Retrial Times Research Articles

Optimal capacity planning for cloud service providers with periodic, time-varying demand

Allocating capacity to private cloud computing services is challenging because demand is time-varying, there are often no buffers, and customers can re-submit jobs a finite number of times. We model this setting using a multi-station queueing network where servers represent CPU cores and jobs not immediately processed retry several times. Assuming retrial rates are stationary and that there is a maximum number of retrial attempts, we determine an optimal service capacity and retrial interval under an admission control policy employed by our partner institution — the server informs customers when they should next attempt service without enforcement. We introduce a recursive representation of the offered load which approximates the fluid dynamics of the system. We then use this representation to develop a solution technique that minimizes the total variation in the constructed offered load. We prove this approach is linked to maximizing system throughput and that in certain settings, the optimal stationary and time-varying retrial intervals are equivalent. Utilizing a data set of cloud computing requests spanning a 24-hour period, our analysis indicates that the optimal policy prescribes a 10% reduction in capacity. We also investigate the fidelity of the fluid model and the sensitivity of our recommendations to the behavior of retrial jobs. We find that retrial-time announcements allow a provider to satisfy service level agreements while encouraging retrial jobs to be processed during off-peak periods. Further, the policy is suitably robust to a customer’s willingness to comply with the suggested retrial times.

Unreliable M[X]/G(P1,P2)/1 feedback retrial queues with combined working vacation

The study examines M[X]/G(P1,P2)/1 feedback retrial queues coupled with starting failure, repair, delay to repair, working vacation, and general retrial times. Also, it explores how different batch sizes affect performance and how bulk arrival affects system behavior. When the server is not in use, a single customer initiates the system while the remaining customers transition to a state of orbit. A new customer must turn on the server to provide two phases of mandatory service at any time. The server could have starting issues. If the service is successfully started (with likelihood α), the customer receives service immediately. In the absence of that, the likelihood of starting failure happens (with likelihood 1−α=α¯). The server was taken for repair with some delay and that customer was transported to an orbital location. When the server was busy or unavailable, the arriving customers queued by FCFS in the orbit. We also discuss the idea of reworking with probability p, and restarting unsuccessful service attempts to improve customer happiness and service efficiency. We also introduce the concept of working vacation, which permits servers to temporarily stop providing services, affecting system performance and availability at both peak and off-peak times. A supplementary variable technique was adopted for the system's and orbit size's probability-generating function. Various performance measurements were provided with appropriate numerical examples.

Bayesian inference for reliability of K-out-of-N: G repairable retrial systems with mixed standbys, switching failure and reboot delay

Existing reliability studies on K-out-of-N: G repairable retrial systems with mixed standbys mainly consider the case where the standby switching is completely reliable and the repair time follows an exponential distribution, and most of them directly analyze the system reliability indices based on probability theory. Given the existence of such phenomenon as standby switching failure in engineering, this paper proposes a novel reliability model of K-out-of-N: G repairable retrial system with switching failure, and performs statistical inference on the reliability to realize the modeling and evaluation of the system. The considered system consists of mixed standbys and a repairer with vacation. When an operating component fails, the warm standby one is switched to the operating one immediately, and the probability of switching failure during the switchover process is p. In the case where the repair time of failed components follows a general distribution, based on the supplementary variable method and recursive algorithm, the steps for calculating the system steady-state availability are presented. In the case of unknown parameters for repair time and retrial time, the Bayesian estimator of the steady-state availability under the squared error loss function and highest posterior density (HPD) confidence interval are obtained. In numerical simulation, the system reliability inference is realized by using the Monte Carlo (MC) method, and the validity of the inference method is analyzed.

Resource optimization for MMAP[c]/PH[c]/S catastrophic queueing model with PH retrial times

Resource optimization for MMAP[c]/PH[c]/S catastrophic queueing model with PH retrial times

Corrigendum: “Extended generator and associated martingales for M/G/1 retrial queue with classical retrial policy and general retrial times”

Corrigendum: “Extended generator and associated martingales for M/G/1 retrial queue with classical retrial policy and general retrial times”

Resource optimization in $$\textit{MMAP[2]/PH[2]/S}$$ priority queueing model with threshold $$\textit{PH}$$ retrial times and the preemptive resume policy

Resource optimization in $$\textit{MMAP[2]/PH[2]/S}$$ priority queueing model with threshold $$\textit{PH}$$ retrial times and the preemptive resume policy

Resource and traffic control optimization in MMAP[c]/PH[c]/S queueing system with PH retrial times and catastrophe phenomenon

