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26
Reduced Load Equivalence under Subexponentiality
, 2001
"... A+B of a queue with capacity # loaded by two independent processes A and B is investigated. When the probability of load deviation in process A decays slower than both in B and e  , we show that W A+B is asymptotically equal to the reduced load queue W A , where b is the mean rate of B. This co ..."
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Cited by 16 (3 self)
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A+B of a queue with capacity # loaded by two independent processes A and B is investigated. When the probability of load deviation in process A decays slower than both in B and e  , we show that W A+B is asymptotically equal to the reduced load queue W A , where b is the mean rate of B. This complements the known result that this property does not hold when both processes have lighter than e  deviation decay rates. Furthermore, using the same methodology, we show that under an equivalent set of conditions the results on sampling at subexponential times hold. Keywords: Large deviations, nonCramer type conditions, reduced load equivalence, independent sampling, subexponential distributions 1
Tail asymptotics for discriminatory processor sharing queues with heavytailed service requirements
 Performance Evaluation
, 2005
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Characterizing heavytailed distributions induced by retransmissions
, 2007
"... Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel availability dynamics is modeled as an i.i.d. sequence {A, Ai}i�1 that is independent of L. During each period of time that the channel becomes available, say Ai, we attempt to tra ..."
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Cited by 12 (7 self)
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Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel availability dynamics is modeled as an i.i.d. sequence {A, Ai}i�1 that is independent of L. During each period of time that the channel becomes available, say Ai, we attempt to transmit the data unit. If L ≤ Ai, the transmission is considered successful; otherwise, we wait for the next available period Ai+1 and attempt to retransmit the data from the beginning. We investigate the asymptotic properties of the number of retransmissions N and the total transmission time T until the data is successfully transmitted. In the context of studying the completion times in systems with failures where jobs restart from the beginning, it was first recognized in [5, 18] that this model results in power law and, in general, heavytailed delays. The main objective of this paper is to uncover the detailed structure of this class of heavytailed distributions induced by retransmissions. More precisely, we study how the functional dependence (P[L> x]) −1 ≈ Φ((P[A> x]) −1) impacts the distributions of N and T; the approximation ≈ will be appropriately defined
Sojourn time tails in the M/D/1 processor sharing queue
"... We consider the sojourn time V in the MIDl1 processor sharing (PS) queue, and show that P(V> x) is of the form Ce'x as x becomes large. The proof involves a geometric random sum representation of V, and a connection with Yule processes, which also enables us to simplify Ott's (1984) de ..."
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Cited by 8 (2 self)
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We consider the sojourn time V in the MIDl1 processor sharing (PS) queue, and show that P(V> x) is of the form Ce'x as x becomes large. The proof involves a geometric random sum representation of V, and a connection with Yule processes, which also enables us to simplify Ott's (1984) derivation of the Laplace transform of V. Numerical experiments show that the approximation P(V> x) ~ Ce'x is excellent even for moderate values of x.
Resource Sharing with Subexponential Distributions
, 2002
"... We investigate the distribution of the waiting time V in an M/G/1 processor sharing queue with traffic intensity #<1. This queue represents a baseline model for evaluating efficient and fair network resource sharing algorithms, e.g. TCP flow control. When the distribution of job size B belongs t ..."
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Cited by 7 (1 self)
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We investigate the distribution of the waiting time V in an M/G/1 processor sharing queue with traffic intensity #<1. This queue represents a baseline model for evaluating efficient and fair network resource sharing algorithms, e.g. TCP flow control. When the distribution of job size B belongs to a class of subexponential distributions with tails heavier than e  , it is shown that as x P[V>x]=P[B>(1 #)x](1 + o(1)).
Reducedload equivalence for queues with Gaussian input, Queueing
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"... Reducedload equivalence for queues with Gaussian ..."
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Sojourn times in the M/PH/1 processor sharing queue. Queueing Syst
"... Abstract. We give in this paper an algorithm to compute the sojourn time distribution in the processor sharing, single server queue with Poisson arrivals and phase type distributed service times. In a first step, we establish the differential system governing the conditional sojourn times probabilit ..."
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Cited by 4 (0 self)
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Abstract. We give in this paper an algorithm to compute the sojourn time distribution in the processor sharing, single server queue with Poisson arrivals and phase type distributed service times. In a first step, we establish the differential system governing the conditional sojourn times probability distributions in this queue, given the number of customers in the different phases of the PH distribution at the arrival instant of a customer. This differential system is then solved by using a uniformization procedure and an exponential of matrix. The proposed algorithm precisely consists of computing this exponential with a controlled accuracy. This algorithm is then used in practical cases to investigate the impact of the variability of service times on sojourn times and the validity of the socalled reduced service rate (RSR) approximation, when service times in the different phases are highly dissymmetrical. For twostage PH distributions, we give conjectures on the limiting behavior in terms of an M/M/1 PS queue and provide numerical illustrative examples.
Tail asymptotics for cumulative processes sampled at heavytailed random times with applications to queueing models in Markovian environments
 Journal of the Operations Research Society of Japan
, 2013
"... Abstract This paper considers the tail asymptotics for a cumulative process {B(t); t ≥ 0} sampled at a heavytailed random time T. The main contribution of this paper is to establish several sufficient conditions for the asymptotic equality P(B(T)> bx) ∼ P(M(T)> bx) ∼ P(T> x) as x → ∞, wh ..."
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Cited by 2 (2 self)
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Abstract This paper considers the tail asymptotics for a cumulative process {B(t); t ≥ 0} sampled at a heavytailed random time T. The main contribution of this paper is to establish several sufficient conditions for the asymptotic equality P(B(T)> bx) ∼ P(M(T)> bx) ∼ P(T> x) as x → ∞, where M(t) = sup0≤u≤tB(u) and b is a certain positive constant. The main results of this paper can be used to obtain the subexponential asymptotics for various queueing models in Markovian environments. As an example, using the main results, we derive subexponential asymptotic formulas for the loss probability of a singleserver finitebuffer queue with an on/off arrival process in a Markovian environment.