Length Of Busy Period Research Articles

Empirical validation and performance of duty cycle–based DTMC model in channel estimation

This paper explores the learning capability of hidden Markov model (HMM) in capturing the temporal correlation and predicting primary user (PU) activity pattern of real spectrum data of GSM-900 band through an USRP-LabVIEW platform for cognitive radio (CR) systems. The inability of the widely used stationary Markov model in estimating the occupancy pattern of primary channels for a long duration of time has been verified. We proposed an alternative duty cycle (DC)–based two-state discrete-time Markov chain (DTMC-DC) model. Analysis of empirical data indicates that DC required for a non-stationary DTMC-DC model can be well approximated by a trapezoidal shape and the PU spectrum usage pattern estimated using DTMC-DC is capable of learning the statistical behavior (length of idle and busy interval periods) of a real channel accurately with a reduced complexity.

Busy Period Analysis of the Geo/Geo/1/K Queue with a Single Vacation

Discrete-time Queueing models are frequently utilized to analyze the performance of computing and communication systems. The length of busy period is one of important performance measures for such systems. In this paper, we consider the busy period of the Geo/Geo/1/K queue with a single vacation. We derive the moments of the length of the busy (idle) period, the number of customers who arrive and enter the system during the busy (idle) period and the number of customers who arrive but are lost due to no vacancies in the system for both early arrival system (EAS) and late arrival system (LAS). In order to do this, recursive equations for the joint probability generating function of the busy period of the Geo/Geo/1/K queue starting with n, 1≤n≤K, customers, the number of customers who arrive and enter the system, and arrive but are lost during that busy period are constructed. Using the result of the busy period analysis, we also numerically study differences of various performance measures between EAS and LAS. This numerical study shows that the performance gap between EAS and LAS increases as the system capacity K decrease, and the arrival rate (probability) approaches the service rate (probability). This performance gap also decreases as the vacation rate (probability) decrease, but it does not shrink to zero.

A Queueing System with Vacations after a Random Amount of Work

This paper considers an $M/G/1$ queue with the following vacation discipline. The server takes a vacation as soon as it has served a certain amount of work since the end of the previous vacation. If the system becomes empty before the server has completed this amount of work, then it stays idle until the next customer arrival and then becomes active again. Such a vacation discipline arises, for example, in the maintenance of production systems, where machines or equipment mainly degrade while being operational. We derive an explicit expression for the distribution of the time it takes until the prespecified amount of work has been served. For the case the total amount of work till vacation is exponentially distributed, we derive the transforms of the steady-state workload at various epochs, busy period, waiting time, sojourn time, and queue length distributions.

The structure of departure process and optimal control strategy N* for Geo/G/1 discrete-time queue with multiple server vacations and Min(N, V)-Policy

This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n+ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0+, n+] from any initial state. Meanwhile, the relationship among departure process, server’s state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures, including the expected length of server busy period, server’s actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N* for minimizing the system cost under a given cost structure.

In the search for the infinite servers queue with Poisson arrivals busy period distribution exponential behaviour

This paper purpose is to investigate exponential behavior conditions for the infinite servers queue with Poisson arrivals busy period length distribution. It is presented a general theoretical result that is the basis of this work. The complementary analysis rely on the infinite servers queue with Poisson arrivals busy period length distribution moments computation. In infinite servers queue with Poisson arrivals practical applications, in economic, management, and business areas, the management of the effective number of servers is essential since the physical presence of infinite servers is not viable and so it is necessary to create that condition through an adequate management of the number of servers during the busy period.

In the search for the infinite servers queue with Poisson arrivals busy period distribution exponential behaviour

The purpose of this paper is to investigate exponential behaviour conditions for the M|G|∞ queue busy period length distribution. It is presented a general theoretical result that is the basis of this work. The complementary analysis rely on the M|G|∞ queue busy period length distribution moments computation. In M|G+∞ queue practical applications - in economic, management and business areas - the management of the effective number of servers is essential since the physical presence of infinite servers is not viable and so it is necessary to create that condition through an adequate management of the number of servers during the busy period.

Queue Length and Busy Period Analysis for the M/G/1 Queue with Negative Arrivals

In this paper, the queue lengths and the busy period lengths of the M/G/1 queueing systems with negative arrivals are analyzed. Two types of negative arrivals are considered. One type is negative customers and the other type is disasters. When a negative customer arrives to a system, one positive customer is removed if the number of positive customers is more than one. In particular, we assume the RCH (Removal of a Customer at the Head) type of negative customers which represent a kind of work-canceling signal to the positive customer in service. On the other hand, disasters get rid of all customers in the system. In this paper, the Probability Generating Function (PGF) of the stationary queue length and busy period length of M/G/1 queue with both negative customers and disasters are derived.

Steady state analysis of the M/G/1//N queue with orbit of blocked customers

This article considers a finite-source queueing model of M/G/1 type in which a customer, arriving at a moment of a busy server, is not allowed either to queue or to do repetitions. Instead, for an exponentially distributed time interval he is blocked in the orbit of inactive customers. We carry out a steady state analysis of the system and compare it with the corresponding system with retrials. Optimization problems are considered and formulas for the Laplace–Stieltjes transform of the busy period length are obtained.

The Israeli Queue with a general group-joining policy

We consider a single-server multi-queue system with unlimited-size batch service where the next queue to be served is the one with the most senior customer (the so called ‘Israeli Queue’). We study a Markovian system with state-dependent group-joining policy and derive results for various performance measures, such as steady-state distribution of the number of groups in the system, sojourn times, group sizes, and lengths of busy periods. Closed-form expressions are obtained for both the Uniform and the Geometric joining policies. Numerical results are presented.