Resource and traffic control optimization in MMAP[c]/PH[c]/S queueing system with PH retrial times and catastrophe phenomenon

Optimization of traffic control in $ MMAP\mathit{[2]}/PH\mathit{[2]}/S$ priority queueing model with $ PH $ retrial times and the preemptive repeat policy

<p style='text-indent:20px;'>The presented study elaborates a multi-server priority queueing model considering the preemptive repeat policy and phase-type distribution (<inline-formula><tex-math id="M3">\begin{document}$ P\!H $\end{document}</tex-math></inline-formula>) for retrial process. The incoming heterogeneous calls are categorized as handoff calls and new calls. The arrival and service processes of both types of calls follow marked Markovian arrival process (<inline-formula><tex-math id="M4">\begin{document}$ M\!M\!A\!P $\end{document}</tex-math></inline-formula>) and <inline-formula><tex-math id="M5">\begin{document}$ P\!H $\end{document}</tex-math></inline-formula> distribution with distinct parameters, respectively. An arriving new call will be blocked when all the channels are occupied, and consequently will join the orbit (virtual space) to retry following <inline-formula><tex-math id="M6">\begin{document}$ P\!H $\end{document}</tex-math></inline-formula> distribution. When all the channels are occupied and a handoff call arrives at the system, out of the following two scenarios, one might take place. In the first scenario, if all the channels are occupied with handoff calls, the arriving handoff call will be lost from the system. While in the second one, if all the channels are occupied and at least one of them is serving a new call, the arriving handoff call will be provided service by using preemptive priority over that new call and the preempted new call will join the orbit. Behaviour of the proposed system is modelled by the level dependent quasi-birth-death <inline-formula><tex-math id="M7">\begin{document}$ (L\!D\!Q\!B\!D) $\end{document}</tex-math></inline-formula> process. The expressions of various performance measures have been derived for the numerical illustration. An optimization problem for optimal channel allocation and traffic control has been formulated and dealt by employing appropriate heuristic approaches.</p>

Asymptotic Diffusion Analysis of Retrial Queueing System M/M/1 with Impatient Customers, Collisions and Unreliable Servers

In this paper, a retrial queueing system of the M/M/1 type with Poisson flows of arrivals, impatient customers, collisions, and an unreliable service device is considered. To make the problem more realistic and, hence, more complicated, we include the breakdowns and repairs of the service in this research study. The retrial times of customers in the orbit, service time, impatience time of customers in the orbit, server’s lifetime (depending on whether it is idle or busy), and server recovery time are supposed to be exponentially distributed. The problem of finding the stationary probability distribution of the number of customers in orbit is solved by using the method of asymptotic diffusion analyses under the condition of a heavy load of the system and the patience of customers in orbit. Numerical results are presented that demonstrate the effectiveness of the obtained theoretical conclusions, and a comparative analysis of the method of asymptotic analysis and the method of asymptotic diffusion analysis for the considered problem is given.

Availability and cost-benefit evaluation for a repairable retrial system with warm standbys and priority

This paper investigates a warm standby repairable retrial system with two types of components and a single repairman, where type 1 components have priority over type 2 in use. Failure and repair times for each type of component are assumed to be exponential distributions. The retrial feature is considered and the retrial time of each failed component is exponentially distributed. By using Markov process theory and matrix-analytic method, the system steady-state probabilities are derived, and the system steady-state availability and some steady-state performance indices are obtained. Using the Bayesian approach, the system parameters can be estimated. The cost-benefit ratio function of the system is constructed based on the failed components and repairman's states. Numerical experiments are given to evaluate the effect of each parameter on the system steady-state availability and optimize the system cost-benefit ratio with repair rate as a decision variable.

An M/G/1 Retrial G-queue with Multiple Working Vacation and a Waiting Server

An M/G/1 retrial G-queue with multiple working vacation and a waiting server is taken into consideration in this study. Both the retrial times and service times are assumed to follow general distribution and the waiting server follows an exponential distribution. During the working vacation period customers are served at a lesser rate of service. Before switching over to a vacation the server waits for some arbitrary amount of time and so is called a waiting server. We obtain the PGF for the number of customers and the mean number of customers in the invisible waiting area which is acquired by utilizing the supplementary variable technique. We compute the waiting time distribution. Out of interest a few special cases are conferred. Numerical outcomes are exhibited.

