Hyper-exponential Service Time Research Articles

Efficient Distributed Threshold-Based Offloading for Large-Scale Mobile Cloud Computing

Mobile cloud computing enables compute-limited mobile devices to perform real-time intensive computations such as speech recognition or object detection by leveraging powerful cloud servers. An important problem in large-scale mobile cloud computing is computational offloading, where each mobile device decides when and how much computation should be uploaded to cloud servers by considering the local processing delay and the cost of using cloud servers. In this paper, we develop a distributed threshold-based offloading algorithm where it uploads an incoming computing task to cloud servers if the number of tasks queued at the device reaches the threshold and processes it locally otherwise. The threshold is updated iteratively based on the computational load and the cost of using cloud servers. We formulate the problem as a symmetric game, and characterize the sufficient and necessary conditions for the existence and uniqueness of the Nash Equilibrium (NE) assuming exponential service times. Then, we show the convergence of our proposed distributed algorithm to the NE when the NE exists. Further, we characterize the performance gap between cost under our proposed distributed algorithm and the minimum cost in terms of Price of Anarchy (PoA) when the cost of using cloud servers is high. Finally, we perform extensive simulations to validate our theoretical findings, demonstrate the efficiency of our proposed distributed algorithm under various scenarios such as hyperexponential service times, imperfect server utilization estimation, and asynchronous threshold updates, and reveal the superior performance of threshold-based policies over their probabilistic counterpart.

Transient Behavior of a Queueing Model with Hyper-Exponentially Distributed Processing Times and Finite Buffer Capacity.

In the paper, a finite-capacity queueing model is considered in which jobs arrive according to a Poisson process and are being served according to hyper-exponential service times. A system of equations for the time-sensitive queue-size distribution is established by applying the paradigm of embedded Markov chain and total probability law. The solution of the corresponding system written for Laplace transforms is obtained via an algebraic approach in a compact form. Numerical illustration results are attached as well.

Asymptotic Diffusion Analysis of Multi-Server Retrial Queue with Hyper-Exponential Service

A multi-server retrial queue with a hyper-exponential service time is considered in this paper. The study is performed by the method of asymptotic diffusion analysis under the condition of long delay in orbit. On the basis of the constructed diffusion process, we obtain approximations of stationary probability distributions of the number of customers in orbit and the number of busy servers. Using simulations and numerical analysis, we estimate the accuracy and applicability area of the obtained approximations.

The Analysis, Optimization, and Simulation of a Two-Stage Tandem Queueing Model with Hyperexponential Service Time at Second Stage

The aim of this paper is to analyze a tandem queueing model with two stages. The arrivals to the first stage are Poisson stream and the service time at this stage is exponential. There is no waiting room at first stage. The service time is hyperexponential and no waiting is allowed at second stage. The transition probabilities and loss probabilities of this model are obtained. In addition, the loss probability at second stage is optimized. Performance measures and the variance of the numbers of customers of this tandem queueing model are found. It is seen that the numbers of customers in first stage and second stage are dependent. Finally we have simulated this queueing model. For different values of parameters, exact values, simulated values, and optimal values of obtained performance measures of this model are numerically shown in tables and graphs.

Explicit solutions for queues with Hypo- or Hyper-exponential service time distribution and application to product-form approximations

Queueing systems with Poisson arrival processes and Hypo- or Hyper-exponential service time distribution have been widely studied in the literature. Their steady-state analysis relies on the observation that the infinitesimal generator matrix has a block-regular structure and, hence, the matrix analytic method may be applied. Let πnk be the steady-state probability of observing the kth phase of service and n customers in the queue, with 1≤k≤K, and K the number of phases, and let πn=(πn1,…,πnK). Then, it is well-known that there exists a rate matrix R such that πn+1=πnR. In this paper, we give a symbolic expression for such a matrix R for both cases of Hypo- and Hyper-exponential queueing systems. Then, we exploit this result in order to address the problem of approximating a M/HypoK/1 queue by a product-form model. We show that the knowledge of the symbolic expression of R allows us to specify the approximations for more general models than those that have been previously considered in the literature and with higher accuracy.

INFINITE-SERVER QUEUES WITH BATCH ARRIVALS AND DEPENDENT SERVICE TIMES

Motivated by large-scale service systems, we consider an infinite-server queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be time varying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d. and independent of the arrival process of batches, and we require that the service times within different batches be independent. We exploit a recently established heavy-traffic limit for the number of busy servers to determine the performance impact of the dependence among the service times. The number of busy servers is approximately a Gaussian process. The dependence among the service times does not affect the mean number of busy servers, but it does affect the variance of the number of busy servers. Our approximations quantify the performance impact upon the variance. We conduct simulations to evaluate the heavy-traffic approximations for the stationary model and the model with a time-varying arrival rate. In the simulation experiments, we use the Marshall–Olkin multivariate exponential distribution to model dependent exponential service times within a batch. We also introduce a class of Marshall–Olkin multivariate hyperexponential distributions to model dependent hyper-exponential service times within a batch.

