Probability Qi Research Articles

Optimal pure strategies for a discrete search game

Consider a two-person zero-sum search game between a Hider and a Searcher. The Hider chooses to hide in one of n discrete locations (or “boxes”) and the Searcher chooses a search sequence specifying which order to look in these boxes until finding the Hider. A search at box i takes ti time units and finds the Hider—if hidden there—independently with probability qi, for i=1,…,n. The Searcher wants to minimize the expected total time needed to find the Hider, while the Hider wants to maximize it. It is shown in the literature that the Searcher has an optimal search strategy that mixes up to n distinct search sequences with appropriate probabilities. This paper investigates the existence of optimal pure strategies for the Searcher—a single deterministic search sequence that achieves the optimal expected total search time regardless of where the Hider hides. We identify several cases in which the Searcher has an optimal pure strategy, and several cases in which such optimal pure strategy does not exist. An optimal pure search strategy has significant practical value because the Searcher does not need to randomize their actions and will avoid second guessing themselves if the chosen search sequence from an optimal mixed strategy does not turn out well.

Fluid M/M/1 catastrophic queue in a random environment

Our main objective in this study is to investigate the stationary behavior of a fluid catastrophic queue of M/M/1 in a random multi-phase environment. Occasionally, a queueing system experiences a catastrophic failure causing a loss of all current jobs. The system then goes into a process of repair. As soon as the system is repaired, it moves with probability qi ≥ 0 to phase i. In this study, the distribution of the buffer content is determined using the probability generating function. In addition, some numerical results are provided to illustrate the effect of various parameters on the distribution of the buffer content.

Analysis of M<SUP align="right">[X]</SUP>/G<SUB align="right">k/1 retrial queue with Bernoulli-2 vacation, search for customers and hot stand-by

A batch arrival retrial queueing model with k optional stages of service, extended Bernoulli vacation and stand-by is studied. After completion of the ith stage of service, the customer may have the option to choose (i + 1)th stage of service with probability θi or may leave the system with probability qi = 1 − θi, I = 1, 2, …, k − 1 and qk = 1. After service completion of each customer, the server may take a vacation with probability v1 and extend a vacation with probability v2 or rejoin the system after the first vacation with probability 1 − v2. Busy server may get to breakdown and the stand-by server provides service only during the repair times. At the completion of service, vacation or repair, the server searches for the customer in the orbit (if any) with probability α or remains idle with probability 1 − α. By using the supplementary variable method, steady state probability generating function for system size, some system performance measures like Lq, Ls, Wq and Ws are discussed. Simulation results are given using MATLAB.

Analysis of M<SUP align="right">[X]</SUP>/G<SUB align="right">k/1 retrial queue with Bernoulli-2 vacation, search for customers and hot stand-by

A batch arrival retrial queueing model with k optional stages of service, extended Bernoulli vacation and stand-by is studied. After completion of the ith stage of service, the customer may have the option to choose (i + 1)th stage of service with probability θi or may leave the system with probability qi = 1 − θi, I = 1, 2, …, k − 1 and qk = 1. After service completion of each customer, the server may take a vacation with probability v1 and extend a vacation with probability v2 or rejoin the system after the first vacation with probability 1 − v2. Busy server may get to breakdown and the stand-by server provides service only during the repair times. At the completion of service, vacation or repair, the server searches for the customer in the orbit (if any) with probability α or remains idle with probability 1 − α. By using the supplementary variable method, steady state probability generating function for system size, some system performance measures like Lq, Ls, Wq and Ws are discussed. Simulation results are given using MATLAB.

