scholarly journals Estimating the Probability of Ruin for Variable Premiums by Simulation

1996 ◽  
Vol 26 (1) ◽  
pp. 93-105 ◽  
Author(s):  
Frédéric Michaud
Keyword(s):  
Sample Path ◽  
Risk Theory ◽  
Single Server ◽  
Single Server Queue ◽  
Storage Process ◽  
Probability Of Ruin ◽  
Surplus Process ◽  
Special Cases ◽  
Server Queue ◽  
Waiting Time Process

AbstractThere is a duality between the surplus process of classical risk theory and the single-server queue. It follows that the probability of ruin can be retrieved from a single sample path of the waiting time process of the single-server queue. In this paper, premiums are allowed to vary. It has been shown that the stationary distribution of a corresponding storage process is equal to the survival probability (with variable premiums). Thus by simulation of the corresponding storage process, the probability of ruin can be obtained. The special cases where the surplus earns interest and the premiums are charged by layers are considered and illustrated numerically.

A single-server queue with server vacations and a class of non-renewal arrival processes

1990 ◽  
Vol 22 (3) ◽  
pp. 676-705 ◽  
Author(s):  
David M. Lucantoni ◽  
Kathleen S. Meier-Hellstern ◽  
Marcel F. Neuts
Keyword(s):  
Waiting Times ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Arbitrary Time ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Server Queue ◽  
Arrival Processes ◽  
Markov Modulated

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes.Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

On queues with periodic inputs

1989 ◽  
Vol 26 (02) ◽  
pp. 381-389 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Limit Theorems ◽  
Single Server ◽  
Single Server Queue ◽  
Input Process ◽  
Poisson Arrival ◽  
Server Queue ◽  
Asymptotically Periodic ◽  
Waiting Time Process ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single-server queue with a periodic and ergodic input. It is shown that if the traffic intensity is less than 1, then the waiting time process is asymptotically periodic. Limit theorems associated with the asymptotic behavior of the queue are also proven. The results are then extended to acyclic networks of queues with periodic inputs. Particular cases of these results had been previously obtained for a single queue with periodic Poisson arrival input process and with independent and identically distributed service times.

scholarly journals The transient behaviour of the queueing system Gi/M/1

1963 ◽  
Vol 3 (2) ◽  
pp. 249-256 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Distribution Function ◽  
Explicit Expression ◽  
Queueing System ◽  
Single Server ◽  
Single Server Queue ◽  
Transient Behaviour ◽  
Special Cases ◽  
Server Queue ◽  
Interarrival Times ◽  
Lagrange Series

SUMMARYWe consider a single server queue for which the interarrival times are identically and independently distributed with distribution function A(x) and whose service times are distributed independently of each other and of the interarrival times with distribution function B(x) = 1 − e−x, x ≧ 0. We suppose that the system starts from emptiness and use the results of P. D. Finch [2] to derive an explicit expression for qnj, the probability that the (n + 1)th arrival finds more than j customers in the system. The special cases M/M/1 and D/M/1 are considerend and it is shown in the general case that qnj is a partial sum of the usual Lagrange series for the limiting probability .

Assembly-like queues

1973 ◽  
Vol 10 (2) ◽  
pp. 354-367 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Theoretic Model ◽  
Single Server ◽  
Time Process ◽  
Single Server Queue ◽  
Multiple Input ◽  
Server Queue ◽  
Waiting Time Process ◽  
Load Conditions

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.

scholarly journals A Batch Arrival Single Server Queue with Server Providing General Service in Two Fluctuating Modes and Reneging during Vacation and Breakdowns

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Monita Baruah ◽  
Kailash C. Madan ◽  
Tillal Eldabi
Keyword(s):  
Repair Process ◽  
Batch Arrival ◽  
Single Server ◽  
Single Server Queue ◽  
Numerical Illustration ◽  
Random Breakdown ◽  
Special Cases ◽  
Server Queue ◽  
Service Rates ◽  
General Service

