Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

1989 ◽  
Vol 26 (03) ◽  
pp. 552-565 ◽  
Author(s):  
J. George Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Stochastic Process ◽  
Discrete Time ◽  
Queueing Systems ◽  
Random Variables ◽  
Decomposition Theorem ◽  
Asymptotic Decomposition ◽  
Ergodic Decomposition ◽  
Special Cases ◽  
Unified View ◽  
Lindley Process

We consider a discrete-time stochastic process {Wn , n≧0} governed by i.i.d random variables {ξ n } whose distribution has support on (–∞,∞) and replacement random variables {Rn } whose distributions have support on [0,∞). Given Wn, Wn + 1 takes the value Wn + ζ n + 1 if it is non-negative. Otherwise Wn + 1 takes the value Rn + 1 where the distribution of Rn + 1 depends only on the value of Wn + ζn + 1 . This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn = 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

1989 ◽  
Vol 26 (3) ◽  
pp. 552-565 ◽  
Author(s):  
J. George Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Stochastic Process ◽  
Discrete Time ◽  
Queueing Systems ◽  
Random Variables ◽  
Decomposition Theorem ◽  
Asymptotic Decomposition ◽  
Ergodic Decomposition ◽  
Special Cases ◽  
Unified View ◽  
Lindley Process

We consider a discrete-time stochastic process {Wn, n≧0} governed by i.i.d random variables {ξ n} whose distribution has support on (–∞,∞) and replacement random variables {Rn} whose distributions have support on [0,∞). Given Wn, Wn+ 1 takes the value Wn + ζ n+ 1 if it is non-negative. Otherwise Wn+ 1 takes the value Rn +1 where the distribution of Rn+ 1 depends only on the value of Wn + ζn +1. This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn = 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (02) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (2) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

Conditional expected sojourn times in insensitive queueing systems and networks

1984 ◽  
Vol 16 (4) ◽  
pp. 906-919 ◽  
Author(s):  
Uwe Jansen
Keyword(s):  
Queueing Networks ◽  
Point Processes ◽  
Response Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Main Tool ◽  
Closed Queueing Networks ◽  
Conditional Mean ◽  
Sojourn Times ◽  
Special Cases

We consider queueing systems where the stationary state probabilities are insensitive with respect to the distribution of certain basic random variables such as service requirements, interarrival times, repair times, etc. The conditional expected sojourn times are stated as Radon–Nikodym densities of the stationary distribution at jump points of the queueing system. The conditions are the given values of such basic random variables for which the insensitivity is valid. We use stationary point processes as our main tool. This means that dependences between certain basic random variables are permitted. Conditional expected real service times, conditional mean response times in closed queueing networks, and similar conditional expected values, are dealt with as special cases.

scholarly journals A Novel Analytic Technique for the Service Station Reliability in a Discrete-Time Repairable Queue

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Renbin Liu ◽  
Yinghui Tang
Keyword(s):  
Reliability Analysis ◽  
Discrete Time ◽  
Queueing System ◽  
Queueing Systems ◽  
Decomposition Technique ◽  
Service Station ◽  
Numerical Examples ◽  
Reliability Indices ◽  
Special Cases ◽  
Bulk Arrival

This paper presents a decomposition technique for the service station reliability in a discrete-time repairableGeomX/G/1 queueing system, in which the server takes exhaustive service and multiple adaptive delayed vacation discipline. Using such a novel analytic technique, some important reliability indices and reliability relation equations of the service station are derived. Furthermore, the structures of the service station indices are also found. Finally, special cases and numerical examples validate the derived results and show that our analytic technique is applicable to reliability analysis of some complex discrete-time repairable bulk arrival queueing systems.

Conditional expected sojourn times in insensitive queueing systems and networks

1984 ◽  
Vol 16 (04) ◽  
pp. 906-919 ◽  
Author(s):  
Uwe Jansen
Keyword(s):  
Queueing Networks ◽  
Point Processes ◽  
Response Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Main Tool ◽  
Closed Queueing Networks ◽  
Conditional Mean ◽  
Sojourn Times ◽  
Special Cases

We consider queueing systems where the stationary state probabilities are insensitive with respect to the distribution of certain basic random variables such as service requirements, interarrival times, repair times, etc. The conditional expected sojourn times are stated as Radon–Nikodym densities of the stationary distribution at jump points of the queueing system. The conditions are the given values of such basic random variables for which the insensitivity is valid. We use stationary point processes as our main tool. This means that dependences between certain basic random variables are permitted. Conditional expected real service times, conditional mean response times in closed queueing networks, and similar conditional expected values, are dealt with as special cases.

scholarly journals Combined Games with Randomly Delayed Beginnings

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 534
Author(s):  
F. Thomas Bruss
Keyword(s):  
Discrete Time ◽  
Real World ◽  
Optimal Stopping ◽  
Random Variables ◽  
Approximate Solutions ◽  
Real World Problems

This paper presents two-person games involving optimal stopping. As far as we are aware, the type of problems we study are new. We confine our interest to such games in discrete time. Two players are to chose, with randomised choice-priority, between two games G1 and G2. Each game consists of two parts with well-defined targets. Each part consists of a sequence of random variables which determines when the decisive part of the game will begin. In each game, the horizon is bounded, and if the two parts are not finished within the horizon, the game is lost by definition. Otherwise the decisive part begins, on which each player is entitled to apply their or her strategy to reach the second target. If only one player achieves the two targets, this player is the winner. If both win or both lose, the outcome is seen as “deuce”. We motivate the interest of such problems in the context of real-world problems. A few representative problems are solved in detail. The main objective of this article is to serve as a preliminary manual to guide through possible approaches and to discuss under which circumstances we can obtain solutions, or approximate solutions.

Filtering of Markov renewal queues, IV: Flow processes in feedback queues

1985 ◽  
Vol 17 (2) ◽  
pp. 386-407 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Arrival Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Flow Process ◽  
Special Cases ◽  
Flow Processes ◽  
Transition Points ◽  
The Times

This paper is a continuation of the study of a class of queueing systems where the queue-length process embedded at basic transition points, which consist of ‘arrivals’, ‘departures’ and ‘feedbacks’, is a Markov renewal process (MRP). The filtering procedure of Çinlar (1969) was used in [12] to show that the queue length process embedded separately at ‘arrivals’, ‘departures’, ‘feedbacks’, ‘inputs’ (arrivals and feedbacks), ‘outputs’ (departures and feedbacks) and ‘external’ transitions (arrivals and departures) are also MRP. In this paper expressions for the elements of each Markov renewal kernel are derived, and thence expressions for the distribution of the times between transitions, under stationary conditions, are found for each of the above flow processes. In particular, it is shown that the inter-event distributions for the arrival process and the departure process are the same, with an equivalent result holding for inputs and outputs. Further, expressions for the stationary joint distributions of successive intervals between events in each flow process are derived and interconnections, using the concept of reversed Markov renewal processes, are explored. Conditions under which any of the flow processes are renewal processes or, more particularly, Poisson processes are also investigated. Special cases including, in particular, the M/M/1/N and M/M/1 model with instantaneous Bernoulli feedback, are examined.

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