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We analyze the time-dependent behavior of an $M/M/c$ priority queue having two customer classes, class-dependent service rates, and preemptive priority between classes. More particularly, we develop a method that determines the Laplace transforms of the transition functions when the system is initially empty. The Laplace transforms corresponding to states with at least $c$ high-priority customers are expressed explicitly in terms of the Laplace transforms corresponding to states with at most $c - 1$ high-priority customers. We then show how to compute the remaining Laplace transforms recursively, by making use of a variant of Ramaswami's formula from the theory of $M/G/1$-type Markov processes. While the primary focus of our work is on deriving Laplace transforms of transition functions, analogous results can be derived for the stationary distribution: these results seem to yield the most explicit expressions known to date., Comment: 34 pages, 4 figures

A finite-buffer G I / 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.

This note discusses the inter-loss time ofan M/G/1/1 queuing system. The inter-loss time is defined as the time duration between two consecutive losses of arriving customers. In this study, we present the explicit Laplace transform of the inter-loss time distribution of an M/G/1/1 queuing system.

We characterize some natural exponential families in R 2 via diagonal elements of quadratic variance function. Our proofs rely on Laplace transforms, and along the way, we obtain some new characterizations for Laplace transforms of probability measures in certain classes of functions.

This paper considers the Brownian perturbed Cram&eacute, r&ndash, Lundberg risk model with a dividends barrier. We study various types of Pad&eacute, approximations and Laguerre expansions to compute or approximate the scale function that is necessary to optimize the dividends barrier. We experiment also with a heavy-tailed claim distribution for which we apply the so-called &ldquo, shifted&rdquo, Pad&eacute, approximation.

We study the discrete time risk process modelled by the skip-free random walk and derive results connected to the ruin probability and crossing a fixed level for this type of process. We use the method relying on the classical ballot theorems to derive the results for crossing a fixed level and compare them to the results known for the continuous time version of the risk process. We generalize this model by adding a perturbation and, still relying on the skip-free structure of that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk process. We further derive the famous Pollaczek-Khinchine type formula for this generalized process, using the decomposition of the supremum of the dual process at some special instants of time.

A method to determine characteristics of a single-channel closed queueing system with exponential distribution of the customer generation time and arbitrary distribution of service time is proposed. An increase in the capacity of the system is reached using the distribution law of service times that depends on the number of customers in the system at the starting moment of servicing. The Laplace transforms for the distribution of the number of customers in the system during the busy period and the distribution function of the busy period are derived. An algorithm for calculation of the stationary characteristics is tested using a simulation model that is developed with the assistance the GPSS World tools.

A single server Markovian queueing model is considered. The arrivals are allowed to join the queue according to a Poisson distribution and the service takes place according to an exponential distribution. Whenever the system is empty, the server goes for a vacation and return back to the system after N or more customers are found in the system. If the number of customers in the system is less than ‘’ then the server takes another vacation. In this paper, we obtain explicit expressions for the time dependent system size probabilities of such a model using Laplace transform and generating function techniques. Numerical illustrations are added to support the theoretical results obtained.

In this paper we consider the MMAP/PH/1 priority queue, both the case of preemptive resume and the case of non-preemptive service. The main idea of the presented analysis procedure is that the sojourn time of the low priority jobs in the preemptive case (and the waiting time distribution in the non-preemptive case) can be represented by the duration of the busy period of a special Markovian fluid model. By making use of the recent results on the busy period analysis of Markovian fluid models it is possible to calculate several queueing performance measures in an efficient way including the sojourn time distribution (both in the time domain and in the Laplace transform domain), the moments of the sojourn time, the generating function of the queue length, the queue length moments and the queue length probabilities.

We consider an $$\textit{M/G/1}$$ retrial queueing system with two classes of customers, in which the service time distributions are different for both classes of customers. When the server is unavailable, an arriving class-1 customer is queued in the queue with infinite capacity, whereas class-2 customer enters the retrial group. In this paper, we are concerned with the analysis of the waiting time distribution. We obtain the joint transform of the waiting time of a class-2 customer and the number of class-2 customers as well as the Laplace–Stieltjes transform of the waiting time of a class-1 customer. We also obtain all the moments of the waiting time distributions of class-1 and class-2 customers.

