a policy-iteration algorithm. We also discuss the practical applications of this model to tele-service and tele-marketing operations.

Scientia Iranica, 2008

This paper introduces an analytical method for approximating the performance of a two-class priority M=M=1 system. The prioritized class-1 jobs are real-time, served either with the preemptive or non-preemptive Earliest-Deadline-First (EDF) policy and can preempt the non real-time class-2 jobs. The preempted service of the class-2 job is resumed from the time in instances where no class-1 job is in the system. The service discipline of class-2 jobs is FCFS. The required mean service times may depend on the class of the jobs. The real-time jobs have exponentially distributed relative deadlines until the end of service. The system is approximated by a Markovian model in the long run, which can be solved numerically, using standard Markovian solution techniques. The performance measures of the system are the loss probability of the class-1 jobs and the mean sojourn (waiting) time of the class-2 jobs. Comparing numerical and simulation results, it is found that the existing errors are relatively small.

Computers & Industrial Engineering, 2009

We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process ðBMAPÞ: The server serves customers in batches of maximum size 'b' with a minimum threshold size 'a'. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N Ã of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process ðMSPÞ: Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.

1998

This paper deals with a repair shop with multiple parallel servers, which has to carry out planned overhauls, consisting of a lot of maintenance jobs. These overhauls are disturbed by randomly arriving emergency jobs. To control the delivery performance of the overhauls, knowledge about the overhaul makespan distribution should be available. However, past approaches do not have an answer to this problem. Using a 2-dimensional Markov model, we derive theoretical solutions for the first and second moment of the overhaul makespan for the case that the repair times of all overhaul jobs are identically and exponentially distributed. In case of non identical repair time distributions, an approximation for the moments is presented. These moments are used to approximate the makespan distribution by a mixture of two Erlang distributions. We show by simulation that this approximation gives very good results.

IEEE Transactions on Computers, 1989

In this paper, we study the performance of a realtime system in which jobs either all have deadlines until the beginning of service or deadlines until the end of service. In the first case, we analyze the minimum laxity scheduling policy (ML) when there are c processors and, in the latter case, the preemptive resume earliest deadline scheduling policy (ED) when there is one server. In both cases, the analysis assumes a Poisson arrival process, exponential service times that are not known to the scheduler, and exponential laxities or deadlines. In both cases, we develop families of upper and lower hounds on the fraction of jobs that miss their deadlines. The pessimistic bounds are of special interest because they correspond to a family of implementable policies, ML(n) and ED(n) , n = 1,. .. where the performance approaches that of ML and ED as n increases, but at the cost of increasing overhead. Numerical results show that the bounds are tight, that the difference between the performances of policies that do not use deadline information (e.g., FIFO) and ML and ED is nonnegligible, and that, even for small values of n (e.g., n = 3), ML(n) and ED(n) perform well when compared to ML and ED.

2014

Here we will study bulk service to customer under optimal operation of a single removable and non-reliable server in Markovian queueing system under steady-state conditions. The decision maker can turn a server on at customer’s arrival or off at service completion. Here it is assumed that the server may breakdown only if working and requires repair at repair facility. Inter-arrival and service time distributions of the customers are assumed to be exponentially distributed. Breakdown and repair time distributions of the server are assumed to be exponentially distributed. The following cost structure is incurred to be system; a holding cost for each customer in the system per unit time, cost per unit time when a server fails, and fixed costs for turning the server on or off. The expected cost function per unit time is developed to obtain the optimal operating policy at minimum cost.

This paper introduces an analytical method for approximating the performance of a two-class priority M/M/1 system. The system is fully non-preemptive. More specifically, the prioritized class-1 jobs are real-time and served with the non-preemptive earliest-deadline-first (EDF) policy, but despite their priority cannot preempt any non real-time class-2 job. The waiting class-2 jobs can only be served from the time instant that no class-1 job is in the system. The service discipline of the class-2 jobs is FCFS. The required mean service times may depend on the class of the jobs. The real-time jobs have exponentially distributed relative deadlines until the end of service. The system is approximated by a Markovian model in the long run, which can be solved numerically using standard Markovian solution techniques. The performance measures of the system are the loss probability of the class-1 jobs and the mean sojourn (waiting) time of the class-2 jobs. Comparing the numerical and simulation results, we find that the existing errors are relatively small.