Characterizing losses during busy periods in finite buffer systems

2003 ◽  
Vol 40 (1) ◽  
pp. 242-249 ◽  
Author(s):  
Erol A. Peköz ◽  
Rhonda Righter ◽  
Cathy H. Xia
Keyword(s):  
Busy Period ◽  
Arrival Rate ◽  
Batch Size ◽  
Buffer Size ◽  
Service Rate ◽  
Finite Buffer ◽  
Initial Number ◽  
Poisson Arrivals ◽  
The Mean ◽  
Number Of Customers

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ= 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and whenρ> 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, whenρ= 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.

Characterizing losses during busy periods in finite buffer systems

2003 ◽  
Vol 40 (01) ◽  
pp. 242-249 ◽  
Author(s):  
Erol A. Peköz ◽  
Rhonda Righter ◽  
Cathy H. Xia
Keyword(s):  
Busy Period ◽  
Arrival Rate ◽  
Batch Size ◽  
Buffer Size ◽  
Service Rate ◽  
Finite Buffer ◽  
Initial Number ◽  
Poisson Arrivals ◽  
The Mean ◽  
Number Of Customers

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ = 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and when ρ > 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, when ρ = 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.

scholarly journals Transient and Stationary Losses in a Finite-Buffer Queue with Batch Arrivals

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Andrzej Chydzinski ◽  
Blazej Adamczyk
Keyword(s):  
Batch Size ◽  
Buffer Size ◽  
Arrival Process ◽  
Time Interval ◽  
Finite Buffer ◽  
Finite Time Interval ◽  
Batch Arrivals ◽  
Batch Markovian Arrival Process ◽  
Buffer Overflows ◽  
Finite Buffer Queue

We present an analysis of the number of losses, caused by the buffer overflows, in a finite-buffer queue with batch arrivals and autocorrelated interarrival times. Using the batch Markovian arrival process, the formulas for the average number of losses in a finite time interval and the stationary loss ratio are shown. In addition, several numerical examples are presented, including illustrations of the dependence of the number of losses on the average batch size, buffer size, system load, autocorrelation structure, and time.

scholarly journals Perturbation analysis of a variable M/M/1 queue: a probabilistic approach

2006 ◽  
Vol 38 (01) ◽  
pp. 263-283 ◽  
Author(s):  
Nelson Antunes ◽  
Christine Fricker ◽  
Fabrice Guillemin ◽  
Philippe Robert
Keyword(s):  
Power Series ◽  
Busy Period ◽  
Stationary Process ◽  
Probabilistic Approach ◽  
Power Series Expansion ◽  
Mean Value ◽  
Service Rate ◽  
The Mean ◽  
The Impact ◽  
Period Duration

In this paper, motivated by the problem of the coexistence on transmission links of telecommunications networks of elastic and unresponsive traffic, we study the impact on the busy period of an M/M/1 queue of a small perturbation in the service rate. The perturbation depends upon an independent stationary process (X(t)) and is quantified by means of a parameter ε ≪ 1. We specifically compute the two first terms of the power series expansion in ε of the mean value of the busy period duration. This allows us to study the validity of the reduced service rate approximation, which consists in comparing the perturbed M/M/1 queue with the M/M/1 queue whose service rate is constant and equal to the mean value of the perturbation. For the first term of the expansion, the two systems are equivalent. For the second term, the situation is more complex and it is shown that the correlations of the environment process (X(t)) play a key role.

scholarly journals Performance Modelling of Health-care Service Delivery in Adekunle Ajasin University, Akungba-Akoko, Nigeria Using Queuing Theory

2020 ◽  
pp. 119-127
Author(s):  
Orimoloye Segun Michael
Keyword(s):  
Health Care ◽  
Service Delivery ◽  
Queuing Theory ◽  
Health Care Service ◽  
Health Centre ◽  
Arrival Rate ◽  
Queuing System ◽  
Service Rate ◽  
Suitable Model ◽  
The Mean

The queuing theory is the mathematical approach to the analysis of waiting lines in any setting where arrivals rate of the subject is faster than the system can handle. It is applicable to the health care setting where the systems have excess capacity to accommodate random variation. Therefore, the purpose of this study was to determine the waiting, arrival and service times of patients at AAUA Health- setting and to model a suitable queuing system by using simulation technique to validate the model. This study was conducted at AAUA Health- Centre Akungba Akoko. It employed analytical and simulation methods to develop a suitable model. The collection of waiting time for this study was based on the arrival rate and service rate of patients at the Outpatient Centre. The data was calculated and analyzed using Microsoft Excel. Based on the analyzed data, the queuing system of the patient current situation was modelled and simulated using the PYTHON software. The result obtained from the simulation model showed that the mean arrival rate of patients on Friday week1 was lesser than the mean service rate of patients (i.e. 5.33> 5.625 (λ > µ). What this means is that the waiting line would be formed which would increase indefinitely; the service facility would always be busy. The analysis of the entire system of the AAUA health centre showed that queue length increases when the system is very busy. This work therefore evaluated and predicted the system performance of AAUA Health-Centre in terms of service delivery and propose solutions on needed resources to improve the quality of service offered to the patients visiting this health centre.

