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Resource Sharing for BookAhead and InstantaneousRequest Calls
 IEEE/ACM TRANSACTIONS ON NETWORKING
, 1999
"... In order to provide an adequate quality of service to largebandwidth calls, such as video conference calls, service providers of integrated services networks may want to allow some customers to book their calls ahead, i.e., make advance reservations. We propose a scheme for sharing resources among ..."
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Cited by 61 (8 self)
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In order to provide an adequate quality of service to largebandwidth calls, such as video conference calls, service providers of integrated services networks may want to allow some customers to book their calls ahead, i.e., make advance reservations. We propose a scheme for sharing resources among bookahead (BA) calls (that announce their call holding times as well as their call initiation times upon arrival) and nonBA calls (that do not announce their holding times). It is possible to share resources without allowing any calls in progress to be interrupted, but in order to achieve a more efficient use of resources, we think that it may be desirable to occasionally allow a call in progress to be interrupted. (In practice, it may be possible to substitute service degradation, such as bit dropping or coarser encoding of video, for interruption.) Thus, we propose an admission control algorithm in which a call is admitted if an approximate interrupt probability (computed in real time) i...
Nash Equilibria for Combined Flow Control and Routing in Networks: Asymptotic Behavior for a Large Number of Users
 In Proceedings of 38th IEEE Conference on Decision and Control
, 1999
"... We consider a noncooperative game framework for combined routing and flow control in a network of parallel links, where the number of users (players) is arbitrarily large. The utility function of each user is related to the power criterion, and is taken as the ratio of some positive power of the tot ..."
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Cited by 36 (9 self)
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We consider a noncooperative game framework for combined routing and flow control in a network of parallel links, where the number of users (players) is arbitrarily large. The utility function of each user is related to the power criterion, and is taken as the ratio of some positive power of the total throughput of that user to the average delay seen by the user. The utility function is nonconcave in the flow rates of the user, for which we introduce a scaling to make it well defined as the number of users, N,be comes arbitrarily large. In spite of the lack of concavity, we obtain explicit expressions for the flow rates of the users and their associated routing decisions, which are in O(1/N ) Nash equilibrium. This O(1/N ) equilibrium solution, which is symmetric across different users and could be multiple in some cases, exhibits a delayequalizing feature among the links which carry positive flow. The paper also provides the complete optimal solution to the singleuser case, and includes several numerical examples to illustrate different features of the solutions in the single as well as Nuser cases, as N becomes arbitrarily large.
AN INTRODUCTION TO NUMERICAL TRANSFORM INVERSION AND ITS APPLICATION TO PROBABILITY MODELS
, 1999
"... ..."
Heavytraffic limits for the G/H∗ 2 /n/m queue
 Math. Oper. Res
, 2005
"... We establish heavytraffic stochasticprocess limits for queuelength, waitingtime and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit th ..."
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Cited by 28 (12 self)
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We establish heavytraffic stochasticprocess limits for queuelength, waitingtime and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit theorem. In order to capture the impact of the servicetime distribution beyond its mean within a Markovian framework, we consider a special class of servicetime distributions, denoted by H ∗ 2, which are mixtures of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. These servicetime distributions exhibit relatively high variability, having squared coefficients of variation greater than or equal to one. As in Halfin and Whitt (1981), Puhalskii and Reiman (2000) and Garnett, Mandelbaum and Reiman (2000), we consider a sequence of queueing models indexed by the number of servers, n, and let n tend to infinity along with the traffic intensities ρn so that √ n(1 − ρn) → β for − ∞ < β < ∞. To treat finite waiting rooms, we let mn / √ n → κ for 0 < κ ≤ ∞. With the special H ∗ 2 servicetime distribution, the limit processes are onedimensional Markov processes, behaving like diffusion processes with different drift and diffusion functions in two different regions, above and below zero. We also establish a limit for the G/M/n/m + M model, having exponential customer abandonments.
A diffusion approximation for the G/GI/n/m queue
 Operations Research
"... informs ® doi 10.1287/opre.1040.0136 © 2004 INFORMS We develop a diffusion approximation for the queuelength stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution, n servers, and m extra ..."
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Cited by 26 (7 self)
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informs ® doi 10.1287/opre.1040.0136 © 2004 INFORMS We develop a diffusion approximation for the queuelength stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution, n servers, and m extra waiting spaces). We use the steadystate distribution of that diffusion process to obtain approximations for steadystate performance measures of the queueing model, focusing especially upon the steadystate delay probability. The approximations are based on heavytraffic limits in which n tends to infinity as the traffic intensity increases. Thus, the approximations are intended for large n. For the GI/M/n/ � special case, Halfin and Whitt (1981) showed that scaled versions of the queuelength process converge to a diffusion process when the traffic intensity �n approaches 1 with �1 − �n � √ n → � for 0 <�<�. A companion paper, Whitt (2005), extends that limit to a special class of G/GI/n/mn models in which the number of waiting places depends on n and the servicetime distribution is a mixture of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. Finite waiting rooms are treated by incorporating the additional limit mn / √ n → � for 0 <� � �. The approximation for the more general G/GI/n/m model developed here is consistent
A Diffusion Approximation for Markovian Queue with Reneging
, 2002
"... Consider a singleserver queue with a Poisson arrival process and exponential processing times in which each customer independently reneges after an exponentially distributed amount of time. We establish that this system can be approximated by either a reflected OrnsteinUhlenbeck process or a refle ..."
