Stationary Solution of a Fluid Queue Driven by a Queue with Chain Sequence Rates and Controlled Input

2020 ◽  
Vol 57 (4) ◽  
pp. 552-565
Author(s):  
Susairaj Sophia ◽  
Babu Muthu Deepika
Keyword(s):  
Fluid Flow ◽  
Stationary Solution ◽  
Continued Fraction ◽  
Queueing System ◽  
The State ◽  
Fluid Queue ◽  
Queueing Process ◽  
Buffer Content ◽  
Service Rates ◽  
Chain Sequence

A fluid queueing system in which the fluid flow in to the buffer is regulated by the state of the background queueing process is considered. In this model, the arrival and service rates follow chain sequence rates and are controlled by an exponential timer. The buffer content distribution along with averages are found using continued fraction methodology. Numerical results are illustrated to analyze the trend of the average buffer content for the model under consideration. It is interesting to note that the stationary solution of a fluid queue driven by a queue with chain sequence rates does not exist in the absence of exponential timer.

scholarly journals Stationary solution to the fluid queue fed by an M/M/1 queue

2002 ◽  
Vol 39 (2) ◽  
pp. 359-369 ◽  
Author(s):  
N. Barbot ◽  
B. Sericola
Keyword(s):  
Stationary Solution ◽  
Stationary Distribution ◽  
Generating Functions ◽  
The State ◽  
Finite Capacity ◽  
Fluid Queue ◽  
Analytic Expression ◽  
Joint Stationary Distribution

We consider an infinite-capacity buffer receiving fluid at a rate depending on the state of an M/M/1 queue. We obtain a new analytic expression for the joint stationary distribution of the buffer level and the state of the M/M/1 queue. This expression is obtained by the use of generating functions which are explicitly inverted. The case of a finite capacity fluid queue is also considered.

scholarly journals Stationary solution to the fluid queue fed by an M/M/1 queue

2002 ◽  
Vol 39 (02) ◽  
pp. 359-369 ◽  
Author(s):  
N. Barbot ◽  
B. Sericola
Keyword(s):  
Stationary Solution ◽  
Stationary Distribution ◽  
Generating Functions ◽  
The State ◽  
Finite Capacity ◽  
Fluid Queue ◽  
Analytic Expression ◽  
Joint Stationary Distribution

We consider an infinite-capacity buffer receiving fluid at a rate depending on the state of an M/M/1 queue. We obtain a new analytic expression for the joint stationary distribution of the buffer level and the state of the M/M/1 queue. This expression is obtained by the use of generating functions which are explicitly inverted. The case of a finite capacity fluid queue is also considered.

Asymptotics of palm-stationary buffer content distributions in fluid flow queues

1999 ◽  
Vol 31 (01) ◽  
pp. 235-253 ◽  
Author(s):  
Tomasz Rolski ◽  
Sabine Schlegel ◽  
Volker Schmidt
Keyword(s):  
Distribution Function ◽  
Fluid Flow ◽  
Asymptotic Representation ◽  
Regular Variation ◽  
Queueing System ◽  
Activity Period ◽  
Buffer Content ◽  
Special Case ◽  
Tail Function

We study a fluid flow queueing system with m independent sources alternating between periods of silence and activity; m ≥ 2. The distribution function of the activity periods of one source, is supposed to be intermediate regular varying. We show that the distribution of the net increment of the buffer during an aggregate activity period (i.e. when at least one source is active) is asymptotically tail-equivalent to the distribution of the net input during a single activity period with intermediate regular varying distribution function. In this way, we arrive at an asymptotic representation of the Palm-stationary tail-function of the buffer content at the beginning of aggregate activity periods. Our approach is probabilistic and extends recent results of Boxma (1996; 1997) who considered the special case of regular variation.

Asymptotics of palm-stationary buffer content distributions in fluid flow queues

1999 ◽  
Vol 31 (1) ◽  
pp. 235-253 ◽  
Author(s):  
Tomasz Rolski ◽  
Sabine Schlegel ◽  
Volker Schmidt
Keyword(s):  
Distribution Function ◽  
Fluid Flow ◽  
Asymptotic Representation ◽  
Regular Variation ◽  
Queueing System ◽  
Activity Period ◽  
Buffer Content ◽  
Special Case ◽  
Tail Function

We study a fluid flow queueing system with m independent sources alternating between periods of silence and activity; m ≥ 2. The distribution function of the activity periods of one source, is supposed to be intermediate regular varying. We show that the distribution of the net increment of the buffer during an aggregate activity period (i.e. when at least one source is active) is asymptotically tail-equivalent to the distribution of the net input during a single activity period with intermediate regular varying distribution function. In this way, we arrive at an asymptotic representation of the Palm-stationary tail-function of the buffer content at the beginning of aggregate activity periods. Our approach is probabilistic and extends recent results of Boxma (1996; 1997) who considered the special case of regular variation.

scholarly journals Transient analysis of a fluid queue driven by a birth and death process suggested by a chain sequence

