Generating Function Technique Research Articles

Exact spectral solution of two interacting run-and-tumble particles on a ring lattice

Exact solutions of interacting random walk models, such as 1D lattice gases, offer precise insight into the origin of nonequilibrium phenomena. Here, we study a model of run-and-tumble particles on a ring lattice interacting via hardcore exclusion. We present the exact solution for one and two particles using a generating function technique. For two particles, the eigenvectors and eigenvalues are explicitly expressed using two parameters reminiscent of Bethe roots, whose numerical values are determined by polynomial equations which we derive. The spectrum depends in a complicated way on the ratio of direction reversal rate to lattice jump rate, . For both one and two particles, the spectrum consists of separate real bands for large , which mix and become complex-valued for small . At exceptional values of , two or more eigenvalues coalesce such that the Markov matrix is non-diagonalizable. A consequence of this intricate parameter dependence is the appearance of dynamical transitions: non-analytic minima in the longest relaxation times as functions of (for a given lattice size). Exceptional points are theoretically and experimentally relevant in, e.g. open quantum systems and multichannel scattering. We propose that the phenomenon should be a ubiquitous feature of classical nonequilibrium models as well, and of relevance to physical observables in this context.

Retrial Queuing System with Randomized Push-Out Mechanism and Non-Preemptive Priority

A single-server retrial queueing system with finite buffer size, Poisson arrivals, an exponentially distributed service time is considered. If an arriving customer finds the queue completely occupied, it joins a special retrial waiting group (named orbit) in order to seek service over again after some period of time which have exponential distribution. The primary customers take non-preemptive priority over secondary customers. We also introduce so-called randomized push-out buffer management mechanism. It allows you push secondary customers out of the system to free up space that could be taken by primary customers. It is shown that such a queueing system can be reduced to analogous model without retrials. Using generating function technique, the loss probabilities for both types of customers are obtained. Theoretical results allow to investigate the dependency of loss probabilities on the main model parameters (like push-out and retrial probabilities). Special areas of loads are found, where the model locks itself up for secondary customers or follows linear loss law as well. A detailed comparison is made with the case of preemptive priority, which was studied by the authors earlier.

Transient analysis of an M/M/c queuing system with retention of reneging customers

In this paper, we study the transient behaviour of an M/M/c queuing system with reneging and retention of reneging customers. The probability generating function technique along with Bessel function properties is used to derive the time-dependent state probabilities explicitly. The transient behaviour of system size probabilities, the expected system size, the average reneging rate, and the average retention rate is studied with the help of a numerical example.

Transient analysis of an M/M/c queuing system with retention of reneging customers

In this paper, we study the transient behaviour of an M/M/c queuing system with reneging and retention of reneging customers. The probability generating function technique along with Bessel function properties is used to derive the time-dependent state probabilities explicitly. The transient behaviour of system size probabilities, the expected system size, the average reneging rate, and the average retention rate is studied with the help of a numerical example.

Impact of customers’ impatience on an M/M/1 queueing system subject to differentiated vacations with a waiting server

ABSTRACTImpatience behavior of an M/M/1 queueing system, which has differentiated server vacations, is considered. Arrivals follow a Poisson process, service is exponentially distributed, as are both vacation types. When the system is empty after waiting for a random period of time, the server takes a vacation and returns after a random duration. If there are still no customers in the system, server can go for a vacation of shorter duration. If the server is in a vacation state, arriving customers become impatient with an individual timer which is exponentially distributed. Explicit expressions for time-dependent probabilities of the system size are obtained in terms of the modified Bessel function of first kind by making use of Laplace transforms and probability generating function techniques along with continued fractions and the confluent hypergeometric function. Explicit expressions for the time-dependent mean and variance of the system size are also derived. Moreover, steady-state probabilities for the system size are derived from the transient state probabilities. Finally, a numerical example is presented to study the behavior of the system.

A New Class of Hermite-Apostol Type Frobenius-Euler Polynomials and Its Applications

The article is written with the objectives to introduce a multi-variable hybrid class, namely the Hermite–Apostol-type Frobenius–Euler polynomials, and to characterize their properties via different generating function techniques. Several explicit relations involving Hurwitz–Lerch Zeta functions and some summation formulae related to these polynomials are derived. Further, we establish certain symmetry identities involving generalized power sums and Hurwitz–Lerch Zeta functions. An operational view for these polynomials is presented, and corresponding applications are given. The illustrative special cases are also mentioned along with their generating equations.

