scholarly journals On the Modelling of the Mobile WiMAX (IEEE 802.16e) Uplink Scheduler

2010 ◽  
Vol 2010 ◽  
pp. 1-7
Author(s):  
Darmawaty Mohd Ali ◽  
Kaharudin Dimyati
Keyword(s):  
Waiting Time ◽  
Arrival Rate ◽  
Mobile Wimax ◽  
Single Server ◽  
Service Time Distribution ◽  
Ieee 802.16E ◽  
Delay Constraint ◽  
Threshold Policy ◽  
Polling Model ◽  
Mean Waiting Time

Packet scheduling has drawn a great deal of attention in the field of wireless networks as it plays an important role in distributing shared resources in a network. The process involves allocating the bandwidth among users and determining their transmission order. In this paper an uplink (UL) scheduling algorithm for the Mobile Worldwide Interoperability for Microwave Access (WiMAX) network based on the cyclic polling model is proposed. The model in this study consists of five queues (UGS, ertPS, rtPS, nrtPS, and BE) visited by a single server. A threshold policy is imposed to the nrtPS queue to ensure that the delay constraint of real time traffic (UGS, ertPS, and rtPS) is not violated making this approach original in comparison to the existing contributions. A mathematical model is formulated for the weighted sum of the mean waiting time of each individual queues based on the pseudo-conservation law. The results of the analysis are useful in obtaining or testing approximation for individual mean waiting time especially when queues are asymmetric (where each queue may have different stochastic characteristic such as arrival rate and service time distribution) and when their number is large (more than 2 queues).

The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part II.

1975 ◽  
Vol 7 (3) ◽  
pp. 647-655 ◽  
Author(s):  
John Dagsvik
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Time Process ◽  
Waiting Time Process ◽  
Bulk Queue ◽  
Erlang Distributions ◽  
Matrix Factorisation

In a previous paper (Dagsvik (1975)) the waiting time process of the single server bulk queue is considered and a corresponding waiting time equation is established. In this paper the waiting time equation is solved when the inter-arrival or service time distribution is a linear combination of Erlang distributions. The analysis is essentially based on algebraic arguments.

On queues with periodic Poisson input

1981 ◽  
Vol 18 (04) ◽  
pp. 889-900 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Direct Proof ◽  
Poisson Model ◽  
Fixed Time ◽  
Time Of Day ◽  
Asymptotic Distributions ◽  
Single Server ◽  
Service Time Distribution ◽  
Asymptotic Results ◽  
Poisson Input

This paper is concerned with asymptotic results for a single-server queue having periodic Poisson input and general service-time distribution, and carries forward the analysis of this model initiated in Harrison and Lemoine. First, it is shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it is then pointed out that Hooke's theorem leads to the extension (developed in [3]) of a related theorem of Takács. Second, it is demonstrated that the asymptotic distribution for the server-load process at a fixed ‘time of day' coincides with the distribution for the supremum, over the time horizon [0,∞), of the sum of a stationary compound Poisson process with negative drift and a continuous periodic function. Some implications of this characterization result for the computation and approximation of the asymptotic distributions are then discussed, including a direct proof, for the periodic Poisson case, of a recent result of Rolski comparing mean asymptotic customer waiting time with that of a corresponding M/G/1 system.

scholarly journals Mean Waiting Time in Large-Scale and Critically Loaded Power of d Load Balancing Systems

2021 ◽  
Vol 5 (2) ◽  
pp. 1-34
Author(s):  
Tim Hellemans ◽  
Benny Van Houdt
Keyword(s):  
Load Balancing ◽  
Waiting Time ◽  
Large Scale ◽  
Mean Field ◽  
Arrival Rate ◽  
Mean Field Limit ◽  
Mean Waiting Time ◽  
Constant Rate ◽  
Finite Set ◽  
The Mean