Cell loss probability for M/G/1 and time-slotted queues

2000 ◽  
Vol 37 (04) ◽  
pp. 1149-1156
Author(s):  
David McDonald ◽  
François Théberge
Keyword(s):  
Continuous Time ◽  
Time Slot ◽  
Arrival Rate ◽  
Cell Loss ◽  
Loss Probability ◽  
Finite Buffer ◽  
Asymptotic Results ◽  
Overflow Probability ◽  
The Mean

It is common practice to approximate the cell loss probability (CLP) of cells entering a finite buffer by the overflow probability (OVFL) of a corresponding infinite buffer queue, since the CLP is typically harder to estimate. We obtain exact asymptotic results for CLP and OVFL for time-slotted queues where block arrivals in different time slots are i.i.d. and one cell is served per time slot. In this case the ratio of CLP to OVFL is asymptotically (1-ρ)/ρ, where ρ is the use or, equivalently, the mean arrival rate per time slot. Analogous asymptotic results are obtained for continuous time M/G/1 queues. In this case the ratio of CLP to OVFL is asymptotically 1-ρ.

Multiple channel queues in heavy traffic. I

1970 ◽  
Vol 2 (01) ◽  
pp. 150-177 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Arrival Rate ◽  
Standard System ◽  
Interarrival Time ◽  
Service Rate ◽  
Service Channel ◽  
Multiple Channel ◽  
System A ◽  
The Mean ◽  
Service Channels

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving customers form a single queue and are served in the order of their arrival without defections. We shall treat two distinct modes of operation for the service channels. In the standard system a waiting customer is assigned to the first available service channel and the servers (servers ≡ service channels) are shut off when they are idle. Thus the classical GI/G/s system is a special case of our standard system. In the modified system a waiting customer is assigned to the service channel that can complete his service first and the servers are not shut off when they are idle. While the modified system is of some interest in its own right, we introduce it primarily as an analytical tool. Let λ i denote the arrival rate (reciprocal of the mean interarrival time) in the ith arrival channel and μ j the service rate (reciprocal of the mean service time) in the jth service channel. Then is the total arrival rate to the system and is the maximum service rate of the system. As a measure of congestion we define the traffic intensity ρ = λ/μ.

Optimal control of service rates in networks of queues

1987 ◽  
Vol 19 (1) ◽  
pp. 202-218 ◽  
Author(s):  
Richard R. Weber ◽  
Shaler Stidham
Keyword(s):  
Rate Control ◽  
Queueing Networks ◽  
Arrival Rate ◽  
Service Rate ◽  
Discounted Cost ◽  
Holding Costs ◽  
Optimal Service ◽  
Monotonicity Result ◽  
Service Rates ◽  
Number Of Customers

We prove a monotonicity result for the problem of optimal service rate control in certain queueing networks. Consider, as an illustrative example, a number of ·/M/1 queues which are arranged in a cycle with some number of customers moving around the cycle. A holding cost hi(xi) is charged for each unit of time that queue i contains xi customers, with hi being convex. As a function of the queue lengths the service rate at each queue i is to be chosen in the interval , where cost ci(μ) is charged for each unit of time that the service rate μis in effect at queue i. It is shown that the policy which minimizes the expected total discounted cost has a monotone structure: namely, that by moving one customer from queue i to the following queue, the optimal service rate in queue i is not increased and the optimal service rates elsewhere are not decreased. We prove a similar result for problems of optimal arrival rate and service rate control in general queueing networks. The results are extended to an average-cost measure, and an example is included to show that in general the assumption of convex holding costs may not be relaxed. A further example shows that the optimal policy may not be monotone unless the choice of possible service rates at each queue includes 0.

Multiple channel queues in heavy traffic. I

1970 ◽  
Vol 2 (1) ◽  
pp. 150-177 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Arrival Rate ◽  
Standard System ◽  
Interarrival Time ◽  
Service Rate ◽  
Service Channel ◽  
Multiple Channel ◽  
System A ◽  
The Mean ◽  
Service Channels

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving customers form a single queue and are served in the order of their arrival without defections. We shall treat two distinct modes of operation for the service channels. In the standard system a waiting customer is assigned to the first available service channel and the servers (servers ≡ service channels) are shut off when they are idle. Thus the classical GI/G/s system is a special case of our standard system. In the modified system a waiting customer is assigned to the service channel that can complete his service first and the servers are not shut off when they are idle. While the modified system is of some interest in its own right, we introduce it primarily as an analytical tool. Let λi denote the arrival rate (reciprocal of the mean interarrival time) in the ith arrival channel and μj the service rate (reciprocal of the mean service time) in the jth service channel. Then is the total arrival rate to the system and is the maximum service rate of the system. As a measure of congestion we define the traffic intensity ρ = λ/μ.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (3) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

scholarly journals A fluid queue with a finite buffer and subexponential input

2000 ◽  
Vol 32 (1) ◽  
pp. 221-243 ◽  
Author(s):  
A. P. Zwart
Keyword(s):  
Stationary Distribution ◽  
Fluid Model ◽  
Buffer Size ◽  
Finite Buffer ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Buffer Content ◽  
The Mean ◽  
Heavy Tailed ◽  
Finite Dam

We consider a fluid model similar to that of Kella and Whitt [32], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established by Kella and Whitt are extended to the finite buffer case: it is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N On-Off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue.

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