Residence time near an absorbing set

We determine how long a diffusing particle spends in a given spatial range before it dies at an absorbing boundary. In one dimension, for a particle that starts at x0 and is absorbed at x = 0, the average residence time of the particle in the range is for x < x0 and for x > x0, where D is the diffusion coefficient. We extend our approach to biased diffusion, to a particle confined to a finite interval, and to general spatial dimensions. We then use the generating function technique to derive parallel results for the average number of times that a one-dimensional symmetric nearest-neighbor random walk visits site x when the walk starts at x0 = 1 and is absorbed at x = 0. We also determine the distribution of times when the random walk first revisits x = 1 before being absorbed.

Performance analysis of a discrete-time &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ Geo/G/1$&lt;/tex-math&gt;&lt;/inline-formula&gt; retrial queue with non-preemptive priority, working vacations and vacation interruption

This paper is concerned with a discrete-time $ Geo/G/1$ retrial queueing system with non-preemptive priority, working vacations and vacation interruption where the service times and retrial times are arbitrarily distributed. If an arriving customer finds the server free, his service commences immediately. Otherwise, he either joins the priority queue with probability $ α$, or leaves the service area and enters the retrial group (orbit) with probability $ \mathit{\bar{\alpha }}\left( = 1-\alpha \right)$. Customers in the priority queue have non-preemptive priority over those in the orbit. Whenever the system becomes empty, the server takes working vacation during which the server can serve customers at a lower service rate. If there are customers in the system at the epoch of a service completion, the server resumes the normal working level whether the working vacation ends or not (i.e., working vacation interruption occurs). Otherwise, the server proceeds with the vacation. Employing supplementary variable method and generating function technique, we analyze the underlying Markov chain of the considered queueing model, and obtain the stationary distribution of the Markov chain, the generating functions for the number of customers in the priority queue, in the orbit and in the system, as well as some crucial performance measures in steady state. Furthermore, the relation between our discrete-time queue and its continuous-time counterpart is investigated. Finally, some numerical examples are provided to explore the effect of various system parameters on the queueing characteristics.

Number of spin- J states and odd-even staggering for identical particles in a single- j shell

In this work, new recursion relations for the number of spin-J states for identical particles in a single-j shell are presented. Such relations are obtained using the generating-function technique, which enables one to exhibit an odd-even staggering in the spin distribution of an even number of fermions in a single-j shell: the number of states with an even value of J is larger than the number of states with an odd value of J. An analytical expression of the excess of states with an even value of J is provided, and its asymptotic behavior for large values of j is discussed.

Approximating the tail probabilities of the longest run in a sequence of Bernoulli trials

ABSTRACTWe consider the longest run of either successes or failures in a sequence of Bernoulli trials. The exact distribution of this random variable is obtained using probability generating function techniques. We consider approximating the tail areas of this random variable using a partial fraction approximation and a saddlepoint approximation with various continuity corrections. We investigate and compare the performance of the approximations for various values of sequence length and various values of the probability of success.

Optimizing Dynamic Performance of Multistate Systems With Heterogeneous 1-Out-of-&lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;${N}$ &lt;/tex-math&gt; &lt;/inline-formula&gt; Warm Standby Components

This paper models and optimizes dynamic performance of multistate systems with a general series parallel structure. Each system component is a 1-out-of- ${N}$ warm standby configuration of heterogeneous functional elements, which can be characterized by different time-to-failure distributions, performances, and costs. The entire system must satisfy a random demand specified by a time-dependent distribution. An iterative algorithm is developed for determining performance stochastic processes of particular components. A universal generating function technique is used for evaluating expected system availability and unsupplied demand over a particular mission time for the considered system. Two types of optimization problems are then identified and solved, with the objective of finding component structures and element activation sequences to maximize system availability, or minimize unsupplied system demand, or minimize total cost. Optimization results can facilitate the optimal decision on design and operation of multistate series parallel systems. A practical example of a power station coal transportation system is provided to illustrate application of the proposed methodology and optimization problems.

Transient Analysis of an M/M/1 Queue with Reneging, Catastrophes, Server Failures and Repairs

An M / M / 1 queue with reneging, catastrophes, server failures and repairs is considered. The arrivals follow a Poisson process and the servers serve according to an exponential distribution. On arrival a customer decides to join the queue and after joining the queue if a customer has to wait for the service longer than his expectation, he may renege. Explicit expression for the time-dependent probabilities of the system size is obtained in terms of the modified Bessel function of first kind by making use of Laplace transform and probability generating function techniques. The system queue length and failure distribution for steady state are derived. Additionally, time-dependent mean and variance are obtained. A numerical example is presented to study the behavior of the system.

