Sojourn Time Approximations for a Discriminatory Processor Sharing Queue

Abstract

We study a multi-class time-sharing discipline with relative priorities known as Discriminatory Processor Sharing (DPS), which provides a natural framework to model service differentiation in systems. The analysis of DPS is extremely challenging and analytical results are scarce. We develop closed-form approximations for the mean conditional (on the service requirement) and unconditional sojourn times. The main benefits of the approximations lie in its simplicity, the fact that it applies for general service requirements with finite second moments, and that it provides insights into the dependency of the performance on the system parameters. We show that the approximation for the mean conditional and unconditional sojourn time of a customer is decreasing as its relative priority increases. We also show that the approximation is exact in various scenarios, and that it is uniformly bounded in the second moments of the service requirements. Finally we numerically illustrate that the approximation for exponential, hyperexponential and Pareto service requirements is accurate across a broad range of parameters.

Analysis of QoS in Heterogeneous Networks with Clustered Deployment and Caching Aware Capacity Allocation

“…We need to obtain mean sojourn time of DPS queue described thus far; however, the total class of users is large, and this makes deriving the exact mean sojourn time intractable. To this end, we use the best approximated mean sojourn time of DPS queue (see [14]). We assume that the traffic load of the tier is less than the critical traffic value, which is less than 1, and all weights are strictly positive, which implies the queue is stable.…”

“…We assumed that the arrival of the requests is ergodic. The approximated mean unconditional sojourn time when service time is exponentially distributed is given by using light and heavy traffic approximation order 2 (see [14]). The equation (28) follows from Little's law, and ( 29) and ( 30) are by the definition (see Proposition 1 [8]).…”

“…Mathematically, DPS queue is complex and some tractability is lost, and many analyses are limited under special circumstances [11]- [13]. Instead of using the exact analysis of the DPS queue, we use the best approximation derived by [14] to show the possible improvement of a cellular network traffic efficiency by the allocation system. To the best of our knowledge, this is the first application DPS queue while taking an account of the effect of spatial stochastic patterns of users and base stations to analyze the capacity allocation performance of HetNets.…”

Heavy-traffic limits for Discriminatory Processor Sharing models with joint batch arrivals

“…Finally, the asymptotic results form open up possibilities for the development of approximations for the queue-length and sojourn-time distributions for arbitrary values of the load, for the models batched arrivals. To this end, the elegant interpolation technique proposed in [9] forms an excellent basis.…”

“…Grishechkin [7] allows for batch arrivals and explored the relationship between Processor Sharing models and Crump-Mode-Jagers branching processes. Recently, Izagirre et al [9] proposed an approximation for the mean sojourn time in a DPS queue with Poisson arrivals and general service times by interpolating between heavy traffic and light traffic.…”