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This paper studies the impatient behaviour in an infinite server queue with additional tasks assigned to the system. Whenever the system becomes empty, the system as a whole is assigned a secondary task of duration U whose distribution is exponential. Any arrival during the period U becomes impatient due to the unavailability of service facility. Each individual waiting customer activates an independent impatience timer of duration T which is exponentially distributed. When the system comes back after the completion of U, before T expires, the waiting customers are simultaneously taken for service and they leave the system after the completion of service. If T expires before the completion of task U, the customers abandon the system and never to return. The transient system size probabilities of this model are derived explicitly for both single and multiple task cases. The time-dependent mean and variance of system size are also derived. Further, numerical simulations are also presented to analyse the effect of system indices.

We consider an M / G / ∞ model with exponentially distributed setup times. Arrivals result in an off server starting a setup. Upon completing a setup, a server begins serving a waiting customer. Upon completing service, with probability p the server becomes eligible to serve another waiting customer. Among other things, we show, in steady state, that the number of busy servers is Poisson distributed and is independent of the number of servers in setup.

The present study deal with the development of statistical methodology for updating the basic arrival and service time distributions in respect of prior variations. These updated arrival and services time distributions provide us the modified form of traffic intensity as the ratio of updated mean service time to updated mean inter-arrival time. As such all the basic queue characteristics will be developed in the changed scenario.

The Mθ/G/1/m and Mθ/G/1 queueing systems, in which the service time depends on the queue length and is determined at the beginning of service of a customer, are investigated. The average busy period and the stationary distribution of the number of customers in the system have been found via the approach based on the idea of V.S. Korolyuk’s potential method. As a special case, the Mθ/G/1 queue with a single threshold of switching of service modes is examined. The results are checked using the simulation model constructed with the use of GPSS World software tools.

This paper considers a fluid queue driven by an M/M/1 disaster queue. We derive explicit expression for the stationary distribution function of the buffer content in terms of modified Bessel function of first kind by using generating function technique. Numerical results are presented to illustrate the behavior of the buffer content distribution.

It has been shown by (R.O. Al-Seedy, A.A. El-Sherbiny, S.A. El-Shehawy, S.I. Ammar, Transient solution of the M/M/c queue with balking and reneging, Comput. Math. Appl. 57 (2009) 1280–1285) that a generating function technique can be successfully applied to derive the transient solution for an M/M/c queueing system. In this paper, we further illustrate how this technique can be used to obtain the busy period density function of an M/M/1 queue with balking and reneging. Finally, numerical calculations are presented.

We consider a compound Poisson risk model in which the premium rate changes when the surplus exceeds a threshold. The explicit form of the ruin probability for the risk model is obtained by deriving and using the overflow probability of the workload process in the corresponding M=G=1 queueing model.

In this paper, we are concerned with the analysis of the waiting time distribution in the M/M/m retrial queue. We give expressions for the Laplace-Stieltjes transform (LST) of the waiting time distribution and then provide a numerical algorithm for calculating the LST of the waiting time distribution. Numerical inversion of the LSTs is used to calculate the waiting time distribution. Numerical results are presented to illustrate our results.

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 Uniform Minimum Variance Unbiased (UMVU) estimators of ρl, the probability of having l or more customers, L, the expected system size, L q , the expected number of customers in the queue, and , the expected number of customers in a non empty queue, are derived based on a random sample of fixed size n on system size at departure points from the geometric distribution on the support {0, 1, 2,…} with mean , which is the distribution of system size in M/M/1 queueing system in equilibrium. The derivations are based on application of Lehmann-Scheffe theorem. Also, CAN estimators of performance measures are derived. In addition the probability distribution of UMVU estimators are obtained.

In this article, maximum likelihood estimator (MLE) as well as Bayes estimator of traffic intensity (ρ) in an M/M/1/∞ queueing model in equilibrium based on number of customers present in the queue at successive departure epochs have been worked out. Estimates of some functions of ρ which provide measures of effectiveness of the queue have also been derived. A comprehensive simulation study starting with the transition probability matrix has been carried out in the last section.

