Queues with hysteretic control by vacation and post-vacation periods

The paper investigates the queueing process in stochastic systems with bulk input, batch state dependent service, server vacations, and three post-vacation disciplines. The policy of leaving and entering busy periods is hysteretic, meaning that, initially, the server leaves the system on multiple vacation trips whenever the queue falls below r (⩾1), and resumes service when during his absence the system replenishes to N or more customers upon one of his returns. During his vacation trips, the server can be called off on emergency, limiting his trips by a specified random variable (thereby encompassing several classes of vacation queues, such as ones with multiple and single vacations). If by then the queue has not reached another fixed threshold M (⩽ N), the server enters a so-called “post-vacation period” characterized by three different disciplines: waiting, or leaving on multiple vacation trips with or without emergency. For all three disciplines, the probability generating functions of the discrete and continuous time parameter queueing processes in the steady state are obtained in a closed analytic form. The author uses a semi-regenerative approach and enhances fluctuation techniques (from his previous studies) preceding the analysis of queueing systems. Various examples demonstrate and discuss the results obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fluctuation Analysis in Parallel Queues with Hysteretic Control

Methodology and Computing in Applied Probability - We study an enhanced hysteretic control system, with primary and secondary queues and random batch service. When the primary queue down-crosses r,...     

Fluctuation analysis in queues with several operational modes and priority customers

We analyze a complex queueing system with a single server operating in three different modes and dependent on circumstances, servicing two different queues simultaneously. There are different switching policies that specify when the server takes one or two queues. Main techniques are based on fluctuation analysis. One of the objectives is to model processes that occur in software, computer, and electrical engineering, and to argue that methods of fluctuation theory produce closed form functionals.     

A Versatile Stochastic Maintenance Model with Reserve and Super-Reserve Machines

This article analyzes the maintenance of production system with unreliable machines. This system includes a repair facility and three types of unreliable machines: the main facility of working and reserve machines, and an auxiliary facility of super-reserve machines. Operating times of working machines are exponentially distributed. Upon failure, a working machine is immediately replaced by reserve machines available. Defective machines line up for repair, whose duration is arbitrarily distributed. Refurbished machines return to the main facility. If the main facility is restored to its original quantity (i.e., all machines are intact), the repair facility leaves on routine maintenance; all w+1 reserve machines are temporarily blocked and renewals come from the super-reserve group until the latter becomes exhausted. Then, the busy period is regenerated. The techniques include two-variate Markov and semi-regenerative processes, and a duality principle, to find the probability distribution of the number of intact machines. Explicit formulas obtained demonstrate a relatively effortless use of functionals of the main stochastic characteristics (such as expenses due to repair, maintenance, waiting, and rewards for higher reliability) and optimization of their objective function. Applications of such models include computer networking, human resources, and manufacturing processes.     

On Exit Times of a Multivariate Random Walk and Its Embedding in a Quasi Poisson Process

Abstract

We introduce and analyze a delayed renewal process  = {τ₀,τ₁,…} marked by a multivariate random walk (,) and its behavior about fixed levels to be crossed by one of the components of (,). We derive the joint distribution of first passage time τ_ρ, pre-exit time τ_ρ?1 (i.e., the instant one phase prior to the first passage time), and the respective values of (,) at τ_ρ and τ_ρ?1 in a closed form. The results obtained are then applied to a multivariate quasi Poisson process Π, forming a random walk ((Π),) embedded in Π over . Processes like these can model various phenomena including stock market and option trading.

One of the central issues in the investigation of ((Π),) is to obtain the information about Π at any moment of time in random vicinities of τ_ρ and τ_ρ?1 previously available only upon . The results offer, again, closed form functionals. Numerous examples throughout the paper illustrate introduced constructions and connect the results with real-world applications, most prominently the stock market.     

On single-server closed queues with priorities and state dependent parameters

A wide class of closed single-channel queues is considered. The more general model involvesm +w + 1 “permanent” customers that occasionally require service. Them customers are of the first priority and the rest are of the second priority. The input rate and service of customers depend upon the total number of customers waiting for service. Such a system can also be described in terms of servicing machines processes with reserve replacement and multi-channel queues with finite waiting room. Two dual models, with and without idle periods, are treated. An explicit relation between the servicing processes of both models is derived. The semi-regenerative techniques originally developed in the author's earlier work [4] are extended and used to derive the probability distribution of the processes in equilibrium. Applications and examples are discussed. This paper is a part of work supported by the National Science Foundation under Grant No. DMS-8706186.     

On multivariate antagonistic marked point processes

This paper investigates two bivariate “antagonistic” processes in the framework of noncooperative games that find applications in economics, engineering, and warfare. These processes represent two players with entirely opposite interests. Hostile actions, associating with one of the components (such as deliberate sellout of stocks of the opponent), take place at random times having a random effect to various vital areas. A consequence of these actions is material damage (such as financial losses) which is associated with the second component. So, in general, each player has two (dependent) ways of striking, while his opponent can sustain multiple damages (of two kinds and interdependent) until his threshold of endurance is reached. Then, the player is ruined. One of the goals is to predict the ruin time of each player as well as the collateral damage to the loser and winner at the ruin time. Furthermore, it is also of vital interest to relate these random elements to the continuous time parameter processes referred to as time sensitive analysis. We succeed in these goals by arriving at closed form joint functionals of the named elements and processes.     

Random Walk Analysis in Antagonistic Stochastic Games

Abstract

This article deals with two “antagonistic random processes” that are intended to model classes of completely noncooperative games occurring in economics, engineering, natural sciences, and warfare. In terms of game theory, these processes can represent two players with opposite interests. The actions of the players are manifested by a series of strikes of random magnitudes imposed onto the opposite side and rendered at random times. Each of the assaults is aimed to inflict damage to vital areas. In contrast with some strictly antagonistic games where a game ends with one single successful hit, in the current setting, each side (player) can endure multiple strikes before perishing. Each player has a fixed cumulative threshold of tolerance which represents how much damage he can endure before succumbing. Each player will try to defeat the adversary at his earliest opportunity, and the time when one of them collapses is referred to as the “ruin time”. We predict the ruin time of each player, and the cumulative status of all related components for each player at ruin time. The actions of each player are formalized by a marked point process representing (an economic) status of each opponent at any given moment of time. Their marks are assumed to be weakly monotone, which means that each opposite side accumulates damages, but does not have the ability to recover. We render a time-sensitive analysis of a bivariate continuous time parameter process representing the status of each player at any given time and at the ruin time and obtain explicit formulas for related functionals.     

Bulk input queues with hysteretic control

This paper deals with a bulk input-batch service queueing system and (r,N)-hysteretic control, i.e., under N-policy and r-quorum discipline. The model also includes state dependent service. The goal of such global control is to reduce the waste of server capacity (r-quorum), an unwanted number of switchovers between idle and busy modes (N-policy), and the queue length (by means of variable service rates). The analysis of the system is based on first excess level technique developed by the second author. This approach enables the authors to obtain major characteristics for the queueing process in a closed analytical form. This revised version was published online in June 2006 with corrections to the Cover Date.