Mean field models are a popular tool used to analyse load balancing policies. In some exceptional cases the waiting time distribution of the mean field limit has an explicit form. In other cases it can be computed as the solution of a set of differential equations. In this paper we study the limit of the mean waiting time E[Wλ] as the arrival rate λ approaches 1 for a number of load balancing policies in a large-scale system of homogeneous servers which finish work at a constant rate equal to one and exponential job sizes with mean 1 (i.e. when the system gets close to instability). As E[Wλ] diverges to infinity, we scale with -log(1-λ) and present a method to compute the limit limλ-> 1- -E[Wλ]/l(1-λ). We show that this limit has a surprisingly simple form for the load balancing algorithms considered. More specifically, we present a general result that holds for any policy for which the associated differential equation satisfies a list of assumptions. For the well-known LL(d) policy which assigns an incoming job to a server with the least work left among d randomly selected servers these assumptions are trivially verified. For this policy we prove the limit is given by 1/d-1. We further show that the LL(d,K) policy, which assigns batches of K jobs to the K least loaded servers among d randomly selected servers, satisfies the assumptions and the limit is equal to K/d-K. For a policy which applies LL(di) with probability pi, we show that the limit is given by 1/ ∑i pi di - 1. We further indicate that our main result can also be used for load balancers with redundancy or memory. In addition, we propose an alternate scaling -l(pλ) instead of -l(1-λ), where pλ is adapted to the policy at hand, such that limλ-> 1- -E[Wλ]/l(1-λ)=limλ-> 1- -E[Wλ]/l(pλ), where the limit limλ-> 0+ -E[Wλ]/l(pλ) is well defined and non-zero (contrary to limλ-> 0+ -E[Wλ]/l(1-λ)). This allows to obtain relatively flat curves for -E[Wλ]/l(pλ) for λ ∈ [0,1] which indicates that the low and high load limits can be used as an approximation when λ is close to one or zero. Our results rely on the earlier proven ansatz which asserts that for certain load balancing policies the workload distribution of any finite set of queues becomes independent of one another as the number of servers tends to infinity.

Waiting time and workload in queues with periodic Poisson input

1989 ◽  
Vol 26 (02) ◽  
pp. 390-397 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Differential Equation ◽  
Waiting Time ◽  
Arrival Rate ◽  
Time Of Day ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Service Distribution ◽  
Server Queue ◽  
General Service

This paper develops moment formulas for asymptotic workload and waiting time in a single-server queue with periodic Poisson input and general service distribution. These formulas involve the corresponding moments of waiting-time (workload) for the M/G/1 system with the same average arrival rate and service distribution. In certain cases, all the terms in the formulas can be computed exactly, including moments of workload at each ‘time of day.' The approach makes use of an asymptotic version of the Takács [12] integro-differential equation, together with representation results of Harrison and Lemoine [3] and Lemoine [6].

scholarly journals An M/G/1 queue with multiple types of feedback and gated vacations

1997 ◽  
Vol 34 (03) ◽  
pp. 773-784 ◽  
Author(s):  
Onno J. Boxma ◽  
Uri Yechiali
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Poisson Arrivals

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

scholarly journals On Consistency of Absolute Deviations Estimators of Convex Functions

2016 ◽  
Vol 5 (2) ◽  
pp. 1
Author(s):  
Yao Luo ◽  
Eunji Lim
Keyword(s):  
Waiting Time ◽  
Convex Functions ◽  
Service Rate ◽  
Data Sets ◽  
Single Server ◽  
Computationally Efficient ◽  
Data Set ◽  
Least Absolute Deviations ◽  
Long Run ◽  
Mean Waiting Time

When estimating an unknown function from a data set of n observations, the function is often known to be convex. For example, the long-run average waiting time of a customer in a single server queue is known to be convex in the service rate (Weber 1983) even though there is no closed-form formula for the mean waiting time, and hence, it needs to be estimated from a data set. A computationally efficient way of finding the best fit of the convex function to the data set is to compute the least absolute deviations estimator minimizing the sum of absolute deviations over the set of convex functions. This estimator exhibits numerically preferred behavior since it can be computed faster and for a larger data sets compared to other existing methods (Lim & Luo 2014). In this paper, we establish the validity of the least absolute deviations estimator by proving that the least absolute deviations estimator converges almost surely to the true function as n increases to infinity under modest assumptions.

