Exact Solutions for M/M/c/Setup Queues

Recently multiserver queues with setup times have been extensively studied because they have applications in power-saving data centers. The most challenging model is the M/M/$c$/Setup queue where a server is turned off when it is idle and is turned on if there are some waiting jobs. Recently, Gandhi et al.~(SIGMETRICS 2013, QUESTA 2014) present the recursive renewal reward approach as a new mathematical tool to analyze the model. In this paper, we derive exact solutions for the same model using two alternative methodologies: generating function approach and matrix analytic method. The former yields several theoretical insights into the systems while the latter provides an exact recursive algorithm to calculate the joint stationary distribution and then some performance measures so as to give new application insights.Comment: Submitted for revie

Sleep Mode Analysis via Workload Decomposition

The goal of this paper is to establish a general approach for analyzing queueing models with repeated inhomogeneous vacations. The server goes on for a vacation if the inactivity prolongs more than the vacation trigger duration. Once the system enters in vacation mode, it may continue for several consecutive vacations. At the end of a vacation, the server goes on another vacation, possibly with a different probability distribution; if during the previous vacation there have been no arrivals. However the system enters in vacation mode only if the inactivity is persisted beyond defined trigger duration. In order to get an insight on the influence of parameters on the performance, we choose to study a simple M/G/1 queue (Poisson arrivals and general independent service times) which has the advantage of being tractable analytically. The theoretical model is applied to the problem of power saving for mobile devices in which the sleep durations of a device correspond to the vacations of the server. Various system performance metrics such as the frame response time and the economy of energy are derived. A constrained optimization problem is formulated to maximize the economy of energy achieved in power save mode, with constraints as QoS conditions to be met. An illustration of the proposed methods is shown with a WiMAX system scenario to obtain design parameters for better performance. Our analysis allows us not only to optimize the system parameters for a given traffic intensity but also to propose parameters that provide the best performance under worst case conditions

Stationary Analysis of a Multiserver queue with multiple working vacation and impatient customers

We consider an M/M/c queue with multiple working vacation and impatient customers. The server serves the customers at a lower rate rather than completely halts the service during this working vacation period. The impatience of the customer’s arises when they arrive during the working vacation period, where the service rate of the customer’s is lower than the normal busy period. The queue is analyzed for multiple working vacation policies. The policy of a MWV demands the server to keep taking vacation until it finds at least a single customer waiting in the system at an instant vacation completion. On returning of the server from his vacation along with finding at least one customer in the system, the server changes its service rate, thereby giving rise to a non-vacation period; otherwise the server immediately goes for another WV. We formulate the probability generating function for the number of customers present when the server is both in a service period as well as in a working vacation period. We further derive a closed-form solution for various performance measures such as the mean queue length and the mean waiting time. The stochastic decomposition properties are verified for the model

The analysis of M/M/1 queue with working vacation in fuzzy environment

This study investigates the FM/FM/1 queue with working vacation. For this fuzzy queuing model, the researcher obtains some performance measure of interest such as the regular busy period, working vacation period, stationary queue length and waiting time. Finally, numerical results are presented to show the effects of system parameters

Analysis of an M/M/1 Queue With Working Vacation and Vacation Interruption

In this paper, an M/M/1 queue with working vacation and vacation interruption is investigated. The server is supposed to interrupt the vacation and return back to the normal working period, if there are at least N customers waiting in the system at a service completion instant during the working vacation period. Otherwise, the server continues the vacation until the system is nonempty after a vacation ends or there are at least N customers after a service ends. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions for the stationary queue length. Moreover, we demonstrate stochastic decomposition structures of the queue length and waiting time, and obtain the distributions of the additional queue length and additional delay for the case N = 2 . Finally, numerical examples are presented

Analysis and optimization of vacation and polling models with retrials

We study a vacation-type queueing model, and a single-server multi-queue polling model, with the special feature of retrials. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. Our main focus is on queue length analysis, both at embedded time points (beginnings of glue periods, visit periods and switch- or vacation periods) and at arbitrary time points.Comment: Keywords: vacation queue, polling model, retrials Submitted for review to Performance evaluation journal, as an extended version of 'Vacation and polling models with retrials', by Onno Boxma and Jacques Resin