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Cited by 20 (1 self)
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Consider a singleserver queue with a Poisson arrival process and exponential processing times in which each customer independently reneges after an exponentially distributed amount of time. We establish that this system can be approximated by either a reflected OrnsteinUhlenbeck process or a reflected affine diffusion when the arrival rate exceeds or is close to the processing rate and the reneging rate is close to 0. We further compare the quality of the steadystate distribution approximations suggested by each diffusion.
Control of Communication Networks
 Perspectives in Control Engineering: Technologies, Applications, New Directions
, 1999
"... this paper have been studied for the GPS policy, and for the Generalized Longest Queue First (GLQF) policy (see references in [32]). While most of these deal with the issue of computing rare buffer overflow, the similar problem for delay has been addressed in [32]. When using the effective bandwidth ..."
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Cited by 18 (1 self)
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this paper have been studied for the GPS policy, and for the Generalized Longest Queue First (GLQF) policy (see references in [32]). While most of these deal with the issue of computing rare buffer overflow, the similar problem for delay has been addressed in [32]. When using the effective bandwidthtype results as above, one should also keep in mind that these 16
Diffusion Approximations for a Single Node Accessed by CongestionControlled Sources
 IEEE Transactions on Automatic Control
, 1999
"... We consider simple models of congestion control in highspeed networks and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ONOFF type with exponential ON and OFF times, then, under a certain scaling, the steadystate distribution ..."
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Cited by 11 (5 self)
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We consider simple models of congestion control in highspeed networks and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ONOFF type with exponential ON and OFF times, then, under a certain scaling, the steadystate distribution of the number of active sources can be described by a combination of two appropriately truncated and renormalized normal distributions. For the case where the source arrival process is Poisson and the service times are exponential, the steadystate distribution consists of appropriately normalized and truncated Gaussian and exponential distributions. We then consider the case where the arrival process is a general renewal process with finite coefficient of variation and servicetime distributions that are phasetype and show the impact of these distributions on the steadystate distribution of the number of sources in the system. We also establish an insensitivity to servicetime distributi...
CALCULATING TRANSIENT CHARACTERISTICS OF THE ERLANG LOSS MODEL BY NUMERICAL TRANSFORM INVERSION
 Stochastic Models
"... In this paper we consider the classical Erlang loss model, i.e., the M/M/c/0 system with Poisson arrival process, exponential service times, c servers and no extra waiting space, where blocked calls are lost. We let the individual service rate be 1 and the arrival rate (which coincides with the offe ..."
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Cited by 10 (6 self)
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In this paper we consider the classical Erlang loss model, i.e., the M/M/c/0 system with Poisson arrival process, exponential service times, c servers and no extra waiting space, where blocked calls are lost. We let the individual service rate be 1 and the arrival rate (which coincides with the offered load) be a. We show how to compute several transient characteristics by numerical transform inversion. Transience arises by considering arbitrary fixed initial states.
Variance reduction in simulation of loss models
 Operations Research
, 1999
"... We propose a new estimator of steadystate blocking probabilities for simulations of stochastic loss models that can be much more efficient than the natural estimator (ratio of losses to arrivals). The proposed estimator is a convex combination of the natural estimator and an indirect estimator base ..."
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Cited by 10 (8 self)
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We propose a new estimator of steadystate blocking probabilities for simulations of stochastic loss models that can be much more efficient than the natural estimator (ratio of losses to arrivals). The proposed estimator is a convex combination of the natural estimator and an indirect estimator based on the average number of customers in service, obtained from Little’s law (L = λW). It exploits the known offered load (product of the arrival rate and the mean service time). The variance reduction is dramatic when the blocking probability is high and the service times are highly variable. The advantage of the combination estimator in this regime is partly due to the indirect estimator, which itself is much more efficient than the natural estimator in this regime, and partly due to strong correlation (most often negative) between the natural and indirect estimators. In general, when the variances of two component estimators are very different, the variance reduction from the optimal convex combination is about 1 − ρ 2, where ρ is the correlation between the component estimators. For loss models, the variances of the natural and indirect estimators are very different under both light and heavy loads. The combination estimator is effective for estimating multiple blocking probabilities in loss networks with multiple traffic classes, some of which are in normal