On the increasing convex order of generalized aggregation of dependent random variables

In this article, we study stochastic properties of the generalized sum of right tail weakly stochastic arrangement increasing (RWSAI) nonnegative random variables accompanied with stochastic arrangement increasing (SAI) Bernoulli variables. In terms of monotonicity, supermodularity/submodularity, and convexity of the bivariate kernel function, sufficient conditions are developed for the increasing convex ordering on the generalized aggregation. Applications in actuarial science including the individual risk model and the reserving capital allocation are presented to highlight our results.     

Tail asymptotics for the queue length in an M/G/1 retrial queue

In this paper, we study the tail behavior of the stationary queue length of an M/G/1 retrial queue. We show that the subexponential tail of the stationary queue length of an M/G/1 retrial queue is determined by that of the corresponding M/G/1 queue, and hence the stationary queue length in an M/G/1 retrial queue is subexponential if the stationary queue length in the corresponding M/G/1 queue is subexponential. Our results for subexponential tails also apply to regularly varying tails, and we provide the regularly varying tail asymptotics for the stationary queue length of the M/G/1 retrial queue. AMS subject classifications: 60J25, 60K25     

Moment inequalities for the discrete-time bulk service queue

For the discrete-time bulk service queueing model, the mean and variance of the steady-state queue length can be expressed in terms of moments of the arrival distribution and series of the zeros of a characteristic equation. In this paper we investigate the behaviour of these series. In particular, we derive bounds on the series, from which bounds on the mean and variance of the queue length follow. We pay considerable attention to the case in which the arrivals follow a Poisson distribution. For this case, additional properties of the series are proved leading to even sharper bounds. The Poisson case serves as a pilot study for a broader range of distributions.     

A unified algorithm for computing the stationary queue length distributions inM(k)/G/1/N and GI/M(k)/1/N queues

An iterative algorithm is developed for computing numerically the stationary queue length distributions in M/G/1/N queues with arbitrary state-dependent arrivals, or simply M(k)/G/1/N queues. The only input requirement is the Laplace-Stieltjes transform of the service time distribution.In addition, the algorithm can also be used to obtain the stationary queue length distributions in GI/M/1/N queues with state-dependent services, orGI/M(k)/1/N, after establishing a relationship between the stationary queue length distributions inGI/M(k)/1/N and M(k)/G/1/N+1 queues.Finally, we elaborate on some of the well studied special cases, such asM/G/1/N queues,M/G/1/N queues with distinct arrival rates (which includes the machine interference problems), andGI/M/C/N queues. The discussions lead to a simplified algorithm for each of the three cases.     

A discrete-time priority queue with two-class customers and bulk services

This paper studies the behavior of a discrete queueing system which accepts synchronized arrivals and provides synchronized services. The number of arrivals occurring at an arriving point may follow any arbitrary discrete distribution possessing finite first moment and convergent probability generating function in ¦ z ¦ 1 + with > 0. The system is equipped with an infinite buffer and one or more servers operating in synchronous mode. Service discipline may or may not be prioritized. Results such as the probability generating function of queue occupancy, average queue length, system throughput, and delay are derived in this paper. The validity of the results is also verified by computer simulations.The work reported in this paper was supported by the National Science Council of the Republic of China under Grant NSC1981-0404-E002-04.     

Multi-server retrial queue with negative customers and disasters

We consider a multi-server retrial queue with waiting places in service area and four types of arrivals, positive customers, disasters and two types of negative customers, one for deleting customers in orbit and the other for deleting customers in service area. The four types of arrivals occur according to a Markovian arrival process with marked transitions (MMAP) which may induce the dependence among the arrival processes of the four types. We derive a necessary and sufficient condition for the system to be positive recurrent by comparing sample paths of auxiliary systems whose stability conditions can be obtained. We use a generalized truncated system that is obtained by modifying the retrial rates for an approximation of stationary queue length distribution and show the convergence of approximation to the original model. An algorithmic solution for the stationary queue length distribution and some numerical results are presented.      