Batch processor sharing with hyper-exponential service time

We study the Processor Sharing queueing model with a hyper-exponential service time distribution and Poisson batch arrival process. In the case of the hyper-exponential service time distribution we find an analytical expression for the expected conditional response time function and obtain an alternative proof of its concavity with respect to the service time.

A queueing model of activities in an intensive care unit

Activities in an intensive care unit (ICU) at a major teaching hospital are modelled by means of a queue-theoretic approach. Using data supplied by the ICU relating to the admissions process, the bed availability and the length of stay of patients, it was possible to fit theoretical distributions to the observed ‘arrival’ and ‘service’ distributions. Queueing equations relevant to a multi-channel system having random arrivals and hyper-exponential service times for each channel are set up, and solved iteratively. Results obtained match well with observations, and the model is then utilised to investigate several ‘what if? ’ scenarios. Reference is made to a simulation model developed in conjunction with the queueing model.

Optimal control of an M/H k /1 queueing system with a removable server

This paper studies a single removable server queueing system with Poisson arrivals and k-type hyper-exponential service time distribution operating under the N-policy. The server may be turned on at arrival epochs or off at departure epochs. We show that the probability that the server is busy in the steady-state is equal to ∑i=1k qi ρi. Various system performance measures, such as the expected number of customers in the system, the expected waiting time in the queue, and the expected waiting time in the system are obtained. A total expected cost function per unit time is developed to determine the optimal operating N-policy at minimum cost.

The sojourn time distribution in an infinite server resequencing queue with dependent interarrival and service times

We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.

The sojourn time distribution in an infinite server resequencing queue with dependent interarrival and service times

We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.

Allocation of service time in a two-server system

Consider a two-server FCFS queueing model where servers are arranged in series. All arrivals join the first service center where they receive a maximum of T units of service. Arrivals with service requirements that exceed the threshold T join the second queue where they receive their remaining service. We show that when the service requirements have hyper-exponential service times with large coefficient of variation our scheme provides better system performance than the standard two parallel server model in the sense of reducing the mean delay per customer in the system. Our model is likely to be useful in systems where high variability is a cause of performance degradation and where numerous service interruptions are not desirable. Scope and purpose Reducing congestion is a primary concern in the design and analysis of queueing models, especially in systems where the distribution of service times is characterized by high variability. As an example consider a two server repair facility where job arrivals have one of two types of defects and where the type of defect is not known a priori. Each type of defect requires an exponential service, but with different service rate. Thus, the overall repair time has a hyper-exponential distribution. The purpose of this article is to present a model that is likely to be useful in systems where high variability is a cause of performance degradation. Our policy is to arrange servers in series and to service an arrival by the first server for a maximum of T units of service. Arrivals join the second queue, if necessary, where they receive their remaining service. It is shown that this simple scheme outperforms the standard two parallel server model.

An analysis of virtual circuits with parallel links

Computer networks supporting virtual circuits deliver packets at the destination in the same sequence as they are received at the source. But packets may arrive at the destination out of sequence if the source is connected to the destination by multiple links. Consequently, in addition to queueing delay and service time (transmission delay) the packets suffer a resequencing delay. The sum of the queueing delay, service time, and resequencing delay is called the total delay. Assuming a Poisson stream of packets, a two-stage hyperexponential service time distribution and m equal capacity links connecting the source to the destination, the distribution and expected value of resequencing delay are derived here. With numerical examples it is shown that the mean total delay decreases very rapidly with the number of links m. The mean queueing delay increases rapidly with the mean service time, and the mean resequencing delay increases slowly with the mean service time.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Insensitivity in models of flexible manufacturing systems with zero-buffer stations and reversible routing

In a recent paper, Yao and Buzacott have shown that, for models of flexible manufacturing systems with zero-buffer stations, the state distribution has a product-form and that it is valid even for hyperexponential service times. In this article, we prove that this state distribution actually holds for general service time distributions.

A numerical solution for the multi-server queue with hyper-exponential service times

In this paper we present a numerical method for the queue GI/ H 2/ s, which is based on general results for GI/ H m / s. We give a complete description of the algorithm which yields exact results for the steady distributions of the actual waiting time, the virtual waiting time and the number of customers in the system both at arrival epochs and in continuous time.

Single server queues with hyperexponential service times

AbstractSingle server queues with hyperexponential service times are studied. Explicit expressions are derived for the waiting time and state distributions.

Queues with batch poisson arrivals and hyper-exponential service time distribution

We consider in this paper the transient behaviour of the queuing system in which (i) the input, following a Poisson distribution, is in batches of variable numbers; (ii) queue discipline is ‘first come first served’, it being assumed that the batches are pre-ordered for service purposes; and (iii) service time distribution is hyper-exponential withn branches. The Laplace transform of the system size distribution is determined by applying the method of generating functions, introduced in queuing theory byBailey [1]. However, assuming steady state conditions to obtain, the problem is completely solved and it is shown that by suitably defining the traffic intensity factor,ϱ, the value,p 0, of the probability of no delay, remains the same in this case of batch arrivals also as in the case of single arrivals. The Laplace transform of the waiting time distribution is also calculated in steady state case from which the mean waiting time may be calculated. Some of the known results are derived as particular cases.