Idle and Busy Period Analysis of Two Class Data Traffic through Queueing Technique

of Idle and busy period of any communication system gives the overall information about system behavior when system is empty and data present in the system. In this paper we use a queueing theory approach to model the system with two class data traffic.we develop and analyze the idle and busy period of two class data traffic through queueing system using Markov chain. We also develop the markov chain for calculating the number of customers served during busy period. The length of busy period is also calculated through the construction of Markov chain. The cumulative distribution function of the busy period for each state is also calculated for the various arrival rates.

Stochastic law of the busy period for the simple machine interference problem

In this not, the Laplaces transform of the distribution of the length of the busy period for the simple machine interference problem has been obtained. It is shown that the mean length of the busy period under suitable limiting conditions corresponds to the mean length of the busy period for the infinite case.

Non-Markovian bulk queueing system with state dependent arrivals and multiple vacations - a simulation approach

We considered a non-Markovian bulk arrival general bulk service queueing system with state dependent arrivals and multiple vacations in this paper. The arrivals occur in bulk with a rateλ, when the server is busy and with a rateλ0, when the server is on vacation. Using a supplementary variable technique and difference differential equations method we derive the queue size distribution, expected length of idle and busy periods and expected queue size of a Mx/G(a, b)/1 queueing system with state dependent arrivals and multiple vacations. We study the same through simulation using ARENA, a powerful manufacturing system simulation software and obtained some performance measures, which are mandatory for the study and analysis of the said queueing model. The simulated results for a few cases are presented.

A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

A working vacation queue with priority customers and vacation interruptions

In this paper, we analyse a working vacation queueing model with priority customers where the customer service times follow phase-type distributions. The server serves the customers at a lower rate during a working vacation period than in a non-vacation one. After serving a customer in working vacation, if the server finds any customer waiting in queue, the vacation is interrupted and the server switches its service rate from working vacation to the non-vacation service rate. The length of busy period, busy cycle and the distributions of queue length as well as of waiting times are derived for two types of priority customers. The response times are also analysed. With the help of numerical experiments the effects of system parameters on its performance is demonstrated.

Infinite servers queue systems busy period length moments

Infinite servers queue systems busy period length moments

Infinite servers queue systems busy period length moments

Infinite servers queue systems busy period length moments

Cost analysis of a bulk service retrial queue

This paper studies a batch arrival general bulk service retrial queueing model with constant retrial rate. The primary customers arrive in bulk according to Poisson process and they get service under general bulk service rule with minimum of one customer and maximum of ‘b’ customers. If the arriving batch of customers, of size ‘ζ ’, 1≤ζ ≤ b , finds the server free, then all of them get service immediately; while, if the size of the arriving batch is more than ‘b’, then, ‘b’ customers enter the service station and the remaining ζ - b customers join the orbit. However, if an arriving batch of customers finds the server busy, then the entire batch joins the orbit in order to seek service again. The customers in the orbit will try for service one by one with a constant retrial rate ‘v’ when the server is idle. For the proposed model, the probability generating function of the steady-state queue size distribution at an arbitrary time, expected number of customers in the orbit, expected waiting time, expected length of busy period and expected length of busy cycle are obtained. The cost analysis of the queueing system is discussed. The effects of several parameters on the system are analysed numerically.

Analysis of minimal backlogging-based available bandwidth estimation mechanism

This paper analyzes the minimal backlogging-based available bandwidth estimation mechanism to strengthen the theory behind the mechanism. The minimal backlogging method estimates the available bandwidth using the statistic of the probing traffic service rate. We show that the statistic of the probing traffic service rate is a consistent estimator of the available bandwidth for a G/G/1 queueing system under minimal backlogging condition to support the minimal backlogging method theoretically. In order to emulate the minimal backlogging method in a real multi-hop network, we detect the minimal backlogging condition or closeness of the probing rate to the available bandwidth based on the busy period length, and change the probing rate adaptively to maintain the minimal backlogging condition. We explain that the minimal backlogging condition or available bandwidth might be detected more accurately by the busy period of probing packets than by the gap response curve or rate response curve, and enhance the minimal backlogging method further by introducing a new initial probing rate estimation mechanism. A reasonable range of available bandwidth for a short time interval can be obtained using the mean and variance of the estimated available bandwidth, since the proposed mechanism can estimate the available bandwidth quickly and track it adaptively. The proposed mechanism is implemented in a Linux environment. The performance of our scheme is compared to those of conventional available bandwidth estimation mechanisms through experiments on a test-bed with single-hop or multiple-hop topologies.

Working vacations queueing model with multiple types of server breakdowns

This paper deals with a single server working vacation queueing model with multiple types of server breakdowns. In a working vacations queueing model, the server works at a different rate instead of being completely idle during the vacation period; the arrival rate varies according to the server’s status. It is assumed that the server is subject to interruption due to multiple types of breakdowns and is sent immediately for repair. Each type of breakdown requires a finite random number of stages of repair. The life time of the server and the repair time of each phase are assumed to be exponentially distributed. We propose a matrix–geometric approach for computing the stationary queue length distribution. Various performance indices namely the expected length of busy period, the expected length of working vacation period, the mean waiting time and average delay, etc. are established. In order to validate the analytical approach, by taking illustration, we compute numerical results. The sensitivity analysis is also performed to explore the effect of different parameters.