A complete and simple solution for a discrete-time multi-server queue with bulk arrivals and deterministic service times

A complete distribution for the system content of a discrete-time multi-server queue with an infinite buffer is presented, where each customer arriving in a group requires a deterministic service time that could be greater than one slot. In addition, when the service time equals one slot, a complete distribution for the delay is also presented.     

Testing for increasing convex order in several populations

Increasing convex order is one of important stochastic orderings. It is very often used in queueing theory, reliability, operations research and economics. This paper is devoted to studying the likelihood ratio test for increasing convex order in several populations against an unrestricted alternative. We derive the null asympotic distribution of the likelihood ratio test statistic, which is precisely the chi-bar-squared distribution. The methodology for computing critical values for the test is also discussed. The test is applied to an example involving data for survival time for carcinoma of the oropharynx.     

A GI/Geo/1 queue with negative and positive customers

The arrival of a negative customer to a queueing system causes one positive customer to be removed if any is present. Continuous-time queues with negative and positive customers have been thoroughly investigated over the last two decades. On the other hand, a discrete-time Geo/Geo/1 queue with negative and positive customers appeared only recently in the literature. We extend this Geo/Geo/1 queue to a corresponding GI/Geo/1 queue. We present both the stationary queue length distribution and the sojourn time distribution.     

Stochastic comparisons for bulk queues

We consider two important classes of single-server bulk queueing models: M^(X)/G^(Y)/1 with Poisson arrivals of customer groups, and G^(X)/m^(Y)1 with batch service times having exponential density. In each class we compare two systems and prove that one is more congested than the other if their basic random variables are stochastically ordered in an appropriate manner. However, it must be recognized that a system that appears congested to customers might be working efficiently from the system manager's point of view. We apply the results of this comparison to (i) the family {M/G^(s)/1,s 1} of systems with Poisson input of customers and batch service times with varying service capacity; (ii) the family {G^(s)/1,s 1} of systems with exponential customer service time density and group arrivals with varying group size; and (iii) the family {M/D/s,s 1} of systems with Poisson arrivals, constant service time and varying number of servers. Within each family, we find the system that is the best for customers, but this turns out to be the worst for the manager (or vice versa). We also establish upper (or lower) bounds for the expected queue length in steady state and the expected number of batches (or groups) served during a busy period. The approach of the paper is based on the stochastic comparison of random walks underlying the models.This research was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.     

From regulatory life tables to stochastic mortality projections: The exponential decline model

Often in actuarial practice, mortality projections are obtained by letting age-specific death rates decline exponentially at their own rate. Many life tables used for annuity pricing are built in this way. The present paper adopts this point of view and proposes a simple and powerful mortality projection model in line with this elementary approach, based on the recently studied mortality improvement rates. Two main applications are considered. First, as most reference life tables produced by regulators are deterministic by nature, they can be made stochastic by superposing random departures from the assumed age-specific trend, with a volatility calibrated on market or portfolio data. This allows the actuary to account for the systematic longevity risk in solvency calculations. Second, the model can be fitted to historical data and used to produce longevity forecasts. A number of conservative and tractable approximations are derived to provide the actuary with reasonably accurate approximations for various relevant quantities, available at limited computational cost. Besides applications to stochastic mortality projection models, we also derive useful properties involving supermodular, directionally convex and stop-loss orders.     

Ordering Gini indexes of multivariate elliptical risks

Gini index is a well-known tool in economics that is often used for measuring income inequality. In insurance, the index and its modifications have been used to compare the riskiness of portfolios, to order reinsurance contracts, and to summarize insurance scores (relativities). In this paper, we establish several stochastic orders between the Gini indexes of multivariate elliptical risks with the same marginals but different dependence structures. This work is motivated by the applied studies of Brazauskas et al. (2007) and Samanthi et al. (2015), who employed the Gini index to compare the riskiness of insurance portfolios. Based on extensive Monte Carlo simulations, these authors have found that the power function of the associated hypothesis test increases as portfolios become more positively correlated. The comparison of the Gini indexes (of empirically estimated risk measures) presented in this paper provides a theoretical explanation to this statistical phenomenon. Moreover, it enriches the studies of the problem of central concentration of elliptical distributions and generalizes the pd-1 order proposed by Shaked and Tong (1985).     

