Retrial BMAP/PH/N Queueing System with a Threshold-Dependent Inter-Retrial Time Distribution

A tandem retrial queueing system with two Markovian flows and reservation of channels

To improve the service quality and save the system cost of the cloud computing center, this paper studies the joint optimization problem of energy consumption and performance of cloud computing centers with a batch Markovian arrival process. The system has multiple parallel processors and the processing time of each processor follows phase type distribution. The system has finite buffer. We construct the system as a BMAP/PH/N/M queueing system and analyze the performance of the system based on queuing theory. An optimization model is established by combing the energy consumption and system performance measures. The optimal number of servers is analyzed. Some managerial insights are given by numerical analysis.

In this paper we assume that the customers arrive at a single server queueing system according to Markovian Arrival process. When the system is empty, the server goes for vacation and produces inventory for future use during this period. The maximum processed inventory at a stretch is L. The inventory processing time follows phase type distribution. These are required for the service of customers-one for each customer. The server returns from vacation when there are N customers in the system. The service time follows two distinct phase type distributions depending on whether there is processed items or no processed item available at service commencement epoch. We analyse the distributions of time till the number of customers hit N or the inventory level reaches L, idle time, the distribution of time until the number of customers hit N and also the distribution of the number of inventory processed before the arrival of first customer. Also we provide the distribution of a busy cycle, LSTs of busy cycles in which no item is left in the inventory and at least one item is left in the inventory. We perform some numerical experiments to evaluate the expected idle time, standard deviation and coefficient of variation of idle time of the server.

This paper studies a production inventory model with retrial of customers under (s, S) policy. The arrival of customers is according to a Markovian Arrival Process with representation (D0, D1) and service times follow an exponential distribution. The production process follows a phase-type distribution. When the inventory level reduces to a pre-assigned level s due to demands, production starts and service is given at a reduced rate. This reduced rate continuous up to the zero level of inventory. The arriving customers are directed to a buffer of finite capacity equal to the current inventory level. An arriving customer, who notices the buffer full, proceeds to an orbit of infinite capacity with some probability and decides to leave the system with the complementary probability. An orbiting customer may retry from the orbit and inter-retrial times are exponentially distributed with linear rate. Various system performance measures of the model are defined. A suitable cost function is constructed and analyzed algorithmically. The optimum (s, S) pair is obtained. The effect of correlation between two successive inter-arrival times is also analyzed.

Analysis of queueing model with processor sharing discipline and customers impatience

In this paper, we consider two queueing models. Model I is on a single-server queueing system in which the arrival process follows MAP with representation D = (D0,D1) of order m and service time follows phase-type distribution (β, S) of order n. When a customer enters into service, a generalized Erlang clock is started simultaneously. The clock has k stages. The pth stage parameter is θp for 1 ≤ p ≤ k. If a customer completes the service in between the realizations of stages k1 and k2 (1 < k1 < k2 < k) of the clock, it is a perfect one. On the other hand, if the service gets completed either before the kth1 stage realization or after the kth2 stage realization, it is discarded because of imperfection. We analyse this model using the matrix-geometric method. We obtain the expected service time and expected waiting time of a tagged customer. Additional performance measures are also computed. We construct a revenue function and numerically analyse it. In Model II, a single server queueing system in which all assumptions are the same as in Model I except the assumption on service time, is considered. Up to stage k1 service time follows phase-type distribution (α′ , T′) of order n1 and beyond stage k1, the service time follows phase type distribution (β′ , S′) of order n2. We compare the values of the revenue function of the two models

This paper considers a single server retrial queue in which a state-dependent service policy is adopted to control the service rate. Customers arrive in the system according to a Poisson process and the service times and inter-retrial times are all exponentially distributed. If the number of customers in orbit is equal to or less than a certain threshold, the service rate is set in a low value and it also can be switched to a high value once this number exceeds the threshold. The stationary distribution and two performance measures are obtained through the partial generating functions. It is shown that this state-dependent service policy degenerates into a classic retrial queueing system without control policy under some conditions. In order to achieve the social optimal strategies, a new reward-cost function is established and the global numerical solutions, obtained by Canonical Particle Swarm Optimization algorithm, demonstrate that the managers can get more benefits if applying this state-dependent service policy compared with the classic model.

This paper analyses a production inventory model with varying production rates where demand is according to Marko­vian arrival process (MAP). The time between additions of two successive items by production to the inventory follows Phase type distribution. When the inventory level decreases to a predetermined level production starts and the rate of production is higher until the number of items in the inventory crosses that level. The arriving customers are directed to a buffer of finite capacity equal to the current inventory level and service times are exponentially distributed. An arriving customer who notices the buffer full, proceeds to an orbit of infinite capacity and retry from there. Inter-retrial times follow an exponential distribution. Various system performance measures of the model are defined. A suitable cost function is also constructed.

Phase-type distributions in stochastic automata networks

We analyze and compare three (s,S) inventory systems with positive service time and retrial of customers. In all of these systems, arrivals of customers form a Poisson process and service times are exponentially distributed. When the inventory level depletes to s due to services, an order of replenishment is placed. The lead-time follows an exponential distribution. In model I, an arriving customer, finding the inventory dry or server busy, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the inventory dry or server busy, returns to the orbit with probability δ and is lost forever with probability (1−δ). In addition to the description in model I, we provide a buffer of varying (finite) capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III. In models II and III, an arriving customer, finding the buffer full, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the buffer full, returns to the orbit with probability δ and is lost forever with probability (1−δ). In all these models, the interretrial times are exponentially distributed with linear rate. Using matrix-analytic method, we study these inventory models. Some measures of the system performance in the steady state are derived. A suitable cost function is defined for all three cases and analyzed using graphical illustrations.

We consider the stability of N-model systems that consist of two customer classes and two server pools. Servers in one of the pools can serve both classes, but those in the other pool can serve only one of the classes. The standard fluid models in general are not sufficient to establish the stability region of these systems under static priority policies. Therefore, we use a novel and a general approach to augment the fluid model equations based on induced Markov chains. Using this new approach, we establish the stability region of these systems under a static priority rule with thresholds when the service and interarrival times have phase-type distributions. We show that, in certain cases, the stability region depends on the distributions of the service and interarrival times (beyond their mean), on the number of servers in the system, and on the threshold value. We also show that it is possible to expand the stability region in these systems by increasing the variability of the service times (without changing their mean) while keeping the other parameters fixed. The extension of our results to parallel server systems and general service time distributions is also discussed.

Predicting departure times in multi-stage queueing systems

Bayesian estimation for the M/ G/1 queue using a phase-type approximation

A multi-server queueing system with a Markovian arrival process and finite and infinite buffers to model a call center with a call-back option is investigated. If all servers are busy during the customer arrival epoch, the customer may leave the system forever or move to the buffer (such a customer is referred to as a real customer), or, alternatively, request for call-back (such a customer is referred to as a virtual customer). During a waiting period, a real customer can be impatient and may leave the system without service or request for call-back (becomes a virtual customer). The service time of a customer and the dial time to a virtual customer for a server have a phase-type distribution. To simplify the investigation of the system we introduce the notion of a generalized phase-type service time distribution. We determine the stationary distribution of the system states and derive the Laplace–Stieltjes transforms of the sojourn and waiting time distributions for real and virtual customers. Some key performance measures are calculated and numerical results are presented.

An analysis of the equilibrium strategies for route-choosing customers in a two-station queueing system