Analysis of a Discrete-Time Geo/G/1 Queue in a Multi-Phase Service Environment with Disasters

Abstract This paper considers a discrete-time Geo/G/1 queue in a multi-phase service environment, where the system is subject to disastrous breakdowns, causing all present customers to leave the system simultaneously. At a failure epoch, the server abandons the service and the system undergoes a repair period. After the system is repaired, it jumps to operative phase i with probability qi, i = 1, 2 ⋯, n. Using the supplementary variable technique, we obtain the distribution for the stationary queue length at the arbitrary epoch, which are then used for the computation of other performance measures. In addition, we derive the expected length of a cycle time, the generating function of the sojourn time of an arbitrary customer, and the generating function of the server’s working time in a cycle. We also give the relationship between the discrete-time queueing system to its continuous-time counterpart. Finally, some examples and numerical results are presented.

On an M/G/1 queue in random environment with Min(N, V) policy

In this paper, we analyze an M∕G∕1 queue operating in multi-phase random environment with Min(N, V) vacation policy. In operative phase i, i = 1, 2, …, n, customers are served according to the discipline of First Come First Served (FCFS). When the system becomes empty, the server takes a vacation under the Min(N, V) policy, causing the system to move to vacation phase 0. At the end of a vacation, if the server finds no customer waiting, another vacation begins. Otherwise, the system jumps from the phase 0 to some operative phase i with probability qi, i = 1, 2, …, n. And whenever the number of the waiting customers in the system reaches N, the server interrupts its vacation immediately and the system jumps from the phase 0 to some operative phase i with probability qi, i = 1, 2, …, n, too. Using the method of supplementary variable, we derive the distribution for the stationary system size at arbitrary epoch. We also obtain mean system size, the results of the cycle analysis and the sojourn time distribution. In addition, some special cases and numerical examples are presented.

An unreliable M[X]/G/1 retrial Queue with multi optional stages of service and delay in repair

Unreliable vacation retrial queue and multi stages of service delay in repair is studied. After completion of the ith(i=1,2,…k) stage of service, the unit may have the option to choose (i+1)th stage of service with probability θi, or with pi may join into orbit to give feedback or may leave the station with probability qi = 1 – pi – θi, (i = 1,2,…k –1) and qi = 1 – pi, (i = k). After service completion if the orbit has no units, server takes avacation. During repair, the unit waiting in the system to complete the remaining service (delay time) is discussed. We analyzed the system using the method of supplementary variable. Simulation results are given using MATLAB.

Multi stage unreliable retrial Queueing system with Bernoulli vacation

In this work we considered the Bernoulli vacation in group arrival retrial queues with unreliable server. Here, a server providing service in k stages. Any arriving group of units finds the server free, one from the group entering the first stage of service and the rest are joining into the orbit. After completion of the ith, (i=1,2,…k) stage of service, the customer may go to (i+1)th stage with probability θi, or leave the system with probability qi = 1 – θi, (i = 1,2,…k – 1) and qi = 1, (i = k). The server may enjoy vacation (orbit is empty or not) with probability v after finishing the service or continuing the service with probability 1-v. After finishing the vacation, the server search for the customer in the orbit with probability θ or remains idle for new arrival with probability 1-θ. We analyzed the system using the method of supplementary variable.

On the GI/M/1 Queue with Vacations and Multiple Service Phases

This paper considers a GI/M/1 queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase 0. If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase 0 to some service phase i with probability qi, i=1,2,…,N. Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the type-i cycle, and mean number of customers in the system at the end of phase 0. Finally, some numerical examples are presented.

Analysis of the M/G/1 queue in multi-phase random environment with disasters

This paper studies an M/G/1 queue in a multi-phase random environment. When in operative phase i, i=1,2,…,n, the system is subject to disastrous interruptions, causing all present customers (waiting and served) to leave the system. At an exponential failure instant, the server abandons the service and the system goes directly to repair phase. After an exponential repair time, the system moves to operative phase i with probability qi, i=1,2,…,n. Using the supplementary variable technique, we obtain the distribution for the stationary queue at an arbitrary epoch. We also derive results of the cycle analysis, the sojourn time distribution and the length of the server's working time in a service cycle. In addition, some special cases and numerical examples are presented.