We study the behavior of a batch arrival queuing system equipped with a single server providing general arbitrary service to customers with different service rates in two fluctuating modes of service. In addition, the server is subject to random breakdown. As soon as the server faces breakdown, the customer whose service is interrupted comes back to the head of the queue. As soon as repair process of the server is complete, the server immediately starts providing service in mode 1. Also customers waiting for service may renege (leave the queue) when there is breakdown or when server takes vacation. The system provides service with complete or reduced efficiency due to the fluctuating rates of service. We derive the steady state queue size distribution. Some special cases are discussed and numerical illustration is provided to see the effect and validity of the results.

On queues with periodic inputs

1989 ◽  
Vol 26 (2) ◽  
pp. 381-389 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Limit Theorems ◽  
Single Server ◽  
Single Server Queue ◽  
Input Process ◽  
Poisson Arrival ◽  
Server Queue ◽  
Asymptotically Periodic ◽  
Waiting Time Process ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single-server queue with a periodic and ergodic input. It is shown that if the traffic intensity is less than 1, then the waiting time process is asymptotically periodic. Limit theorems associated with the asymptotic behavior of the queue are also proven. The results are then extended to acyclic networks of queues with periodic inputs. Particular cases of these results had been previously obtained for a single queue with periodic Poisson arrival input process and with independent and identically distributed service times.

scholarly journals Sample-Path Analysis of Single-Server Queue with Multiple Vacations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Muhammad El-Taha
Keyword(s):  
Stochastic Process ◽  
Conservation Law ◽  
Sample Path ◽  
Single Server ◽  
Queueing Model ◽  
Single Server Queue ◽  
Sample Path Analysis ◽  
Multiple Vacation ◽  
Server Queue ◽  
Background Material

We a give deterministic (sample path) proof of a result that extends the Pollaczek-Khintchine formula for a multiple vacation single-server queueing model. We also give a conservation law for the same system with multiple classes. Our results are completely rigorous and hold under weaker assumptions than those given in the literature. We do not make stochastic assumptions, so the results hold almost surely on every sample path of the stochastic process that describes the system evolution. The article is self contained in that it gives a brief review of necessary background material.

Assembly-like queues

1973 ◽  
Vol 10 (02) ◽  
pp. 354-367 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Theoretic Model ◽  
Single Server ◽  
Time Process ◽  
Single Server Queue ◽  
Multiple Input ◽  
Server Queue ◽  
Waiting Time Process ◽  
Load Conditions

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.

scholarly journals PROPERTIES OF THE GITTINS INDEX WITH APPLICATION TO OPTIMAL SCHEDULING

2011 ◽  
Vol 25 (3) ◽  
pp. 269-288 ◽  
Author(s):  
Samuli Aalto ◽  
Urtzi Ayesta ◽  
Rhonda Righter
Keyword(s):  
Specific Problem ◽  
Optimal Scheduling ◽  
Single Server ◽  
Processor Sharing ◽  
Gittins Index ◽  
Single Server Queue ◽  
Index Policy ◽  
Special Cases ◽  
The Family ◽  
Server Queue

We consider the optimal scheduling problem for a single-server queue without arrivals. We allow preemptions, and our purpose is to minimize the expected flow time. The optimal nonanticipating discipline is known to be the Gittins index policy, which, however, is defined in an implicit way. Until now, its general behavior in this specific problem has been characterized only in a few special cases. In this article, we give as complete a characterization as possible. It turns out that the optimal policy always belongs to the family of multilevel processor sharing disciplines.

A single-server queue with server vacations and a class of non-renewal arrival processes

1990 ◽  
Vol 22 (03) ◽  
pp. 676-705 ◽  
Author(s):  
David M. Lucantoni ◽  
Kathleen S. Meier-Hellstern ◽  
Marcel F. Neuts
Keyword(s):  
Waiting Times ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Arbitrary Time ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Server Queue ◽  
Arrival Processes ◽  
Markov Modulated

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes. Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

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