Quality of Real-Time Streaming in Wireless Cellular Networks : Stochastic Modeling and Analysis

Transcription

1 Quality of Real-Time Streaming in Wireless Cellular Networs : Stochastic Modeling and Analysis B. Blaszczyszyn, M. Jovanovic and M. K. Karray Based on paper [1] WiOpt/WiVid Mai 16th, 2014

2 Outline Introduction Model System metrics User metrics Wireless cellular networs Numerical results Conclusion

3 Introduction Objective : build an analytical approach for the evaluation of the quality of real-time streaming service (video call, TV on mobile devices etc.) perceived by the users in wireless cellular networs New model for real-time streaming based on queueing theory comprises Poisson call arrivals and arbitrarily distributed call durations Outage periods do not alter the call sojourn times in the system We evaluate the ey performance characteristics of the new service model based only on stationary probabilities of the (free) traffic demand process

4 Model User classes Consider J 1 classes of users characterized by different radio conditions SINR 6 SINR 1 SINR 2 SINR 5 SINR 3 SINR 4

5 Model Traffic demand Users of class arrive in time according to a Poisson process with intensity λ > 0 and stay in the system for independent requested streaming times having some general distribution with mean 1/µ < Number of users in each class X(t) = (X 1 (t),..., X J (t)) called user configuration at time t Stationary distribution π of X(t) : distribution of a vector (X 1,..., X J ) of independent Poisson random variables with means ρ := λ µ, = 1, 2,..., J called traffic demand for class

6 Model Resource constraints and outage policy Users requested streaming times are not altered by the eventual outages of their services For class, let F N J be a subset of user configurations such that users of class are served if and only if X(t) F Its complement F is called the th outage set No user departure can cause outage of any class However a user departure may mae some class switch from outage to service Example in wireless cellular networs F = x = (x 1,..., x J ) N J : ϕ j x j 1 j=1 for some coefficients ϕ j (will be specified later)

7 System metrics User configuration X(t) alternates visits in F and F Notation B = n Z ɛ τ n point process counting exit instants from F σ n duration of the nth visit to F σ n duration of the nth visit to F X(t) F F F F F F σ σ 1 σ1 0 σ0 σ 2 τ0 0 τ1 τ2 t

8 System metrics Intensity of outage incidents 1 Λ := lim T T B ([0, T [) Mean service time between two outage incidents Mean outage duration 1 σ := lim N N σ 1 := lim N N N n=1 N n=1 σ n σ n

9 System metrics Intensity of outage incidents Λ = J j=1 λ j π { x F, x + ε j F } (1) Proof : N := J j=1 N j point process of all arrival timess. By Campbell s formula [2, Eq (1.2.19)] [ ] Λ = E 1 F F (X (t ), X (t)) N (dt) [0,1) = λp 0 { N X(0 ) F, X(0) F } where P 0 N Palm probability associated to N. Then invoe PASTA (Poisson Arrivals See Time Averages) property [2, Eq (3.3.4)]

10 System metrics Mean service time between two outage incidents and mean outage duration σ = π (F ) Λ, σ := π(f ) Λ Proof : By ergodicity [2, Eq (1.6.8) ] σ = E 0 B [σ 0 By the inverse formula of Palm calculus [2, Eq (1.2.27)] ] E 0 B [τ0 = 1/Λ Applying the mean value formula [2, Eq (1.3.2)] [ ] π(f ) = E0 B σ [ ] [ 0 ] E 0 = Λ B τ E 0 B σ0 = Λ σ 0 ]

11 User metrics Notation N = {Tn : n N } be arrival times {W n : n N } requested streaming times X(t) F F F F F F W 2 W 1 0 T 1 T 2 t

12 User metrics P : probability of outage at the arrival epoch 1 P = lim N N N 1 F (X(Tn )) n=1 D : mean total time in outage per call of type 1 D = lim N N N [Tn,Tn +Wn[ 1 F (X(t)) dt n=1 M : mean number of outage incidents per call of type 1 M = lim N N N [) B (]Tn, Tn + Wn n=1