We consider a 2-class, single-server, preemptive priority queueing model in which the high-priority customers form a classical M/M/1 queue, while the low-priority customers form the so-called Israeli Queue with at most N different groups and unlimited-size batch service. We provide an extensive probabilistic analysis and calculate key performance measures. Special cases are analyzed and numerical examples are presented and discussed.

We first analyze a multi-server M / M / c queue with working vacations. Using a quasi birth-and-death (QBD) process and a matrix-geometric solution method, the steady state distribution of the queue length is derived. Furthermore, a fluid flow model driven by this M / M / c working vacation queue is discussed, we obtain the sets of differential equations satisfied by the stationary joint distribution of the buffer content, by which we gain the simple structure of the Laplace transform (LT) of the stationary distribution of the buffer content. Finally, we give the probability of empty buffer content and the mean of the buffer content based on the relationship between the LT and the Laplace–Stieltjes transform (LST) of the stationary distribution.

In Sigman (J. Appl. Probab. 48A:209–216, 2011b), a first exact simulation algorithm was presented for the stationary distribution of customer delay for FIFO M/G/c queues in which ρ=λ/μ

This article investigates the behavior of the cognitive queueing system in a spectrum-sharing environment under the primary outage probability constraint. The queueing model is investigated under two cognitive transmit modes. A variable transmit rate that is equal to the cognitive channel's capacity is first revisited, then the case of constant transmit rate is considered. Both transmit modes are found to lead to an $$M/G/1$$ M / G / 1 queueing model. The performance measures of the cognitive queue under these two modes are investigated and numerically compared. The performance measures include the mean transmit rate, mean service time, server utilization, mean waiting and transit times of the packets in the queue, mean number of waiting and transit packets in the queue, mean duration of the server's busy period, and mean number of packets served during the server's busy period. The effects of changing the primary outage probability constraint and the primary transmit rate on the performance measures are investigated as well.

This paper considers an infinite buffer M / M / c queueing system in which servers follow a multi-threshold vacation policy. With such a policy, at a service completion instant, if the number of customers in the system is less than a prefixed threshold value, part of servers together take a single vacation (or leave for a random amount of time doing other secondary job). At the vacation completion instant, they return to the system for serving the customers. Some practical production and inventory systems or call centers could be modeled as this Markovian queue with a multi-threshold vacation policy. Using the Markovian process model, we obtain the exact closed-form expression of rate matrix and the stationary distribution of the number of customers in the system. A cost model is developed to search the joint optimal values of the thresholds of vacation policy and service rate of each server, which minimizes the long-term average cost. Some numerical results are presented to illustrate the optimization procedures.

Queueing models create lot of importance due to their ready applicability. Much work has reported in literature regarding queueing models with homogeneous Poisson arrivals. But in many practical situations arising at places like transportation, communication networks, production processes, etc., the arrivals are non homogeneous and time dependent. Hence, in this paper we develop and analyze a queueing model with non homogeneous (time dependent) bulk arrivals under parallel and series configuration. Using the difference differential equations the joint probability density function of the number of customers in each queue is obtained. The performance of the model is evaluated by deriving the explicit expressions of the system characteristics like the probability of the system emptiness, average number of customers in each queue, the mean waiting time of the customer, the throughput of the nodes, etc. The sensitivity analysis of the model revealed that the bulk size distribution parameters and non homogeneous arrivals have significant influence on the system performance measures. The utility of the model in congestion control is demonstrated through a numerical illustration.

Queueing models provide the basic frame work to get the optimum operating policies in production processes and service facilities. In transportation processes involving buses, aero planes, trains, ships, elevators and so on, bulk service queueing models are often encountered. Control chart technique analyzes the behavior of services and provides the prior idea about expected waiting time, maximum waiting time and minimum waiting time which in turn guarantees customer's satisfaction. Keeping this in view, the construction of control charts for waiting time in system of M /M (a, b) /1 bulk service queueing model is proposed in this paper.

