Poisson hail on a hot ground

Using a known fact that a Galton–Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.

Metamodeling and multi-objective optimization for buffer allocation in multi-product serial production lines

This paper presents a flow approximation of acyclic queueing networks with multiple classes of jobs, unreliable servers, and finite buffers. Different jobs compete for service before a server, or they may combine before service to yield a composite job. Up times and down times of servers are random and service times have gamma distributions or may be constant quantities. Random service times are approximated by piecewise deterministic variables. The model is a combination of discrete event simulation and analysis. The simulator keeps track of the queue and server states. Between two consecutive state transitions the flow rates remain constant and the model calculates the cumulative flow of each job. When a transition takes place the model modifies the flow rates of the affected servers and schedules new events. Compared to a conventional simulator, the model is quite fast and accurate.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This article presents a perishable stochastic inventory system under continuous review at a service facility in which the waiting hall for customers is of finite size M. The service starts only when the customer level reaches N (< M), once the server has become idle for want of customers. The maximum storage capacity is fixed as S. It is assumed that demand for the commodity is of unit size. The arrivals of customers to the service station form a Poisson process with parameter λ. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The items of inventory have exponential life times. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The demands that occur during stock out periods are lost.The joint probability distribution of the number of customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived. The results are illustrated with numerical examples.

A multidimensional version of a result of Sakhanenko for the Gaussian approximation of sequences of successive sums of independent nonidentically distributed random vectors with finite exponential moments is obtained.

The time-constrained traveling salesman problem (TCTSP) is a variant of the classical traveling salesman problem, where only a subset of the customers can be visited due to the time limit constraint. In this paper, we consider the TCTSP with stochastic travel and service times. Given the normal working hours T and a tolerance time ΔT, the total travel and service times of a route can exceed T as long as it is within T+ΔT, though a penalty proportional to the amount in excess of T will be imposed. The problem consists of optimally selecting and sequencing a subset of customers to visit in the presence of random travel and service times to maximize the expected profit while satisfying the time limit constraint. We formulate the problem as a two-stage stochastic program with recourse, and propose an integer L-shaped solution method for solving it. Computational results show that the algorithm can solve problems with moderate size to optimality within reasonable time.

Virtualization technologies are being actively used to design infrastructure of cloud computing systems. In this case applications can be duplicated and hosted in different virtual machines on different physical nodes. That defines various performance of applications which causes the problem of managing performance of the entire heterogeneous system. There are different ways of solving this problem, including queuing theory methods. However research of the threshold discipline in scope of queuing theory is not complete because of difficulty of gathering precise analytic values and building of precise mathematic model of the system. Another feature of heterogeneous systems is the finite random time of system functioning which is defined by random endogenous and exogenous factors. This paper gives an overview on a functional model of the system with two heterogeneous devices with random functioning time and different service disciplines. In scope of simulation statistic experiments for different service disciplines at random time interval an average time needed to process a single request is measured. A comparison of service disciplines is conducted. Authors also provide a working software implementation of the heterogeneous system and experiments with use of service disciplines is performed.

We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

We consider the GI/G/1 queue where customers are served in random order and the service time distribution has a finite exponential moment. We derive the large deviations result for the waiting time distribution by showing that the asymptotic decay rate of the waiting time distribution is the same as that of the busy period distribution.

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.

We study the first eigenfunction φ 1 of the Dirichlet Laplacian on a convex domain in Euclidean space. Elementary properties of Bessel functions yield that $$\left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2 \to \infty$$ if D is a sector in Euclidean plane with area 1 and the angle tends to 0. We aim to characterize those domains D such that $$(vol(D))^{1/2} \left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2$$ is large in terms of the ratio of the first eigenvalue of D and the infimum of the first eigenvalues of all subdomains D≈ of D with given volume.

We obtain dimension-free concentration inequalities for $\ell^p$-norms, $p\geq2$, of infinitely divisible random vectors with independent coordinates and finite exponential moments. Besides such norms, the methods and results extend to some other classes of Lipschitz functions.

This paper presents sharp inequalities for deviation probability of a general quadratic form of a random vector ξ with finite exponential moments. The obtained deviation bounds are similar to the case of a Gaussian random vector. The results are stated under general conditions and do not suppose any special structure of the vector ξ. The obtained bounds are exact (non-asymptotic), all constants are explicit and the leading terms in the bounds are sharp.