13 User metrics Probability of outage at the arrival epoch P = π { x + ε F } Proof :By ergodicity P = P 0 N { X(0) F } By PASTA property the configuration of users X(0 ) has distribution π. Once the user enters the system, the users configuration becomes X(0 ) + ε, whence the result

14 User metrics Mean total time in outage Proof : By ergodicity D = 1 π { x + ε F } P = (2) µ µ [ D = E 0 N [0,W0 [ 1 F (X(t)) dt Let Y(t) := X(t) ε 1 [T 0,T0 +W 0 [ (t). By Slivnya theorem [3, Th 1.4.8], distribution of {Y(t)} under P 0 N equals distribution of {X(t)} under P. Moreover, under P 0 N, W0 and Y(t) are independent D = 0 ] ] E 0 N [1 [0,W 0 [ (t)1 F (Y(t) + ε ) dt = 1 µ π { x + ε F )]

15 User metrics Mean number of outage incidents M = 1 J λ j π { x + ε F, x + ε + ε j F } µ Proof : By ergodicity j=1 M = E 0 N [ B (]0, W 0 [)] Note that B counts exit epochs of Y(t) from F = {x : x + ε F }, then [ [) ] M = E 0 N B (]0, W0 W0 = w P W (dw) 0 = E 0 N [B (]0, w[)] P W (dw) 0 = E [B (]0, w[)] P W0 (dw) = Λ µ where B counts exit epochs of X(t) from F and Λ its intensity given by (1) with F replaced by F

16 Wireless cellular networs Wireless resource constraint r : bit-rate requested by the users of class : maximal bit-rate of a user of class r max achievable when user is served alone by the base station In AWGN r max = W log(1 + SINR ) where W is the frequency bandwidth Let ν be the portion of resource (time or frequency) allocated to class, the number of served users x satisfies x r ν r max Since J =1 ν = 1, we get the resource constraint where ϕ := r /r max J x ϕ 1 =1 called resource demand

17 Wireless cellular networs Service policies If the requested bit-rates are not achievable then some classes of users will be temporarily put in outage i.e., they will receive some smaller bit-rates If best-effort bit-rates vanish, then users are in deep-outage Assume (without loss of generality) that ϕ 1 < ϕ 2 <... < ϕ J Family of service polices for which classes with smaller resource demands have higher priority : for given δ F δ = { x = (x 1,..., x J ) N J : j=1 ϕ jx j + ϕ J j=+1 x j1 {ϕ j ϕ (1 + δ)} 1 } called least-effort-served-first policy with δ-margin (LESF(δ) for short) δ = 0 : optimal, serve the maximal subset of users δ = : practical,currently implemented 0 < δ < : intermediate, suboptimal

18 Wireless cellular networs Service policies Class is served at requested bit-rate iff X (t) F δ Let { } K = max : X (t) F δ C = K j=1 ϕ jx j (t) 1 fraction of server capacity consumed by users which are not in outage Note that 1 C ϕ K J j=k+1 X j (t) 1 {ϕ j ϕ K (1 + δ)} i.e. we may allocate service capacity ϕ K for all users in outage in classes whose service demand exceeds ϕ K by no more than δ 100% If > K and ϕ (1 + δ)ϕ K then r = r max 1 C J j=k+1 X j (t) 1 {ϕ j ϕ K (1 + δ)}

19 Wireless cellular networs Mean throughput during the typical call of class T = r (1 P ) + T where { }] T [r = E (X + ε ) 1 X + ε / F δ Proof : Similar to proof of (2). By ergodicity [ ] W T = µ E 0 0 ( { } { }) N r 1 X(t) F δ + r (X(t))1 X(t) F δ dt 0 Let Y(t) := X(t) ε 1 [T 0,T0 +W 0 [ (t). By Slivnya theorem [3, Th 1.4.8], distribution of {Y(t)} under P 0 N equals distribution of {X(t)} under P. Moreover, under P 0 N, W0 and Y(t) are independent T = r π { } [ { }] X + ε F δ + E r (X + ε ) 1 X + ε / F δ