Optimal component allocation in a multi-state system with hierarchical performance sharing groups

Performance sharing is a common technique to save energy and increase system reliability. This paper considers a multi-state system with hierarchical performance sharing groups. The system has several subsystems connected with a primary common bus, and each subsystem has some groups of components connected with a secondary common bus. Each component group has a demand to satisfy. The groups within the same subsystem can share performance through the secondary common bus, whereas the different subsystems can share capacity with each other through the primary common bus. Both primary and secondary common buses have a maximum transmission capacity. The system fails if the demand for any group cannot be satisfied. The system reliability is evaluated with a universal generating function technique, and the optimal allocation of components which maximises the system reliability is studied.

Queuing loss models with more alternative heterogeneous groups

SummaryThis paper considers queuing model with 2 Poisson traffics offered to the primary group, where the rejected demands of one of them are overflowed to the alternative groups in sequence and leave the system without being served when all channels are busy. Here, serving intensity from primary group is changed and gets various values in alternative groups. Generating function technique is used for the analytical solution of steady‐state equations system. Common solution obtained for the state probabilities permits efficient procedure for model parameters calculation. Next, explicit solutions in the case with secondary and tertiary groups are derived. Moreover, obtained solutions for binomial moments are of significant importance for treating more complex models. Some numerical results are also presented through illustrative examples justifying the need to dedicate special attention to the cases where phenomenon of heterogeneous alternative channels or groups is remarked. In addition, as an example, explicit solution for the model with primary group and 3 heterogeneous alternate channels is obtained. This leads to the comparative support when solving 4 channels system with losses and renewal arrivals. These solutions represent a prospective tool applicable in different serving situations, particularly considering traditional and next generation communications networks.

Computation of k-ary Lyndon words using generating functions and their differential equations

By using generating functions technique, we investigate some properties of the k-ary Lyndon words. We give an explicit formula for the generating functions including not only combinatorial sums, but also hypergeometric function. We also derive higher-order differential equations and some formulas related to the k-ary Lyndon words. By applying these equations and formulas, we also derive some novel identities including the Stirling numbers of the second kind, the Apostol-Bernoulli numbers and combinatorial sums. Moreover, in order to compute numerical values of the higher-order derivative for the generating functions enumerating k-ary Lyndon words with prime number length, we construct an efficient algorithm. By applying this algorithm, we give some numerical values for these derivative equations for selected different prime numbers.

Transient performance analysis of a single server queuing model with retention of reneging customers

In this paper, we study a single server queuing model with retention of reneging customers. The transient solution of the model is derived using probability generating function technique. The time-dependent mean and variance of the model are also obtained. Some important special cases of the model are derived and discussed. Finally, based on the numerical example, the transient performance analysis of the model is performed.

A transient solution to the M/M/c queuing model equation with balking and catastrophes

In this paper, we consider a Markovian multi-server queuing system with balking and catastrophes. The probability generating function technique along with the Bessel function properites is used to obtain a transient solution to the queuing model. The transient probabilities for the number of customers in the system are obtained explicitly. The expressions for the time-dependent expected number of customers in the system are also obtained. Finally, applications of the model are also discussed.

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.

Reliability evaluation for wireless sensor networks with chain structures using the universal generating function

AbstractIn different protocols, wireless sensor networks (WSNs) form different topologies. The reliability evaluation of a WSN whose topology is a chain is studied in this paper. A bidirectional data transmission tree (BDTT) model is established to describe the process of data fusion and transmission on a WSN with a chain structure. From the characteristics of the measurement errors, the number of fusions, and measurement credibility, 3 definitions for the reliability of a BDTT are presented. Based on the universal generating function technique, an algorithm is suggested for evaluating the reliability of a BDTT. In the algorithm, a BDTT model is divided into 2 branches, and 2 new composition operators are defined according to data fusion and transmission. We use the first operator to obtain the u‐function of each branch and the second one to combine the u‐functions of 2 branches and obtain the u‐function of the BDTT. The process of calculating the reliability of a BDTT is illustrated with an example, and the effects of measurement errors, the number of fusions, and measurement credibility on the system reliability are analyzed.

Transient analysis of an M/M/c queuing system with balking and retention of reneging customers

ABSTRACTIn this paper, we study an infinite capacity multi-server Markovian queuing system with balking and retention of reneging customers. The transient analysis of the model is performed. The probability generating function technique along with Bessel function properties is used to derive the time-dependent state probabilities explicitly.