AN M/M/1 QUEUE IN RANDOM ENVIRONMENT WITH DISASTERS

We study a M/M/1 queue in a multi-phase random environment, where the system occasionally suffers a disastrous failure, causing all present jobs to be lost. The system then moves to a repair phase. As soon as the system is repaired, it moves to phase i with probability qi ≥ 0. We use two methods of analysis to study the probabilistic behavior of the system in steady state: (i) via probability generating functions, and (ii) via matrix geometric approach. Due to the special structure of the Markov process describing the disaster model, both methods lead to explicit results, which are related to each other. We derive various performance measures such as mean queue sizes, mean waiting times, and fraction of lost customers. Two special cases are further discussed.

Games of chance with multiple objectives

We consider games of chance between two players: Player M can win only by amassing point totals in several categories before player N scores a prescribed total number n of points. Let M have k objectives, with mi points required in category i and probability qi of scoring a point in that category. We resolve certain special cases:

Bold Play Is Best: A Simple Proof

Introduction In the classical ruin problem, a gambler with initial capital i dollars plays against an adversary with initial capital a i dollars. Here i and a are positive integers, i < a. The gambler wins a dollar with probability p and loses a dollar with probability q = 1 p; the game is repeated until one of the players goes broke. (Part of the analysis of the problem shows the game cannot go on forever.) Another interpretation of this model is that of a gambler playing a game in a casino (the adversary) where the gambler plays until she goes broke or until she wins a fixed predetermined amount, at which time she quits. Given a total fixed capital a, we are interested in the probability qi that the gambler starting with initial capital i, 1 < i < a 1, is ruined. This problem is often discussed in a first course in probability and introduces the student to the ideas of random walks and Markov chains. Our main reference is the classic book of Feller [3, chapter 14]; see also [1], [4], and [5]. We will regard the sequence of the gambler's fortune after each play as a random walk in the initerval [0, a], with absorbing barriers at 0 and a. The probability of ruin is the probability of hitting 0 before hitting a. Using a difference equation approach and some algebra, the following solution is derived in the case p = 1/2 (see, e.g., [3]):

Individual Testing of Independent Items in Optimal Group Testing

We consider the group testing problem for a set of independent items I = [I1,… In] where Ii, has probability pi, of being defective and probability qi = 1 – pi of being good. The problem is to classify all items as good or defective with a minimum expected number of group tests where a group test is a test on a subset S of I with two possible outcomes: either S is pure (contains no defective) or S is contaminated (contains at least one defective, with no information provided about which or how many). No polynomial-time algorithm is known for the group testing problem even for the special case pi = p for all i. Hence, any method that reduces the size of the problem is very helpful. In this paper, we give such a method by providing a simple condition to screen items that should be tested (only) individually. This condition leads to a necessary and sufficient condition for the individual testing algorithm to be optimal, generalizing a result of Unger [1] for the special case of identical pi.

A note on the random walk model arising in double diffusion

AbstractThe discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.

Collisional transition probability and non-equilibrium unimolecular rate

The rate constant for a unimolecular reaction at the low-pressure limit under non-equilibrium conditions (k0†) is a function of the collisional transition probability qi→j. For polyatomic molecules, the form of qi→j can only be surmised at present. We have used six models of qi→j, representing energy transfer that can be termed efficient (bimodal gaussian and bimodal gamma transition probability models), inefficient (exponential and Schwartz-Slawsky-Herzfeld models) and intermediate (unimodal gaussian and unimodal gamma models), and have computed k0†/k0 and Ea0=Ea0†−Ea0, where k0 is the equilibrium low-pressure rate constant, and †Ea0 is the temperature coefficient of k0†/k0, i.e., the difference in the corresponding experimental (Arrhenius) activation energies, for two molecular models based on the pyrolysis of H2O2 and the isomerization of CH3NC at low pressures. It turns out that when compared at the same average energy transferred per deactivating collision, the various transition probability models give results for k0†/k0 and Ea0 that are independent of the nature of the decomposing molecule, are virtually indistinguishable experimentally, and in particular yield ∼-2kT as the maximum Ea0 due to inefficient energy transfer.