A generalised Pollaczek–Khinchin formula for the Polya/G/1 queue, with a Polya peaked arrival process, general distributed service times, and infinite number of waiting positions, is obtained. It is shown that the peakedness of the number of arrivals and the variance of the service time lead to a significant increase in the service delay and queue length.

This paper presents a heuristic approach to derive the Laplace-Stieltjes transform (LST) and the probability generating function (PGF) of the waiting time distributions of a continuous- and a discrete-time GI/G/1 queue, respectively. This is a new idea to derive the well-known results, the waiting time distribution of GI/G/1 queue, in a different way.

This article discusses the steady state analysis of the M / G / 2 queuing system with two heterogeneous servers under new queue disciplines when the classical First Come First Served ‘(FCFS)’ queue discipline is to be violated. Customers are served either by server-I according to an exponential service time distribution with mean rate μ or by server-II with a general service time distribution B ( t ) . Sequel to some objections raised in the literature on the use of the classical FCFS queue discipline in heterogeneous service systems, two alternative queue disciplines (Serial and Parallel) are considered in this work with the objective that if the FCFS is violated then the violation is a minimum in the long run. Using the embedded method under the serial queue discipline and the supplementary variable technique under the parallel queue discipline, we present an exact analysis of the steady state number of customers in the system and most importantly, the actual waiting time expectation of customers in the system. Our work shows that one can obtain all stationary probabilities and other vital measures for this queue under certain simple but realistic assumptions.

We present a Markov model to analyze the queueing behavior of the nonstationary G ( t ) / G ( t ) / s ( t ) + G ( t ) queue. We assume an exhaustive service discipline (where servers complete their current service before leaving) and use acyclic phase-type distributions to approximate the general interarrival, service, and abandonment time distributions. The time-varying performance measures of interest are: (1) the expected number of customers in queue, (2) the variance of the number of customers in queue, (3) the expected number of abandonments, and (4) the virtual waiting time distribution of a customer arriving at an arbitrary moment in time. We refer to our model as G-RAND since it analyzes a general queue using the randomization method. A computational experiment shows that our model allows the accurate analysis of small- to medium-sized problem instances.

An operation of a single-machine manufacturing system is modeled by an unreliable finite-buffer-type queuing system with Poisson arrivals, in which service times, failure-free times and times of repairs are totally independent and exponentially distributed random variables. Applying the idea of embedded Markov chain and the formula of total probability a system of integral equations for the transient conditional queue-size distributions of jobs present in the system at fixed time t is built. The solution of the corresponding system written for Laplace transforms is obtained in a compact form using the potential technique.

A single-server queue system with nonordinary input flow that depends on the queue length is considered. Sufficient existence conditions for the stationary distribution of the queue length provided that the process under study is an eigen process are obtained. Recurrence formulas for the Laplace transform of the nonstationary distributions of the queue length are derived.

In this paper we present an exact steady-state analysis of a discrete-time Geo/G/1 queueing system with working vacations, where the server can keep on working, but at a slower speed during the vacation period. The transition probability matrix describing this queuing model can be seen as an M/G/1-type matrix form. This allows us to derive the probability generating function (PGF) of the stationary queue length at the departure epochs by the M/G/1-type matrix analytic approach. To understand the stationary queue length better, by applying the stochastic decomposition theory of the standard M/G/1 queue with general vacations, another equivalent expression for the PGF is derived. We also show the different cases of the customer waiting to obtain the PGF of the waiting time, and the normal busy period and busy cycle analysis is provided. Finally, we discuss various performance measures and numerical results, and an application to network scheduling in the wavelength division-multiplexed (WDM) system illustrates the benefit of this model in real problems.