In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N-(V, E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relationship between the M/G/1/N and GI/M/1/N queues, we also provide asymptotic properties for the standard GI/M/1/N model.

We consider an M/M/1 queueing system with inventory under the $$(r,Q)$$ policy and with lost sales, in which demands occur according to a Poisson process and service times are exponentially distributed. All arriving customers during stockout are lost. We derive the stationary distributions of the joint queue length (number of customers in the system) and on-hand inventory when lead times are random variables and can take various distributions. The derived stationary distributions are used to formulate long-run average performance measures and cost functions in some numerical examples.

This paper considers the M/M/Nrepairable queuing system. The customers' arrival is a Poisson process. The servers are subject to breakdown according to Poisson processes with different rates in idle time and busy time, respectively. The breakdown servers are repaired by repairmen, and the repair time is an exponential distribution. Using probability generating function and transform method, we obtain the steady-state probabilities of the system states, the steady-state availability of the servers, and the mean queueing length of the model.

This paper considers a BMAP/M/∞ queue with a batch Markovian arrival process (BMAP) and an exponential service time distribution. We first prove that the BMAP/M/∞ queue is stable if and only if the expectation of the logarithm of the batch-size distribution is finite. Using this result, we also present the stability condition for an infinite-server queue with a multiclass batch Markovian arrival process and class-dependent exponential service times.

The subject of this paper is the problem of estimating the service time distribution of the M/G/∞ queue from incomplete data on the queue. The goal is to estimate G from observations of the queue-length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. An upper bound on the maximal risk is derived. The problem of estimating the arrival rate is considered as well.

We consider a horizontal traffic queue (HTQ) on a periodic road segment, where vehicles arrive according to a spatio-temporal Poisson process, and depart after traveling a distance that is sampled independently and identically from a spatial distribution. When inside the queue, the motion of vehicles is governed by a car following model. We consider safe first and second order car following models. In the first order models, the speed of a vehicle is proportional to a power m > 0 of the distance to the vehicle in front. For the first order model, we show monotonicity of the service rate in between arrivals and departures. We extend the busy period calculations for M/G/1 queue to our setting, including for non-empty initial conditions in order to derive a probabilistic lower bound on the throughput for the m > 1 case. For the m < 1 case, we study throughput under a release control policy, where the additional expected waiting time caused by the release control policy is interpreted as the magnitude of the perturbation to the arrival process. We derive a lower bound on throughput for a given combination of maximum allowable perturbation and m < 1. For the second order model, we design a safe release control policy which guarantees a provable lower bound on the throughput. Illustrative simulation results are also presented.

Recently, Ammar [1] has discussed the transient behavior of a multiple vacations queue with impatient customers. In this paper, a similar technique is used to derive a new elegant explicit solution for an M/M/1 vacation queue with impatient customers and a waiting server, where the server is allowed to take a vacation whenever the system is empty after waiting for a random period of time. If the server does not return from the vacation before the expiry of the customer impatience time, the customer abandons the system forever. Moreover, the formulas of mean and variance expressed in terms of the obtained possibilities for this model.

We study the age of information, which is a recently introduced metric for measuring the freshness of a continually updated piece of information as observed at a remote monitor. The age of information metric has been studied for a variety of different queuing systems. In this work, we introduce a packet deadline as a control mechanism and study its impact on the average age of information for an M/M/1/2 queuing system. We analyze the system for a fixed deadline and derive a mathematical expression for the average age. We numerically evaluate the expression and show the relationship of the age performance to that of the M/M/1/1 and M/M/1/2 systems. We show that the system with a deadline constraint can outperform both the M/M/1/1 and M/M/1/2 without such a deadline.

