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Large Deviations and the Generalized Processor Sharing Scheduling: Upper and Lower Bounds  Part I: TwoQueue Systems
 Queueing Systems
, 1995
"... We prove asymptotic upper and lower bounds on the asymptotic decay rate of persession queue length tail distributions for a single constant service rate server queue shared by multiple sessions with the generalized processor sharing (GPS) scheduling discipline. The simpler case of a GPS system with ..."
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Cited by 31 (1 self)
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We prove asymptotic upper and lower bounds on the asymptotic decay rate of persession queue length tail distributions for a single constant service rate server queue shared by multiple sessions with the generalized processor sharing (GPS) scheduling discipline. The simpler case of a GPS system with only two queues needs special attention, as under this case, it is shown that the upper bounds and lower bounds match, thus yielding exact bounds. This result is established in this part (Part I) of the paper. The general case is much more complicated, and is treated separately in Part II of the paper [42], where tight upper and lower bound results are proved by examining the dynamics of bandwidth sharing nature of GPS scheduling. The proofs use samplepath large deviation principle and are based on some recent large deviation results for a single queue with a constant service rate server. These results have implications in call admission control for highspeed communication networks. 1 Int...
Effective Bandwidths of Departure Processes from Queues with Time Varying Capacities
 In Proceedings of IEEE INFOCOM'95
, 1995
"... In this paper, we consider a queue with a time varying capacity and identify the effective bandwidth of the stationary departure process from such a queue. Two important observations are made: (i) the effective bandwidths for the transient departure process and the stationary departure process from ..."
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Cited by 29 (1 self)
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In this paper, we consider a queue with a time varying capacity and identify the effective bandwidth of the stationary departure process from such a queue. Two important observations are made: (i) the effective bandwidths for the transient departure process and the stationary departure process from such a queue are in general different, and (ii) sometimes it is necessary to build up the queue first in order to have a large excursion of the stationary departure process. The new result on the effective bandwidth of the stationary departure process is applied to intree networks with time varying capacities and priority tandem queues. Algorithms for approximating the tail distributions of queue lengths in such networks are derived.
A Large Deviation Principle with Queueing Applications
, 1997
"... In this paper we present a large deviation principle, for partial sums processes indexed by the half line, which is particularly suited to queueing applications. ..."
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Cited by 27 (4 self)
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In this paper we present a large deviation principle, for partial sums processes indexed by the half line, which is particularly suited to queueing applications.
Effective bandwidths with priorities
 IEEE/ACM Transactions on Networking
, 1998
"... Abstract — The notion of effective bandwidths has provided a useful practical framework for connection admission control and capacity planning in highspeed communication networks. The associated admissible set with a single linear boundary makes it possible to apply stochasticlossnetwork (general ..."
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Cited by 27 (1 self)
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Abstract — The notion of effective bandwidths has provided a useful practical framework for connection admission control and capacity planning in highspeed communication networks. The associated admissible set with a single linear boundary makes it possible to apply stochasticlossnetwork (generalizedErlang) models for capacity planning. In this paper we consider the case of network nodes that use a priorityservice discipline to support multiple classes of service, and we wish to determine an appropriate notion of effective bandwidths. Just as was done previously for the firstin firstout discipline, we use largebuffer asymptotics (large deviations principles) for workload tail probabilities as a theoretical basis. We let each priority class have its own buffer and its own constraint on the probability of buffer overflow. Unfortunately, however, this leads to a constraint for each priority class. Moreover, the largebuffer asymptotic theory with priority classes does not produce an admissible set with linear boundaries, but we show that it nearly does and that a natural bound on the admissible set does have this property. We propose it as an approximation for priority classes. Then there is one linear constraint for each priority class. This linearadmissibleset structure implies a new notion of effective bandwidths, where a given connection is associated with multiple effective bandwidths: one for the priority level of the given connection and one for each lower priority level. This structure can be used regardless of whether the individual effective bandwidths are determined by largebuffer asymptotics or by some other method. 1
Asymptotic Buffer Overflow Probabilities in Multiclass Multiplexers: An Optimal Control Approach
 IEEE Trans. Automatic Control
, 1997
"... We consider a multiclass multiplexer with support for multiple service classes, and dedicated buffers for each service class. Under specific scheduling policies for sharing bandwidth among these classes, we seek the asymptotic (as the buffer size goes to infinity) tail of the buffer overflow probabi ..."