2005 ◽  
Vol 2005 (2) ◽  
pp. 143-158 ◽  
Author(s):  
P. R. Parthasarathy ◽  
B. Sericola ◽  
K. V. Vijayashree
Keyword(s):  
Performance Measure ◽  
Death Process ◽  
Birth And Death Process ◽  
Fluid Queue ◽  
Transient Behaviour ◽  
Transient Distribution ◽  
Buffer Content ◽  
Chain Sequence ◽  
A Chain ◽  
Stationary Probabilities

We analyse the transient behaviour of a fluid queue driven by a birth and death process (BDP) whose birth and death rates are suggested by a chain sequence. For the BDP suggested by a chain sequence, the stationary probabilities do not exist and hence the stationary buffer content distribution for fluid queues driven by such BDP does not exist. However, their transient distribution is obtained in a simple closed form by two different approaches: the first is the continued fraction approach and the second is an approach in terms of recurrence relation by an analysis similar to that of Sericola (1998). The probability for the buffer content to be empty at an arbitrary time is also studied. The variations in this performance measure are revealed in the form of graphs. Numerical illustrations are included.

scholarly journals M/G/1 vacation model with limited service discipline and hybrid switching-on policy

1997 ◽  
Vol 3 (3) ◽  
pp. 243-253
Author(s):  
Alexander V. Babitsky
Keyword(s):  
Generating Function ◽  
Busy Period ◽  
Optimization Problem ◽  
Queueing System ◽  
Service Discipline ◽  
Multiple Vacations ◽  
Queueing Process ◽  
Vacation Model ◽  
Hybrid Switching ◽  
Limited Service

The author studies an M/G/1 queueing system with multiple vacations. The server is turned off in accordance with the K-limited discipline, and is turned on in accordance with the T-N-hybrid policy. This is to say that the server will begin a vacation from the system if either the queue is empty orKcustomers were served during a busy period. The server idles until it finds at leastNwaiting units upon return from a vacation.Formulas for the distribution generating function and some characteristics of the queueing process are derived. An optimization problem is discussed.

scholarly journals The fluid mechanics of poohsticks

2020 ◽  
Vol 378 (2179) ◽  
pp. 20190522 ◽  
Author(s):  
Julyan H. E. Cartwright ◽  
Oreste Piro
Keyword(s):  
Fluid Flow ◽  
Fluid Mechanics ◽  
Solid Body ◽  
State Of The Art ◽  
The State ◽  
Theme Issue ◽  
Two Fluids ◽  
Stokes Drag ◽  
Complex Fluid

The year 2019 marked the bicentenary of George Gabriel Stokes, who in 1851 described the drag—Stokes drag—on a body moving immersed in a fluid, and 2020 is the centenary of Christopher Robin Milne, for whom the game of poohsticks was invented; his father A. A. Milne’s The House at Pooh Corner , in which it was first described in print, appeared in 1928. So this is an apt moment to review the state of the art of the fluid mechanics of a solid body in a complex fluid flow, and one floating at the interface between two fluids in motion. Poohsticks pertains to the latter category, when the two fluids are water and air. This article is part of the theme issue ‘Stokes at 200 (part 2)’.

Bias optimal admission control policies for a multiclass nonstationary queueing system

2002 ◽  
Vol 39 (01) ◽  
pp. 20-37 ◽  
Author(s):  
Mark E. Lewis ◽  
Hayriye Ayhan ◽  
Robert D. Foley
Keyword(s):  
Optimal Policy ◽  
Queueing System ◽  
Sufficient Conditions ◽  
System Capacity ◽  
Average Reward ◽  
Finite Capacity ◽  
Long Run ◽  
Optimal Policies ◽  
Service Rates ◽  
Long Run Average Reward

We consider a finite-capacity queueing system where arriving customers offer rewards which are paid upon acceptance into the system. The gatekeeper, whose objective is to ‘maximize’ rewards, decides if the reward offered is sufficient to accept or reject the arriving customer. Suppose the arrival rates, service rates, and system capacity are changing over time in a known manner. We show that all bias optimal (a refinement of long-run average reward optimal) policies are of threshold form. Furthermore, we give sufficient conditions for the bias optimal policy to be monotonic in time. We show, via a counterexample, that if these conditions are violated, the optimal policy may not be monotonic in time or of threshold form.

scholarly journals Entropy Analysis of a Flexible Markovian Queue with Server Breakdowns

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 979
Author(s):  
Messaoud Bounkhel ◽  
Lotfi Tadj ◽  
Ramdane Hedjar
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Threshold Level ◽  
Tail Probability ◽  
Probability Vector ◽  
Fixed Size ◽  
Markovian Queue ◽  
Entropy Principle ◽  
Service Rates ◽  
Good Agreement

In this paper, a versatile Markovian queueing system is considered. Given a fixed threshold level c, the server serves customers one a time when the queue length is less than c, and in batches of fixed size c when the queue length is greater than or equal to c. The server is subject to failure when serving either a single or a batch of customers. Service rates, failure rates, and repair rates, depend on whether the server is serving a single customer or a batch of customers. While the analytical method provides the initial probability vector, we use the entropy principle to obtain both the initial probability vector (for comparison) and the tail probability vector. The comparison shows the results obtained analytically and approximately are in good agreement, especially when the first two moments are used in the entropy approach.

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