The formula for the Laplace transform of an exponential function, , can be derived with equal ease if the parameters a and s are complex numbers. This leads to formulas for the Laplace transforms of eatsin (bt) and eatcos (bt) (where a and b are complex) and to calculations of certain inverse Laplace transforms without the need to consider Laplace transforms of derivatives or convolutions. Simpler proofs also follow from coupling the formula , for which a simple proof is given, with the fact that the operator commutes with certain familiar infinite series.

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.

This paper analyzes the waiting time distribution in the M/PH/1 retrial queue. We give expressions for the Laplace-Stieltjes transform of the waiting time. We then provide a numerical algorithm for calculating the Laplace-Stieltjes transform of the waiting time. Numerical inversion of the Laplace-Stieltjes transforms is used to calculate the waiting time distribution. Numerical results are presented to illustrate our results.

We study a generalized risk process $X(t)=Y(t)-C(t)$, $t\in[0,\tau]$, where $Y$ is a L\'evy process, $C$ an independent subordinator and $\tau$ an independent exponential time. Dropping the standard assumptions on the finite expectations of the processes $Y$ and $C$ and the net profit condition, we derive a Pollaczek-Khinchine type formula for the supremum of the dual process $\widehat{X}=-X$ on $[0,\tau]$ which generalizes the results obtained in \cite{HPSV1}. We also discuss which assumptions are necessary for deriving this formula, specially from the point of view of the ladder process., Comment: 14 pages

In this paper, we investigate the process of semi-Markov random walk with a positive drift, negative jumps, and a delay screen at zero. Explicit forms are found of the Laplace transform in time, the Laplace-Stieltjes transform by the state (phase) of the conditional and unconditional distributions of the process states, and the explicit form of the Laplace-Stieltjes transform of the ergodic distribution of the process.

We apply the general theory of stochastic integration to identify a martingale associated with a Levy process modified by the addition of a secondary process of bounded variation on every finite interval. This martingale can be applied to queues and related stochastic storage models driven by a Levy process. For example, we have applied this martingale to derive the (non-product-form) steady-state distribution of a two-node tandem storage network with Levy input and deterministic linear fluid flow out of the nodes.

This short communication considers a tandem queue fed by Lévy input. The main result concerns expressions for the Laplace transform of the transient workload in the downstream queue, under the condition that the system starts off empty. This expression greatly simplifies if the driving Lévy process is spectrally one-sided; in that situation the transform is given explicitly in terms of the Laplace exponent of the Lévy input.

Working vacation queue models are well applied in the modeling and analysis of the router in optical networks. The GI/Geo/c queue with working vacations is studied in this paper. Through establishing two-dimensional Markov chain and using matrix-geometric solution method, the stability condition is derived. Adopting UL-type RG-factorization of irreducible Markov chain, the stationary distribution is given. Based on these, the probability distribution of queue-length and PGF of waiting time are obtained in the end.

The goal is to identify the class of distributions to which the distribution of the maximum of a L\'evy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the L\'evy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing L\'evy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczeck-Khinchine formula for stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion., Comment: 7 pages

By use of a generalization of the Cramer–Rao-type inequality, we establish several inequalities relating characteristic functions and Laplace transforms of lattice distributions. We also indicate how the results can be directly adapted to continuous cases and illustrate some applications.

We study a batch arrival MX/M/1 queue with multiple working vacation. The server serves customers at a lower rate rather than completely stopping service during the service period. Using a quasi upper triangular transition probability matrix of two-dimensional Markov chain and matrix analytic method, the probability generating function (PGF) of the stationary system length distribution is obtained, from which we obtain the stochastic decomposition structure of system length which indicates the relationship with that of the MX/M/1 queue without vacation. Some performance indices are derived by using the PGF of the stationary system length distribution. It is important that we obtain the Laplace Stieltjes transform (LST) of the stationary waiting time distribution. Further, we obtain the mean system length and the mean waiting time. Finally, numerical results for some special cases are presented to show the effects of system parameters.