Nowadays, Local Area Networks (LAN) are one of the most popular networks, and the LAN performance is very important for operators. The LAN method has been applied as an essential infrastructure of numerous companies and organizations for a long time. This study aims to evaluate the M/M/1 queuing model in LAN Based on Poisson and Exponential Distribution and compare the traffic behavior of these Distributions in terms of some essential parameters. Moreover, it also aims to design and implement a model to perform the M/M/1 queuing model with different metrics and finally analyze the results to evaluate traffic behavior of M/M/1 queuing model for Poisson and Exponential Distribution in LAN.

We consider discouraged arrival of Markovian queueing systems whose service speed is regulated according to the number of customers in the system. We will reduce the congestion in two ways. First we attempt to reduce the congestion by discouraging the arrivals of customers from joining the queue. Secondly we reduce the congestion by introducing the concept of service switches. First we consider a model in which multiple servers have three service ratesμ1,μ2, andμ(μ1≤μ2<μ), say, slow, medium, and fast rates, respectively. If the number of customers in the system exceeds a particular pointK1orK2, the server switches to the medium or fast rate, respectively. For this adaptive queueing system the steady state probabilities are derived and some performance measures such as expected number in the system/queue and expected waiting time in the system/queue are obtained. Multiple server discouraged arrival model having one service switch and single server discouraged arrival model having one and two service switches are obtained as special cases. A Matlab program of the model is presented and numerical illustrations are given.

We study the effect of imperfect memory on decision making in the context of a stochastic sequential action-reward problem. An agent chooses a sequence of actions which generate discrete rewards at different rates. She is allowed to make new choices at rate $\beta$, while past rewards disappear from her memory at rate $\mu$. We focus on a family of decision rules where the agent makes a new choice by randomly selecting an action with a probability approximately proportional to the amount of past rewards associated with each action in her memory. We provide closed-form formulae for the agent's steady-state choice distribution in the regime where the memory span is large ($\mu \to 0$), and show that the agent's success critically depends on how quickly she updates her choices relative to the speed of memory decay. If $\beta \gg \mu$, the agent almost always chooses the best action, i.e., the one with the highest reward rate. Conversely, if $\beta \ll \mu$, the agent chooses an action with a probability roughly proportional to its reward rate., Comment: Forthcoming in Mathematics of Operations Research; An extended abstract appeared in the proceedings of ACM SIGMETRICS 2018

The paper deals with a multi-server controllable queueing system M(t)/M/K with time-dependent and, in particular, with periodic arrival rates. The models with homogeneous and heterogeneous servers are of interest. In latter case the fastest free server allocation mechanism is assumed and the preemption is allowed. The control problem consists in evaluation of the optimal number of servers during some specified stages and is solved by finite horizon dynamic programming approach. To calculate the transient solutions we use a forth-order Runge-Kutta method for the system with a truncated queue length. The results are compared with corresponding queues operating in a stationary regime. It is shown that the optimal control policies are also time dependent and periodic as arrival rates and heterogeneous systems are superior in performance comparing to the homogeneous ones.

The present investigation deals with the transient analysis of Markovian queue model with multi stage service. Here, current analysis deals with M/M/1 queueing model with finite capacity. The arrival of the customer in the system follows Poisson process and the customers are served following exponential distribution. The brokendown server is immediately send for repair process so as to become as good as earlier. Transient analysis of finite Markovian queueing model has been studied using Runge-Kutta method. Various performance measures like queue length of the system, waiting time, throughput, service state probabilities have also been obtained.

This paper deals with a single server M/G/1 queue with two phases of heterogeneous service and unreliable server. We assume that customers arrive to the system according to a Poisson process with rate λ. After completion of two successive phases of service the server either goes for a vacation with probability p(0 ⩽ p ⩽ 1) or may continue to serve the next unit, if any, with probability q(=1 − p). Otherwise it remains in the system until a customer arrives. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. For this model, we first derive the joint distribution of state of the server and queue size, which is one of the chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally we obtain some important performance measures and reliability indices of this model.

