Transient Queue-size Distribution Research Articles

A Discrete-Time Queueing Model of a Bottleneck with an Energy-Saving Mechanism Based on Setup and Shutdown Times

Producers are encouraged to reduce their energy consumption of manufacturing systems by applying less-energy-intensive modern technologies and advanced machine tools and operating methods at the system level. In the paper, organizational and analytical solutions are combined to model the sustainable production system. Managers can study the behavior of a production system organized using energy-saving rules by changing key parameters of the input model (arrival intensity, bottleneck service rate, buffer size, setup and shutdown time) to analyze the queue size of the production system and therefore performance. A discrete-time queueing model of a single-bottleneck production line with a finite input buffer capacity is proposed. Jobs occur according to a binomial process and are processed individually, one by one, according to the natural FIFO service discipline, with a general discrete-type cumulative distribution function. The total number of jobs present in the system is bounded by a non-random fixed value N. Every time the system becomes empty, an energy-saving mechanism is started: the processing machine (server) is turned off during a geometrically distributed shutdown time. Similarly, the first job arriving into the empty system initializes a geometrically distributed setup time. Identifying renewal moments in the evolution of the model, a system of difference equations is built for the transient queue-size distribution conditioned by the state of the system at the opening. The solution is obtained explicitly in terms of probability-generating functions. In addition, the Drum-Buffer-Rope concept is proposed to reduce the energy consumption of the production line. The throughput of the production system is maximized by adjusting the time between the order arrivals and the size of the input buffer to the capacity of the bottleneck. Turning off a machine under certain conditions and slowing down non-critical machines are strategies to reduce energy consumption. A detailed illustrating numerical and simulation study of the considered model is attached as well, in which the sensitivity of the queue-size behavior to changes of the key input model parameters is investigated.

On Transient Queue-Size Distribution in a Model of WSN Node with Threshold-Type Power-Saving Algorithm.

This article proposes a queueing model of the operation of a wireless sensor network node, in which a threshold strategy for starting the node after a period of no transmission is used. In this model, transmission of packets is resumed when the number of packets in the accumulation buffer reaches a predefined level. In the literature, most of the results for models with limited access to the service station are obtained in equilibrium. In this paper, a formula for the Laplace transform of the transient queue-size distribution is obtained and written using the key input parameters of the system. The analytical apparatus uses the concept of the embedded Markov chain, the formula for total probability, renewal theory and some supporting algebraic results. Numerical examples are attached as well.

Study on Transient Queue-Size Distribution in the Finite-Buffer Model with Batch Arrivals and Multiple Vacation Policy.

The transient behavior of the finite-buffer queueing model with batch arrivals and generally distributed repeated vacations is analyzed. Such a system has potential applications in modeling the functioning of production systems, computer and telecommunication networks with energy saving mechanism based on cyclic monitoring the queue state (Internet of Things, wireless sensors networks, etc.). Identifying renewal moments in the evolution of the system and applying continuous total probability law, a system of Volterra-type integral equations for the time-dependent queue-size distribution, conditioned by the initial buffer state, is derived. A compact-form solution for the corresponding system written for Laplace transforms is obtained using an algebraic approach based on Korolyuk’s potential method. An illustrative numerical example presenting the impact of the service rate, arrival rate, initial buffer state and single vacation duration on the queue-size distribution is attached as well.

On Time-Dependent Queue-Size Distribution in a Model With Finite Buffer Capacity and Deterministic Multiple Vacations With Applications to LTE DRX Mechanism Modeling

A finite-capacity queueing model with the arrival stream of messages governed by the compound Poisson process and generally-distributed processing times is investigated. Whenever the system becomes empty (the server becomes idle), a number of deterministic independent vacations of equal length are initialized as far as at least one message is detected in the accumulating buffer at the completion epoch of one of them. During vacations the service process is suspended completely, while after finishing the last vacation it restarts normally. Identifying Markov moments in the evolution of the system and using the Korolyuk’s potential method, the compact-form representation for the transient queue-size distribution conditioned by the initial buffer state is found in terms of its Laplace transform. The considered model has potential applications in modeling the energy saving LTE DRX mechanism. A detailed simulation and numerical study is attached.

On transient queue-size distribution in a finite-buffer model with threshold waking and early setup policy

A finite-buffer queueing system with threshold waking and early setup policy is investigated. The arrival stream is governed by a Poisson process while service times are assumed to be generally distributed. The natural FIFO processing discipline is used. Every time when the system empties, a type-specific energy saving policy is initialized that is a mixture of the classical N-type policy and an early setup mechanism. Namely, if the level of accumulated messages reaches M≤N, a generally distributed setup time is started, during which the service station achieves full readiness for processing. If, at the completion epoch of the setup time, the state of the system (the number of accumulated messages) equals at least N, then the service begins immediately. Otherwise, the service station waits (being ready for processing) for the Nth arrival. The representation for the Laplace transform of the transient queue-size distribution is obtained using the analytical approach based on the idea of embedded Markov chain, the formula of total probability, linear algebra and renewal theory. A numerical example and simulational study are attached.