Generalised birth and death queueing processes: recent results

1977 ◽  
Vol 9 (1) ◽  
pp. 125-140 ◽  
Author(s):  
B. W. Conolly ◽  
J. Chan
Keyword(s):  
Waiting Time ◽  
Queueing Systems ◽  
System Size ◽  
Single Server ◽  
Traffic Intensity ◽  
Stable Regime ◽  
Mean Waiting Time ◽  
Service Rates ◽  
Queueing Processes ◽  
Effective Service

The systems considered are single-server, though the theory has wider application to models of adaptive queueing systems. Arrival and service mechanisms are governed by state (n)-dependent mean arrival and service rates λn and µn. It is assumed that the choice of λn and µn leads to a stable regime. Formulae are sought that provide easy means of computing statistics of effectiveness of systems. A measure of traffic intensity is first defined in terms of ‘effective’ service time and inter-arrival intervals. It is shown that the latter have a renewal type connection with appropriately defined mean effective arrival and service rates λ∗ and µ∗ and that in consequence the ratio λ∗/µ∗ is the traffic intensity, equal moreover to where is the stable probability of an empty system, consistent with other systems. It is also shown that for first come, first served discipline the equivalent of Little's formula holds, where and are the mean waiting time of an arrival and mean system size at an arbitrary epoch. In addition it appears that stable regime output intervals are statistically identical with effective inter-arrival intervals. Symmetrical moment formulae of arbitrary order are derived algebraically for effective inter-arrival and service intervals, for waiting time, for busy period and for output.

On queues with periodic Poisson input

1981 ◽  
Vol 18 (4) ◽  
pp. 889-900 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Direct Proof ◽  
Poisson Model ◽  
Fixed Time ◽  
Time Of Day ◽  
Asymptotic Distributions ◽  
Single Server ◽  
Service Time Distribution ◽  
Asymptotic Results ◽  
Poisson Input

This paper is concerned with asymptotic results for a single-server queue having periodic Poisson input and general service-time distribution, and carries forward the analysis of this model initiated in Harrison and Lemoine. First, it is shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it is then pointed out that Hooke's theorem leads to the extension (developed in [3]) of a related theorem of Takács. Second, it is demonstrated that the asymptotic distribution for the server-load process at a fixed ‘time of day' coincides with the distribution for the supremum, over the time horizon [0,∞), of the sum of a stationary compound Poisson process with negative drift and a continuous periodic function. Some implications of this characterization result for the computation and approximation of the asymptotic distributions are then discussed, including a direct proof, for the periodic Poisson case, of a recent result of Rolski comparing mean asymptotic customer waiting time with that of a corresponding M/G/1 system.

ESTIMATING WAITING TIMES WITH THE TIME-VARYING LITTLE'S LAW

2013 ◽  
Vol 27 (4) ◽  
pp. 471-506 ◽  
Author(s):  
Song-Hee Kim ◽  
Ward Whitt
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Arrival Rate ◽  
Rate Function ◽  
Time Varying ◽  
Waiting Time Distribution ◽  
Mean Waiting Time ◽  
Little's Law ◽  
Little’S Law

When waiting times cannot be observed directly, Little's law can be applied to estimate the average waiting time by the average number in system divided by the average arrival rate, but that simple indirect estimator tends to be biased significantly when the arrival rates are time-varying and the service times are relatively long. Here it is shown that the bias in that indirect estimator can be estimated and reduced by applying the time-varying Little's law (TVLL). If there is appropriate time-varying staffing, then the waiting time distribution may not be time-varying even though the arrival rate is time varying. Given a fixed waiting time distribution with unknown mean, there is a unique mean consistent with the TVLL for each time t. Thus, under that condition, the TVLL provides an estimator for the unknown mean wait, given estimates of the average number in system over a subinterval and the arrival rate function. Useful variants of the TVLL estimator are obtained by fitting a linear or quadratic function to arrival data. When the arrival rate function is approximately linear (quadratic), the mean waiting time satisfies a quadratic (cubic) equation. The new estimator based on the TVLL is a positive real root of that equation. The new methods are shown to be effective in estimating the bias in the indirect estimator and reducing it, using simulations of multi-server queues and data from a call center.

scholarly journals SERVER WAITING TIMES IN INFINITE SUPPLY POLLING SYSTEMS WITH PREPARATION TIMES

2015 ◽  
Vol 30 (2) ◽  
pp. 153-184 ◽  
Author(s):  
Jan-Pieter L. Dorsman ◽  
Nir Perel ◽  
Maria Vlasiou
Keyword(s):  
Waiting Time ◽  
Queueing Networks ◽  
Waiting Times ◽  
Fixed Number ◽  
Polling Systems ◽  
Single Server ◽  
Dynamic Allocation ◽  
Mean Waiting Time ◽  
Preparation Phase ◽  
Stationary Waiting Time

We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models.

Sign in / Sign up

Export Citation Format

Share Document