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Cited by 26 (1 self)
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We consider a multiclass multiplexer with support for multiple service classes, and dedicated buffers for each service class. Under specific scheduling policies for sharing bandwidth among these classes, we seek the asymptotic (as the buffer size goes to infinity) tail of the buffer overflow probability for each dedicated buffer. We assume dependent arrival and service processes as is usually the case in models of bursty traffic. In the standard large deviations methodology, we provide a lower and a matching (up to first degree in the exponent) upper bound on the buffer overflow probabilities. We introduce a novel optimal control approach to address these problems. In particular, we relate the lower bound derivation to a deterministic optimal control problem, which we explicitly solve. Optimal state trajectories of the control problem correspond to typical congestion scenarios. We explicitly and in detail characterize the most likely modes of overflow. We specialize our results to the ...
Effective Bandwidth in High Speed Digital Networks
 IEEE Journal on Selected Areas in Communications
, 1999
"... The theory of large deviations provides a simple unified basis for statistical mechanics, information theory and queueing theory. The objective of this paper is to use large deviation theory and the Laplace method of integration to provide an simple intuitive overview of the recently developed theor ..."
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Cited by 18 (5 self)
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The theory of large deviations provides a simple unified basis for statistical mechanics, information theory and queueing theory. The objective of this paper is to use large deviation theory and the Laplace method of integration to provide an simple intuitive overview of the recently developed theory of effective bandwidth for high speed digital networks, especially ATM networks. This includes (i) identification of the appropriate energy function, entropy function and effective bandwidth function of a source, (ii) the calculus of the effective bandwidth functions, (iii) bandwidth allocation and buffer management, (iv) traffic descriptors, and (v) envelope processes and conjugate processes for fast simulations and bounds.
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
On the Large Deviations Behaviour of Acyclic Networks of G/G/1 Queues
 Ann. of Appl. Prob
, 1994
"... We consider a single class, acyclic network of G/G/1 queues. We impose some mild assumptions on the service and external arrival processes and we characterize the large deviations behaviour of all the processes resulting from various operations in the network. For the network model that we are consi ..."
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Cited by 14 (3 self)
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We consider a single class, acyclic network of G/G/1 queues. We impose some mild assumptions on the service and external arrival processes and we characterize the large deviations behaviour of all the processes resulting from various operations in the network. For the network model that we are considering, these operations are passingthroughasingle serverqueue (the process resulting from this operation being the departure process), superposition of independent processes, and deterministic splitting of a process to a number of processes. We also characterize the large deviations behaviour of the waiting time and the queue length observed by a typical customer in a single server queue. We prove that the assumptions imposed on the external arrival processes are preserved by these operations, and we show how to inductively apply these results to obtain the large deviations behaviour of the waiting time and the queue length in all the queues of the network. Our results indicate how thes...
Functional Large Deviation Principles for Waiting and Departure Processes
 Prob. Engin. Info. Sci
, 1998
"... We establish functional large deviation principles (FLDPs) for waiting and departure processes in singleserver queues with unlimited waiting space and the firstin firstout service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space ..."
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Cited by 13 (4 self)
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We establish functional large deviation principles (FLDPs) for waiting and departure processes in singleserver queues with unlimited waiting space and the firstin firstout service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space D with one of the nonuniform Skorohod topologies whenever the arrival and service processes obey FLDPs and the rate function is finite for appropriate discontinuous functions. We apply our previous FLDPs for inverse processes to obtain an FLDP for the waiting times in a queue with a superposition arrival process. We obtain FLDPs for queues within acyclic networks by showing that FLDPs are inherited by processes arising from the network operations of departure, superposition and random splitting. For this purpose, we also obtain FLDPs for split point processes. For the special cases of deterministic arrival processes and deterministic service processes, we obtain convenient explicit express...