An M/G/1-type queuing model with service times depending on queue length

A study is made of an M/G/1-type queuing model in which customers receive one type of service until such time as, at the end of a service, the queue size is found to exceed a given value N, N ≥ 1. Then a second type of service is put into effect and remains in use until the queue size is reduced to a fixed value K, 0 ≤ K ≤ N. Equations are derived for the stationary probabilities both at departure times and at general times. An algorithm is developed that allows the rapid computation of the mean queue length and some important probabilities.     

On stationary queue length distributions for G/M/s/r queues

Consider aG/M/s/r queue, where the sequence{A _n } _n=– of nonnegative interarrival times is stationary and ergodic, and the service timesS _n are i.i.d. exponentially distributed. (SinceA _n =0 is possible for somen, batch arrivals are included.) In caser < , a uniquely determined stationary process of the number of customers in the system is constructed. This extends corresponding results by Loynes [12] and Brandt [4] forr= (with=ES₀/EA₀<s) and Franken et al. [9], Borovkov [2] forr=0 ors=. Furthermore, we give a proof of the relation min(i, s)¯p(i)=p(i–1), 1ir + s, between the time- and arrival-stationary probabilities¯p(i) andp(i), respectively. This extends earlier results of Franken [7], Franken et al. [9].     

Efficient approximations for numbers of survivors in the Lee–Carter model

In portfolios of life annuity contracts, the payments made by an annuity provider (an insurance company or a pension fund) are driven by the random number of survivors. This paper aims to provide accurate approximations for the present value of the payments made by the annuity provider. These approximations account not only for systematic longevity risk but also for the diversifiable fluctuations around the unknown life table. They provide the practitioner with a useful tool avoiding the problem of simulations within simulations in, for instance, Solvency 2 calculations, valid whatever the size of the portfolio.     

Relationships between characteristics in periodic Poisson queues

We study single server periodic queues in the day equilibrium conditions. The following characteristics of interest are considered at time of dayt: V^p(t)-the work load, L^p(t)-the number of customers and u^p(t)-the departure rate. We give relationships between E[V^p(t)], E[L^p(t)] and u^p(t). We also prove that E[V^p(t)] < and E[L^p(t)] < provided the second moment of the service time is finite.     

Prediction of average queue length in manufacturing resources at the time of new lot arrival

This paper presents ways to predict the average queue length, at a manufacturing resource with a constant processing rate, at the time of a new lot arrival. The jobs arrive in constant lot sizes, and the inter arrival time of the lots follows an exponential distribution. Analytical expressions for the queue statistics are developed. Simulation results are provided and are compared to the theoretical predictions. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of the discrete-time bulk-service queue Geo/G/1/N+B

This paper considers a discrete-time bulk-service queueing system with variable capacity, finite waiting space and independent Bernoulli arrival process: Geo/G^Y/1/N+B. Both the analytic and computational aspects of the distributions of the number of customers in the queue at post-departure, random and pre-arrival epochs are discussed.     

Two-stage stochastic optimization problems with stochastic ordering constraints on the recourse

We consider two-stage risk-averse stochastic optimization problems with a stochastic ordering constraint on the recourse function. Two new characterizations of the increasing convex order relation are provided. They are based on conditional expectations and on integrated quantile functions: a counterpart of the Lorenz function. We propose two decomposition methods to solve the problems and prove their convergence. Our methods exploit the decomposition structure of the risk-neutral two-stage problems and construct successive approximations of the stochastic ordering constraints. Numerical results confirm the efficiency of the methods.