Performability analysis of a MMAP[2]/PH[2]/S model with PH retrial times

This work focuses over the performability analysis of a multi-server retrial queueing model with phase-type inter-retrial times in cellular networks. It is considered that the pattern of the new call arrival and handoff call arrival follows marked Markovian arrival process. The service times of both types of calls are phase-type (PH) distributed with different parameters, and inter-failure & inter-repair times of channels are exponentially distributed. For the prioritization of handoff calls, G channels are kept in reserve for handoff calls. When all the available channels, say S , are busy at the arrival epoch of a handoff call, the handoff call will be dropped. Whereas a new call will be blocked and will have an option to join the orbit of infinite capacity or leave the system without getting the connection, if at least S − G channels are busy. A new call in the orbit, termed as retrial call, retries to get the connection after a random interval which follows PH distribution. This model is analyzed as a level-dependent-quasi-birth-death process by applying an efficient method. Through numerical illustrations, the behavior of performance measures depending on the various relevant intensities is discussed.

Retrial queues with constant retrial times

Retrial queues with constant retrial times

Performance analysis of discrete-time GeoX/G/1 retrial queue with various vacation policies and impatient customers

This paper studies the behavior of a batch arrival single server retrial queueing model under three different vacation policies. Three types of vacation policies, single vacation, multiple vacations, and atmost J-vacations with impatient customers in general retrial times are considered. The probability generating function and marginal generating function of orbit size are obtained in a steady state. The stability condition for each vacation model is derived. Performance measures such as mean orbit size, mean system size, mean waiting time of a customer, and the probabilities of the server being in different states have also been determined. Based on performance characteristics, a comparative analysis is performed among the three vacations. Numerical illustrations are displayed to establish the consistency of the theory developed.

Retrial queues with generally distributed retrial times

Retrial queues with generally distributed retrial times

Extended generator and associated martingales for M/G/1 retrial queue with classical retrial policy and general retrial times

AbstractA retrial queue with classical retrial policy, where each blocked customer in the orbit retries for service, and general retrial times is modeled by a piecewise deterministic Markov process (PDMP). From the extended generator of the PDMP of the retrial queue, we derive the associated martingales. These results are used to derive the conditional expected number of customers in the orbit in the transient regime.

Sojourn Times in a Queueing System with Breakdowns and General Retrial Times

This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, and we obtained the stationary distributions of the number of customers in the orbit and in the system by applying the generation function (GF). We provide the stochastic decomposition law and the application bounds for the proximity between the steady-state distributions for the queueing system under consideration and its corresponding standard system. We developed recursive formulae aimed at the calculation of the steady-state of the orbit and the system. We proved that our discrete-time system approximates M/G/1 with breakdowns and repairs. We analyzed the busy period of an auxiliary system, the objective of which was to study the customer’s delay. The stationary distribution of a customer’s sojourn in the orbit and in the system was the object of a thorough and complete study. Finally, we provide numerical examples that outline the effect of the parameters on several performance characteristics and a conclusions section resuming the main research contributions of the paper.

Reliability modeling for repairable circular consecutive-[formula omitted]-out-of-[formula omitted]: F systems with retrial feature

In this paper, we consider a repairable circular consecutive-k-out-of-n: F retrial system, which can be used to simulate the intelligent closed recurring water cooling automation system in industrial production. We assume that the working time and retrial time of each component obey exponential distributions, and the repair time obeys a general distribution. Based on the generalized transition probability and the key component priority repair rule of the retrial system, the availability and reliability indices of the general repairable circular consecutive-k-out-of-n: F retrial system are derived by using the methods of supplementary variable and Laplace transform. Furthermore, we calculate the explicit expressions of the above indices when n and k are fixed at definite values in the system. The repair time is assumed to follow a Weibull distribution in examples demonstration, and the influence of various parameters on system reliability, steady-state availability and other indices are obtained. The sensitivity and relative sensitivity analysis of the steady-state availability, steady-state failure frequency and mean time to first failure (MTTFF) of the system are also performed.

Analysis of M/m/c Retrial Queue with Thresholds, Ph Distribution of Retrial Times and Unreliable Servers

Analysis of M/m/c Retrial Queue with Thresholds, Ph Distribution of Retrial Times and Unreliable Servers

Stochastic monotonicity approach for a non-Markovian priority retrial queue

This paper considers a non-Markovian priority retrial queue which serves two types of customers. Customers in the regular queue have priority over the customers in the orbit. This means that the customer in orbit can only start retrying when the regular queue becomes empty. If another customer arrives during a retrial time, this customer is served and the retrial has to start over when the regular queue becomes empty again. In this study, a particular interest is devoted to the stochastic monotonicity approach based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary joint distribution of the embedded Markov chain of the considered system.