When the service time distribution has a finite exponential moment, we present a large deviations result for the busy period distribution in the M/G/1 queue without the assumption of Abate and Whitt (1997) and Kyprianou (1971).

In the case of M / M / 1 queue with constrained workload (finite dam), we consider two strategies for investing overflow of the dam. Expected value of the present value of each strategy is computed using restricted Laplace transforms and properties of M / M / 1 queues.

We consider the M/M/m retrial queue with nonpersistent customers. Liu et al. (2011) [12] provided the asymptotic lower and upper bounds for the stationary distribution of the number of customers in the orbit. In this paper we strengthen Liu, Wang and Zhao’s result by finding the exact tail asymptotic formula.

M/G/1 queue with server vacations period depending on the previous vacation and customer’s arrival is considered. Most existing studies on M/G/1 queue with server vacations assume that server vacations are independent of customers’ arrival. However, some vacations are terminated by some class o f customers’ arrival in certain queueing systems. In this paper, therefore, we investigate M/G/1 queue with server vacations where each vacation period has different distribution and vacation length is influenced by customers’ arrival. Laplace-Stieltjes transform of the waiting time distribution and the distribution of number of customers waiting for each class of customers are respectively derived. As performance measures, mean waiting time and average number of customers waiting for each class of customers are also derived.

This letter investigates a single-server discrete-time queuing system with single-slot service times. For a broad class of arrival processes, a closed-form expression for the mean queue content in steady state is obtained. Apart from being stationary-ergodic, the arrival process adheres to a regeneration property when there are no arrivals in a slot. Well-studied arrival processes such as autoregressive arrival processes and M / G / ∞ -input or train arrival processes adhere to this property.

This paper analyzes a multi-server queueing system with multiple vacations in which the vacation duration of each server is exponentially distributed. When all the servers are on vacation, customers are impatient if their waiting times exceed a constant value. To derive the stationary distribution of the system, we employ the matrix analytic method. Specifically, our queueing model is represented as an M / G / 1 -type Markov chain, and the stationary distribution is simply obtained by using its transition structures. Furthermore, we explicitly derive the tail decay rate of the stationary distribution, and demonstrate that the decay rate is not always equal to that for the queueing model without vacations.

A recent article on bulk-service queues gives a generating function for the steady-state probabilities of the embedded Markov chain for a single-server infinite-space system in which, the customers arrive according to a Poisson process and are served in batches with quorum a and capacity b, and the service time follows a general distribution dependent on batch size. This system is equivalent to the bulk queueing model M/Gj(a,b)/1, whose general solution requires finding the roots of the denominator of the underlying generating function. The article claims that the use of roots may result in numerical inaccuracies, especially for large values of b. Hence it only solves for the finite-space model M/Gj(a,b)/1/(B + b) using (B + 1) simultaneous linear equations. We present a simple way to obtain the probability distribution of queue length at post-departure epochs for the infinite-space model M/Gj(a,b)/1 using roots, then an alternative method to solve the finite-space queue M/Gj(a,b)/1/(B + b). We d...

This paper illustrates a computable matrix technique that can be used to derive explicit expressions for the transient state probabilities of a finite waiting space single-server queue, namely (M/M/1/N), having discouraged arrivals and reneging. The discipline is the classical first-come, first-served (FCFS). We obtain the transient solution of the system, with results in terms of the eigenvalues of a symmetric tridiagonal matrix. Finally, numerical calculations are given to illustrate the effectiveness of this technique and system behaviour.

