Hyper-exponential Service Research Articles

Optimal management strategies of renewable energy systems with hyperexponential service provisioning: an economic investigation

The current research proposes optimal management strategies for queueing modeling-based renewable energy systems with hyper-exponentially distributed maintenance/repair under the assumption of an admission control policy. Using the concept and steps of the matrix-analytical method, the steady-state probability distribution associated with energy systems is explicitly presented. A relatively straightforward computation that can help with modeling wind energy generation, investigating wind farm performance, optimizing energy based on system storage, reliability inspection, service maintenance planning, and numerous other purposes can be employed to mathematically derive several system performance indicators. The investigation findings are validated via quantitative outcomes, illustrative possesses, and a step-by-step recursive methodology for efficient management of the renewable energy system. Additionally, considering multiple governing parameter values, the nature-inspired optimization technique, Cuckoo Search (CS), is employed to demonstrate the optimum anticipated cost of renewable energy system. A comparison with other metaheuristics and semi-classical approaches is also presented to establish the best convergence results. In order to help system designers, policymakers, engineers, and researchers, several numerical examples are also provided to construct more practical strategies based on the production of energy, storage, and system management. The economic, parametric, and performance investigation findings are highlighted, and the opportunities and recommendations for further research are provided. In a nutshell, the outcomes of the present analysis can be adopted to formulate the most effective economic strategies and regulate decision-making processes in the energy sectors.

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.

Mathematical Model of Call Center in the Form of Multi-Server Queueing System

The paper considers the model of a call center in the form of a multi-server queueing system with Poisson arrivals and an unlimited waiting area. In the model under consideration, incoming calls do not differ in terms of service conditions, requested service, and interarrival periods. It is assumed that an incoming call can use any free server and they are all identical in terms of capabilities and quality. The goal problem is to find the stationary distribution of the number of calls in the system for an arbitrary recurrent service. This will allow us to evaluate the performance measures of such systems and solve various optimization problems for them. Considering models with non-exponential service times provides solutions for a wide class of mathematical models, making the results more adequate for real call centers. The solution is based on the approximation of the given distribution function of the service time by the hyperexponential distribution function. Therefore, first, the problem of studying a system with hyperexponential service is solved using the matrix-geometric method. Further, on the basis of this result, an approximation of the stationary distribution of the number of calls in a multi-server system with an arbitrary distribution function of the service time is constructed. Various issues in the application of this approximation are considered, and its accuracy is analyzed based on comparison with the known analytical result for a particular case, as well as with the results of the simulation.

On Actual Maximum Exposure From 5G Multicolumn Radio Base Station Antennas for Electromagnetic Field Compliance Assessment

The traditional approach of radio frequency electromagnetic field exposure compliance assessment is highly conservative when applied to radio base station antennas implementing dynamic beamforming. In this article, an analytical model based on the queuing theory with a hyperexponential service distribution time is developed to assess the time-averaged actual maximum downlink exposure of 5G multicolumn radio base station antennas by taking into account the effects of beam scanning over time in free space. Using the measured antenna radiation patterns, the 5G downlink antenna precoding codebook, and assuming a conservative user equipment distribution, the ratio of the actual maximum exposure to the theoretical maximum exposure with 100% traffic load and 75% time-division duplex downlink duty cycle is found to be less than 0.5 and 0.3 for four-transmitter and eight-transmitter radio base station antennas, respectively. These results show that assuming constant peak power transmission in a fixed direction leads to an overestimate of downlink exposure also from conventional antennas characterized by only a few transmitters in addition to massive multi-input multi-output products.

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.

The M/G/1 Queue with Quasi-Restricted Accessibility

We consider single-server queues of the M/G/1 kind with a special kind of partial customer rejection called quasi-restricted accessibility (QRA). Under QRA, the actual service time assigned to an arriving customer depends on his service requirement, say x, the current workload, say w, and a prespecified threshold b. If x + w ≤ b the customer is fully served. If w ≤ b < w + x, the customer receives service time b − w + (w + x − b)f for some random number f ∈ [0, 1], while if w > b the actual service time is the fraction fx of the requirement. The random fractions are assumed to be i.i.d. The main aim of this article is to determine some of the central characteristics of this system in closed form. We derive the steady-state distribution of the workload process, which is also the steady-state distribution of the waiting time, and provide explicit results for the case of service times with rational Laplace transforms, in particular for Erlang or hyperexponential service requirements, and uniformly distributed or constant fractions. We also deal with the case of exponential barriers b (instead of one constant threshold). Furthermore, the distribution functions of the length of a busy period and the cycle maximum of the workload are determined. In the case of phase-type service requirements there is an alternative (martingale) technique to derive the busy period distribution; we illustrate this approach in the case of Erlang(2,μ). Finally, we show in the example of the Erlang(2,μ)/M/1-type QRA queue with deterministic fractions (which is non-Markovian) how to compute the busy period distribution via a duality with a Markovian system.

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.

An Equivalent Random Method with hyper-exponential service

The Equivalent Random Method (ERM) has been widely used to predict blocking probabilities at overflow service stations. The method assumes that service times follow an exponential distribution. While this may be a reasonable assumption for voice traffic, it is not a good assumption for dial-up Internet traffic, where service times typically have a coefficient of variation (standard deviation/mean) greater than 1. In this paper, we give a modified ERM for two-term hyper-exponential service distributions. The method is based on an efficient algorithm to estimate the peakedness of the overflow process of an M/ H 2/ S/ S queue. Finally, we investigate the accuracy of the modified ERM using simulation and also compare systems with hyper-exponential service to systems with heavy-tailed service.

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.