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at three epochs of time (arbitrary, pre-arrival, and post-departure) and queueing-time distribution (virtual and actual) of the MAP/R/1 queue, where R represents the class of distributions whose Laplace–Stieltjes transforms are rational functions. Our analysis is based on roots of the associated characteristic equations of the (i) vector-generating function of system-length distribution and (ii) Laplace–Stieltjes transform of the virtual queueing-time distribution. The proposed method for evaluating boundary probabilities is an alternative to the matrix-analytic method as well as spectral method. Numerical aspects have been tested for a variety of arrival and service-time (including matrix-exponential (ME)) distributions and a sample of numerical outputs is presented. The method is analytically quite simple and easy to implement. It is hoped that the results obtained would prove to be beneficial to both theoreticians and practitioners.

A batch arrival single-server queueing system of the M X /G/1 type is considered. At the end of each busy period the server begins a multiple vacation period when the service process is stopped. A new method of the study of departure process in such a system in transient state is proposed. Using the formula of total probability, we direct the analysis to that in the system without vacations. General results are obtained using the renewal-theory approach. An explicit representation for the probability generating function of Laplace transform of departure process is derived. The formula is written down by means of transforms of distributions describing the arrival and service processes, vacation times, and components of certain factorization identity of Wiener-Hopf type connected with these distributions.

In this article, we consider a risk process which exbihits the key features of companies with steady outflows and sporadic inflows (e.g., discoveries, patents). A risk management policy is further implemented stating that the outflow rate is reduced when no revenue (inflow) is generated within an Erlang-n time period. For the surplus process of interest, a Markovian representation is first given which leads to the form of the solution for the Laplace transform of the time to ruin. A homogeneous linear integro-differential equation for the Laplace transform of the time of ruin is later derived. The boundary conditions of the aforementioned integro-differential equation are used to complete the representation of the Laplace transform of the time to ruin. Finally, numerical applications are considered to illustrate the effectiveness of this risk management policy to lower the company's solvency risk.

This work analyzes the waiting time distribution in the M/G/1 retrial queue. The first two moments of the waiting time distribution are known from the literature. In this work we obtain all the moments of the waiting time distribution.

In this paper we study two transient characteristics of a Markov-fluid-driven queue, viz., the busy period and the covariance function of the workload process. Both metrics are captured in terms of their Laplace transforms. Relying on sample-path large deviations, we also identify the logarithmic asymptotics of the probability that the busy period lasts longer than t, as t→∞. Examples illustrating the theory are included.

This paper studies a fluid model driven by an M/M/1 queue with multiple exponential vacations and N-policy. The expression for the Laplace transform of the joint steady-state distribution of the fluid model is of a simple matrix power function form or matrix factorial form. Based on this fact, we introduce a new method of fluid model—modified matrix geometric solution method. The Laplace transform and Laplace-Stieltjes transform of the steady-state distribution of the buffer content are concisely expressed through the minimal positive solution to a crucial quadratic equation. Finally, we give concise expression for the performance measure—mean buffer content, which is useful in parameter design of fluid model and various practical applications.

PurposeThe purpose of this paper is to investigate how queueing theory has been applied to derive results for a Sparre Andersen risk process for which the claim inter‐arrival distribution is hyper Erlang.Design/methodology/approachThe paper exploits the duality results between the queueing theory and risk processes to derive explicit expressions for the ultimate ruin probability and moments of time to ruin in this renewal risk model.FindingsThis paper derives explicit expressions for the Laplace transforms of the idle/waiting time distribution inGI/HEr(ki,λi)/1 and its dual HEr(ki,λi)/G/1. As a consequence, an expression for the ultimate ruin probability is obtained in this model. The relationship between the time of ruin and busy period inM/G/1 queuing system is used to derive the expected time of ruin.Originality/valueThe study of renewal risk process is mostly concentrated on Erlang distributed inter‐claim times. But the Erlang distributions are not dense in the space of all probability distributions and therefore, the paper cannot approximate an arbitrary distribution function by an Erlang one. To overcome this difficulty, the paper uses the hyper Erlang distributions, which can be used to approximate the distribution of any non‐negative random variable.

We study a modified Markovian bulk-arrival and bulk-service queue incorporating state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated and the relationship with our queueing model is examined and exploited. Equilibrium behaviour is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting time and busy period distributions are answered in detail and the Laplace transforms of these distributions are presented. Further properties including expectations of hitting times and busy period are also explored.

