ON AN EQUIVALENCE BETWEEN LOSS RATES AND CYCLE MAXIMA IN QUEUES AND DAMS

2005 ◽  
Vol 19 (2) ◽  
pp. 241-255 ◽  
Author(s):  
René Bekker ◽  
Bert Zwart
Keyword(s):  
Loss Probability ◽  
Single Server ◽  
Finite Buffer ◽  
Single Server Queue ◽  
Release Rates ◽  
Asymptotic Results ◽  
State Dependent ◽  
Tail Distribution ◽  
Server Queue ◽  
Cycle Maximum

We consider the loss probability of a customer in a single-server queue with finite buffer and partial rejection and show that it can be identified with the tail distribution of the cycle maximum of the associated infinite-buffer queue. This equivalence is shown to hold for the GI/G/1 queue and for dams with state-dependent release rates. To prove this equivalence, we use a duality for stochastically monotone recursions, developed by Asmussen and Sigman (1996). As an application, we obtain several exact and asymptotic results for the loss probability and extend Takács' formula for the cycle maximum in the M/G/1 queue to dams with variable release rate.

Loss Probability of a Burst Arrival Finite Queue with Synchronized Service

1992 ◽  
Vol 6 (2) ◽  
pp. 201-216 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Markov Chain ◽  
Loss Probability ◽  
Buffer Size ◽  
Embedded Markov Chain ◽  
Main Concern ◽  
Single Server ◽  
Finite Buffer ◽  
Single Server Queue ◽  
Server Queue ◽  
Computation Algorithm

We are concerned with a burst arrival single-server queue, where arrivals of cells in a burst are synchronized with a constant service time. The main concern is with the loss probability of cells for the queue with a finite buffer. We analyze an embedded Markov chain at departure instants of cells and get a kind of lumpability for its state space. Based on these results, this paper proposes a computation algorithm for its stationary distribution and the loss probability. Closed formulas are obtained for the first two moments of the numbers of cells and active bursts when the buffer size is infinite.

A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

1995 ◽  
Vol 32 (4) ◽  
pp. 1103-1111 ◽  
Author(s):  
Qing Du
Keyword(s):  
Poisson Process ◽  
Arrival Rate ◽  
Loss Probability ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i. The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i. In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

OPTIMALITY OF STATE DEPENDENT SINGLE SERVER QUEUE WITH MMPP INPUT

2020 ◽  
Vol 9 (3) ◽  
pp. 1197-1204
Author(s):  
R. Sakthi ◽  
R. Raghavendran ◽  
V. Vidhya
Keyword(s):  
Single Server ◽  
Single Server Queue ◽  
State Dependent ◽  
Server Queue

A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

1995 ◽  
Vol 32 (04) ◽  
pp. 1103-1111 ◽  
Author(s):  
Qing Du
Keyword(s):  
Poisson Process ◽  
Arrival Rate ◽  
Loss Probability ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i . The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i . In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

Stochastic inventory models with bulk demand and state-dependent leadtimes

1971 ◽  
Vol 8 (3) ◽  
pp. 521-534 ◽  
Author(s):  
Donald Gross ◽  
Carl M. Harris ◽  
James A. Lechner
Keyword(s):  
Single Server ◽  
Single Server Queue ◽  
Inventory Models ◽  
Inventory Cost ◽  
Stochastic Inventory ◽  
State Dependent ◽  
Optimal Value ◽  
Server Queue ◽  
Time Required ◽  
Stochastic Inventory Models

This paper describes two one-for-one ordering (S − 1, S) inventory models in which the time required for order replenishment is state-dependent. The demand is assumed to follow a compound Poisson distribution, and that portion of the leadtime corresponding to the actual filling of orders is assumed to depend on the number of outstanding orders. Since the orders placed are assumed to go into a single-server queue, queuing results are used to obtain the expected inventory cost as a function of S in order to obtain an optimal value of S.

Stochastic inventory models with bulk demand and state-dependent leadtimes

1971 ◽  
Vol 8 (03) ◽  
pp. 521-534
Author(s):  
Donald Gross ◽  
Carl M. Harris ◽  
James A. Lechner
Keyword(s):  
Single Server ◽  
Single Server Queue ◽  
Inventory Models ◽  
Inventory Cost ◽  
Stochastic Inventory ◽  
State Dependent ◽  
Optimal Value ◽  
Server Queue ◽  
Time Required ◽  
Stochastic Inventory Models

This paper describes two one-for-one ordering (S − 1, S) inventory models in which the time required for order replenishment is state-dependent. The demand is assumed to follow a compound Poisson distribution, and that portion of the leadtime corresponding to the actual filling of orders is assumed to depend on the number of outstanding orders. Since the orders placed are assumed to go into a single-server queue, queuing results are used to obtain the expected inventory cost as a function of S in order to obtain an optimal value of S.

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