Though there are conjectures about DPS approximate insensitivity in literature, no analytical study or extensive simulation investigation is given.
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On Processor Sharing (PS) and Its Applications to Cellular Data Network Provisioning Yujing Wu, Carey Williamson , Jingxiang Luo Department of Computer Science University of Calgary
Insensitivity of network performance to the traffic details is a desirable property, since it facilitates robust traffic engineering.
Example: Erlang B call blocking formula
How about 3G cellular data networks? Are performance measures sensitive to the detailed traffic characteristics (e.g., flow size distribution, flow inter-arrival time, number of flows, correlations) or not?
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Downlink Model for a Cellular Data Network flow arrivals feasible rate of flow j at slot t realized throughput of flow j up to slot t schedule flow i at slot t propagation loss, shadowing, fast fading data flow 1 data flow 2 data flow n MS MS MS PF scheduler current feasible rate: r( i ) TDM ... ... 1.667ms frame forward link C(t) CAC
The downlink of the cellular system behaves like a PS queue with respect to the flow-level performance
With different assumptions about rate variations, the system can be abstracted to different models.
Homogenous rate variation (idealized situation): the feasible rate fluctuates around the mean for all active flows, and these fluctuations are statistically identical for all users.
Heterogeneous rate variation: the feasible rate fluctuations around the mean for active flows are statistically different. PF allocates more time to users with lower variability in the feasible rate.
EPS DPS
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Traffic model I: Poisson flow arrivals The flow size distribution is general. Poisson process
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Traffic model II: Poisson session arrivals Flows in a session session arrival epochs (Poisson process)
distribution of number of flows per session
flow size distribution
think time distribution
correlation in successive flow and think time statistics
general session structure flexibility to model more realistic traffic.
Poisson flow arrivals without blocking [Cohen; Kelly]
Poisson session traffic with infinite capacity [Bonald et al. 2001abc; Bonald 2006; Borst 2003]
We prove that the joint queue length distribution, mean number of active flows, and blocking probabilities are insensitive to the session structure in the finite-capacity EPS queue fed by Poisson session arrivals .
Model the system by a queueing network with a restricted state space.
Apply results from stochastic queueing network theory for the proof. (see paper)
Value? Assuming homogenous rate variation in the cellular system, we can replace the complicated Poisson session traffic with simple Poisson flows with exponentially distributed flow sizes. The simplified model will suffice for provisioning purposes.
Single class of traffic, but different flow weights
Finite-capacity: at most 15 concurrent flows
Two types of Poisson session traffic
Type 1: 5 flows/session (deterministic) , LN flow size (mean 2, CV 3), HyperExp thinking time (mean 1, CV 3)
Type 2: Geo dist. for flow/session (mean 10), exp dist. thinking time (mean 0.05), flow sizes being one of five dist. (Deterministic, Exp, HyperExp, LN, Pareto)
Change session details of type 2 and compare the results to those in the case where both types are Poisson flows with exponentially distributed sizes.
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DPS Simulation Results W i =[1, 2], i=1, 2 W i =[1, 10], i=1, 2
Flow details (session structure) have little impact on the first-order system performance unless the weights among different flows are highly skewed (e.g., the weight ratio is 10 or more).
In practical cellular systems, the unequal slot sharing among flows caused by PF scheduling and by heterogeneous rate variations is only modest (e.g., weight ratio is less than 2).
It is conjectured that traffic details do not affect the metrics relevant to network provisioning.
Simulate a shared downlink data channel of the central cell site surrounded by interfering cells (6 direct neighbours, and 12 outer cells).
All BSs transmit at full power on the downlink.
The channel model includes propagation loss, slow fading, and fast fading.
Flows are placed uniformly at random in the center cell, and users do not move during flow transmission. Each active flow has a time-varying SINR updated at every slot.
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Static User Scenario PF unfairness exists, but it is not extreme! BS 13.7% 14.6% 16.9% 17.9% 18.4% 18.4% Slot share 6 5 4 3 2 1 Node ID x x x x x x node 1 node 6
Deliberately treat traffic unequally at the type level (i.e., strict priority)
To what extent does the weight asymmetry among traffic types change the insensitivity property?
A DPS system with two types of Poisson flow arrivals, each with a single subclass.
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DPS with Differentiated Service Change flow size distribution of high priority traffic type Change flow size distribution of low priority traffic type
Compared to the bias among subclasses, the bias among traffic types manifests sensitivity in a much more dramatic way.
Depending on the traffic priority, variability in the flow size distribution has different impacts.
Using simple traffic models may lead to under-estimation or over-estimation of performance in the cellular system when differentiated services are deployed.
Extended the theoretical analysis of the EPS insensitivity to a new finite-capacity case.
Showed that the first-order performance of DPS systems is approximately insensitive to the session structure in relevant regime for practical parameter settings.
Simple and robust traffic engineering is possible for cellular systems using DPS for PF scheduling.
The introduction of differentiated services may pose a great challenge for future cellular network provisioning.
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