A customer in a service system and an outside observer (manager or designer of the system) estimate the system performance differently. Unlike the outside observer, the customer can never find himself in an empty system. Therefore, the sets of scenarios, relevant for the two at a given time, differ. So differ the meanings and values of the performance measures of the queue: expected queue length and expected remaining waiting time (workload). The difference between the two viewpoints can be even more significant when steady-state values of the queue measures are reached slowly, or even are never reached. In this paper, we obtain the relations between the means and variances of the measures in transient time and in steady state for a capacitated FCFS queue with exponentially distributed service time. In particular, a formula similar to Little’s law is derived for the means of the queue measures. Several examples support the validity and significance of the results.

We have introduced excluded volume effect, which is an important factor to model a realistic pedestrian queue, into queueing theory. The probability distributions of pedestrian number and pedestrian waiting time in a queue have been calculated exactly. Due to time needed to close up the queue, the mean number of pedestrians increases as pedestrian arrival probability (λ) and leaving probability (μ) increase even if the ratio between them (i.e., ρ = λ/μ) remains constant. Furthermore, at a given ρ, the mean waiting time does not increase monotonically with the service time (which is inverse to μ), a minimum could be reached instead.

A stepped jump process of a semi-Markov walk with two delay screens at zero and at a is constructed. The Laplace transformation of the distribution of the time of the system sojourn within a given band and its first and second moments are found.

This paper considers the stationary queue length and waiting time distributions in a FIFO BMAP/GI/1 queue with heavy-tailed service times and that with heavy-tailed batch sizes. In each case, we provide sufficient conditions under which the stationary queue length and waiting time distributions are subexponential. Furthermore, we obtain asymptotic relationships between the tail distributions of the stationary queue length and waiting time.

In this paper, we introduce the multiple adaptive vacation policy and the general decrementing service rule based on the classical M/G/1 queueing systems, and obtain the P.G.F. (Probability Generating Function) of stationary queue length by using the embedded Markov chain method and regeneration cycle approach. Then, the LST (Laplace Stieltjes Transform) of stationary waiting time is also derived according to the independence between the waiting time and arrival process. At last some special cases are given to show the general properties of the new model, and some numerical results are shown to compare the mean queue length and waiting time of special cases.

A.K. Erlang introduced the M/D/s queue in 1917, while F. Pollaczek and C.D. Crommelin formalized the theory using complex analysis and transforms. Let D(s,λ) denote the stationary probability of experiencing no waiting time in the M/D/s queue with arrival rate λ and service requirement 1. We use D(s,λ) as a vehicle to give an overview of some of the results we obtained over the last years, including explicit characterizations of the roots, the derivation of infinite series from expressions in terms of roots using Fourier sampling and heavy-traffic limits obtained from square-root staffing. We propose to call D(s,λ) the Erlang D formula, for which several new results are presented and compared with the results of Pollaczek.

In this paper, we illustrate that a power series technique can be used to derive explicit expressions for the transient state distribution of a queueing problem having “chemical” rules with an arbitrary number of customers present initially in the system. Based on generating function and Laplace techniques Conolly et al. (in Math Sci 22:83–91, 1997) have obtained the distributions for a non-empty chemical queue. Their solution enables us only to recover the idle probability of the system in explicit form. Here, we extend not only the model of Conolly et al. but also get a new and simple solution for this model. The derived formula for the transient state is free of Bessel function or any integral forms. The transient solution of the standard M/M/1/∞ queue with λ = μ is a special case of our result. Furthermore, the probability density function of the virtual waiting time in a chemical queue is studied. Finally, the theory is underpinned by numerical results.

In this paper the well-known insurance ruin problem is reconsidered. The ruin probability is estimated in the case of an unknown claims density, assuming a sample of claims is given. An important step in the construction of the estimator is the application of a regularized version of the inverse of the Laplace transform. A rate of convergence in probability for the integrated squared error (ISE) is derived and a simulation study is included.