Asymptotic results for multiplexing subexponential on-off processes

Abstract

Consider an aggregate arrival process A N obtained by multiplexing N On-Off processes with exponential Off periods of rate λ and subexponential On periods τ on . As N goes to infinity, with λN → Λ, A N approaches an M/G/∞ type process. Both for finite and infinite N , we obtain the asymptotic characterization of the arrival process activity period.Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process A ∞ t and capacity c. When On periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable Q P t observed at the beginning of the arrival process activity periodswhere ρ = EA ∞ t < c; r (c ≤ r) is the rate at which the fluid is arriving during an On period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions on On periods than regular variation.In addition, we analyze a queueing system in which one On-Off process, whose On period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate Ee t . This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value Ee t .

Long-run performance analysis of a multi-scale TCP traffic model

“…where factor b represents the coefficient part on the righthand side of either (5) or (8). The similarity of this result with those from the on-off model and the M/G/N model [6] suggests the persistence of the power-law queueing behaviour in various traffic environments. Compared with the latter two, this result displays the roles of different traffic elements.…”

“…Proof: It is straightforward by comparing the level I of the HOO process with definition 3. & It should be pointed out that in previous literature about scaling traffic modelling the M/G/N model is primarily used as the limiting process of the aggregation of on-off flows [6,8]. Proposition 1 indicates that it applies in a more natural sense as a profile of the connection level structure.…”

“…Proof: From lemma 2 and definition 4, the queue tail of a HOO process is the same as that of its M/G/N companion process with r ¼ r on g 1 g 2 . For a queue fed by an M/G/N process, if lmo1, r4c, and the distribution of t is FAV, the following relation holds [6]:…”

“…Traffic models proposed with regard to scaling behaviours largely fall into two categories. One large family of models are based on the on-off model and view the traffic as the aggregation of many on-off flows [1,6,7]. In the other category are more or less mathematically driven models that are intended to approximate the statistics of the traffic rather than mimic the structure of it.…”

The effect of multiple time scales and subexponentiality in MPEG video streams on queueing behavior

“…However, there is no explicit algorithm to extract model parameters from the second-order statistics of an empirical trace. (Precise fluid queue asymptotics with arrivals were obtained in [13], [2], and [25].) Fourth, in the presence of long-tailed renewal times, Theorem 2 (see Section IV-B) further justifies the usage of the fluid type models.…”