Transient solution for the queue-size distribution in a finite-buffer model with general independent input stream and single working vacation policy

• Transient queue-size distribution in GI/M/1/N-type model with working vacation policy is studied. • The idea of embedded Markov chain, total probability law and linear algebra are applied. • Final result is given for the Laplace transform in a compact form. • Illustrative numerical examples are attached. A single-channel finite-buffer queueing model with a general independent input stream of customers, exponential processing times and a working vacation policy is considered. Every time , when the server becomes idle, an exponentially distributed single working vacation period is being initialized, during which the processing is provided with another (slower) rate. After the completion of the vacation period, the service is being continued normally, with the original speed. Using the idea of an embedded Markov chain, the systems of Volterra-type integral equations for the time-dependent queue-size distributions, conditioned by the initial buffer state and related to each other, are built for models beginning the operation in normal and working vacation modes, separately. The solutions of the corresponding systems written for the Laplace transforms are obtained in compact forms using the linear algebraic approach. The numerical illustrative examples are attached as well.

N-policy for redundant repairable system with multiple types of warm standbys with switching failure and vacation

This investigation is concerned with the reliability analysis of redundant repairable system which is supported by mixed standby and two repairmen who turn on according to a threshold N-policy. The first repairman never takes a vacation while the second repairman leaves for a vacation of random length when the number of failed components is less than N. The concepts of standby switching failure, degradation and common cause failure are incorporated to predict the performance metrics of the real time redundant repairable system. By developing Markov model, the transient queue size distribution and expressions for the system reliability, mean time to system failure and other performance measures are obtained. The sensitivity analysis is performed by taking numerical illustration. The model is also examined computationally by employing the adaptive network-based fuzzy interference system (ANFIS) approach to compute the system descriptors. Using supervised learning process, the comparison between the ANFIS results and analytical results are made.

A comprehensive study on the queue-size distribution in a finite-buffer system with a general independent input flow

A finite-buffer GI/M/1/N−type queueing model is considered. The explicit formula for the Laplace transform of the transient queue-size distribution, conditioned by the number of packets present in the system at the starting time, is derived. The shape of the formula allows for finding the stationary distribution by applying the key renewal theorem. Moreover, the convergence rate of the transient queue-size distribution to the stationary one is determined with the constant value given explicitly. Numerical example is attached as well.

Analysis of a discrete-time $$Geo^{\lambda _1,\lambda _2}/G/1$$ G e o λ 1 , λ 2 / G / 1 queue with N-policy and D-policy

This paper explores a discrete-time Geo / G / 1 queueing system with N-policy and D-policy wherein customers arrive at the system in variable input rates according to the states of the server. When the system becomes empty, the server stays idle until the number of waiting customers reaches N or the sum of the service times of the waiting customers in the system reaches or exceeds a predetermined positive integer D, whichever happens first (referred to as Min(N, D)-policy). Employing the renewal theory, the total probability decomposition law and the probability generating function technique we obtain the probability generating function for the transient queue size distribution at time epoch \(n^+\). Also, the explicit recursive formulas of the steady-state queue length distribution at time epochs \(n^+\), n, \(n^-\) and outside observer’s time epoch are derived, respectively. Finally, some numerical experiments are conducted to investigate the sensitivity of system performance measures and the optimization of system capacity.

Analytical Solution for Time-Dependent Queue-Size Behavior in the Manufacturing Line with Finite Buffer Capacity and Machine Setup and Closedown Times

A single reliable-machine manufacturing line with finite buffer capacity is considered, in which each idle period is preceded by a random closedown time, and each busy period is preceded by a random setup time. The stream of arriving jobs is described by a simple Poisson process, while processing (service), closedown and setup times are generally distributed random variables. A system of Volterra-type integral equations for the transient queue-size distribution conditioned by the initial level of buffer saturation is found by using the Markov property of successive service completion epochs and the continuous version of total probability law. A parallel system written for Laplace transforms is obtained and written in a specific form. The solution of the latter system is derived in a compact-form applying the linear algebraic approach. The final representations are found in terms of “input” system parameters (transforms of processing, closedown and setup times’ distributions) and certain sequence defined recursively.

On transient queue-size distribution in the batch-arrivals system with a single vacation policy

A queueing system with batch Poisson arrivals and single vacations with the exhaustive service discipline is investigated. As the main result the representation for the Laplace transform of the transient queue-size distribution in the system which is empty before the opening is obtained. The approach consists of few stages. Firstly, some results for a ``usual'' system without vacations corresponding to the original one are derived. Next, applying the formula of total probability, the analysis of the original system on a single vacation cycle is brought to the study of the ``usual'' system. Finally, the renewal theory is used to derive the general result. Moreover, a numerical approach to analytical results is discussed and some illustrative numerical examples are given.

A Direct Approach to Transient Queue-Size Distribution in a Finite-Buffer Queue with AQM

A finite-buffer M=G=1-type queueing model is considered in which the level of saturation of the buffer is controlled by a dropping function. A direct analytical method to the study of the transient queue-size distribution is proposed. Applying the embedded Markov chain paradigm and the formula of total probability, a specific-type system of integral equations for the transient queue-size distributions, conditioned by the number of packets present in the system at the opening, is derived. The corresponding system of linear equations built for the Laplace transforms is written in a matrix form and solved directly. The M=M=1=2 type system is analyzed as a special case separately. Numerical utility of the approach is illustrated as well.