The M/M/1 model is the most frequently used model in queueing problems. The research on the M/M/1 model with single queue and finite/infinite buffer length is already well developed. However, combinations of multiple queues with limited buffer capacity are rarely discussed. The reason for this deficiency is the lack of queue selection probability with the involved polling strategies. In this article, we propose to apply the multi-dimensional state diagram in analyzing the queueing behaviors of the q-M/M/1/Ki /FCFS/QSR model, where q is the number of queues, Ki the limited length of the ith queue, and QSR the dedicated queue selection rule. The considered QSRs can be either first-come-first-served polling or priority polling in this study. The state probabilities are solved by a set of linear equations, based on steady-state distributions. In order to verify the accuracy and performance of the proposed approach, we conducted several experiments to estimate the state probabilities, queue lengths, and waitin...

Abtsrcat This paper analyzes an M/M/2 queueing system with het- erogeneous servers, one of which is always available but the other goes on vacation in the absence of customers waiting for service. The system is analyzed in the steady state using matrix geometric method. Busy period of the system is analyzed and mean waiting time in the stationary regime com- puted. Conditional stochastic decomposition of stationary queue length is obtained. An illustrative example is also provided.

We examine an M=M=1 retrial queue with an unreliable server whose arrival, service, failure, repair, and retrial rates are all modulated by an exogenous random environment. Provided are conditions for stability, the (approximate) orbit size distribution, and mean queueing performance measures which are obtained via matrix-analytic methods. Additionally, we consider the problem of choosing arrival and service rates for each environment state with the objective of minimizing the steady state mean time spent in orbit by an arbitrary customer, subject to cost and revenue constraints. Two numerical examples illustrate the main results.

Present paper deals a M/M/1:(∞; GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy. Keeping in view the general trend of interdependent arrival and service processes, it is presumed that random variables of arrival and service processes follow a bivariate poisson distribution and the server provides his services under general discipline of service rule in an infinitely large waiting space. In this paper, our central attention is to explore the probability generating functions using Rouche’s theorem in both cases of slower and faster arrival rates of the queueing model taken into consideration; which may be helpful for mathematicians and researchers for establishing significant performance measures of the model. Moreover, for the purpose of high-lighting the application aspect of our investigated result, very recently Maurya [1] has derived successfully the expected busy periods of the server in both cases of slower and faster arrival rates, which have also been presented by the end of this paper.

In this article an M/G/1 queueing model with single server, Poisson input, k-phases of heterogeneous services and Bernoulli feedback design has been considered. For this model, we derive the steady-state probability generating function (PGF) of queue size at the random epoch and at the service completion epoch. Then, we derive the Laplace-Stieltjes Transform (LST) of the distribution of response time, the means of response time, number of customers in the system and busy period.

We consider a variant of M/M/1 where customers arrive singly or in pairs. Each single and one member of each pair is called primary; the other member of each pair is called secondary. Each primary joins the queue upon arrival. Each secondary is delayed in a separate area, and joins the queue when “pushed” by the next arriving primary. Thus each secondary joins the queue followed immediately by the next primary. This arrival/delay mechanism appears to be new in queueing theory. Our goal is to obtain the steady-state probability density function (pdf) of the workload, and related quantities of interest. We utilize a typical sample path of the workload process as a physical guide, and simple level crossing theorems, to derive model equations for the steady-state pdf. A potential application is to the processing of electronic signals with error free components and components that require later confirmation before joining the queue. The confirmation is the arrival of the next signal.

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.

This articles focuses on M t /M t /N queue with catastrophes and obtains stability bounds for the main characteristics of the respective queue-length process.

Estimation of waiting time distribution in the form of its right tail area, known as exceedance probability is carried out in an M/M/1/FCFS/ ∞ / ∞ queue with the help of data consists of non zero waiting time of randomly chosen customer from n independent queues and number of queues with zero waiting time. In the first section, MLE of rate parameters and exceedance probability are obtained and their large sample behaviour is studied. Next section, Bayes estimators of the same parametric function along with the minimum Bayes risks have been derived under squared error loss using McKay’s Bivariate Gamma as prior and also independent Gamma as prior with the prior for arrival rate being truncated between 0 and μ. Some numerical calculations are shown in the last section.