20 Wireless cellular networs Numerical methods Let F δ (t) := P j=1 X δ, j ϕ j t where X δ, j = X j for j = 1,..., 1 and X δ, = J j= X j 1 {ϕ j ϕ (1 + δ)} Probability of outage at arrival Mean total time in outage P = 1 F δ (1 ϕ ) D = P µ Mean number of outage incidents M = 1 J ] λ j [F δ µ (1 ϕ ) F δ (1 ϕ ϕ j ) j=1

21 Wireless cellular networs Numerical methods Laplace transform of F δ(t) L δ (θ) = 1 θ exp where ρ δ, j ρ δ, j=1 ρ δ, j = ρ j for j = 1,..., 1 and = J j= ρ j 1 {ϕ j ϕ (1 + δ)} ( e θϕ j 1) F δ (t) may be retreived from its Laplace transform using Abate and Whitt algorithm [4, Equation (15)] F δ (t) 2 m e A 2 t m =0 ( ) n+ m ( 1) l Re [ L δ ( A+2iπl )] 2t {l = 0} l=0 with a typical choice A = 18.4, n = 15, m = 11

22 Wireless cellular networs Numerical methods Mean number of outage incidents with M = F δ (1 ϕ ) µ J λ j b (j) b (j) = F δ (1 ϕ ) F δ (1 ϕ ϕ j ) F δ (1 ϕ = P (X F, X + ε j / F) ) P (X F) { where F = X R J : } j=1 Xδ, j ϕ j 1 ϕ j=1 The above expression may be seen as the blocing probability for class j in a classical multi-class Erlang loss system with the admission condition X F b ( ) may be calculated by using the Kaufman-Roberts algorithm [5, 6]

23 Numerical setting CDF of SINR obtained from simulation compliant with the 3GPP recommendation in the so-called calibration case [7, Figure A.2.2-1(right)] CDF SINR [db]

24 Numerical setting All calls require the same streaming rate r = 256 bit/s and have the same streaming (sojourn) time distribution Spatially uniform traffic demand : 900 Erlang/m 2 Lin performance where r max = γw log(1 + SINR ) W = 10MHz is the frequency bandwidth γ = 0.5 accounts for practical codes performance compared to ultimate Shannon s bound

25 Numerical results Outage times Mean fraction of the requested streaming time spent in outage, µ D as function of the SINR value characterizing class Fraction of time in outage Optimal policy; δ=0 Itermediate policies δ=0.5 δ=1 δ=2 δ=4 Fair policy; δ= SINR [db]

26 Numerical results Number of outage incidents Mean number of outage incidents per service time, M as function of the SINR value characterizing class Number of outage incidents Optimal policy; δ=0 Itermediate policies δ=0.5 δ=1 δ=2 δ=4 Fair policy; δ= SINR [db]

27 Numerical results User s throughput Mean total throughput T normalized to its maximal value 256bit/s obtained during the service time (upper curves) and its fraction T obtained when a user is in outage (lower curves) Normalized througput Optimal policy (δ= 0) service time outage Intermediate policies: δ=0.5, service time δ=0.5, outage δ=1, service time δ=1, outage δ=2, service time δ=2, outage δ=4, service time δ=4, outage Fair policy (δ= ) service time outage SINR [db]

28 Conclusion A real-time streaming (RTS) traffic, as e.g. mobile TV, is analyzed in the context of wireless cellular networs An adequate stochastic model is proposed to evaluate user performance metrics, such as duration and number of outage periods in function of user radio conditions Despite some fundamental similarities to the classical Erlang loss model : performance expressed in terms of the stationary probabilities of the traffic demand process, a new model was required for this type of service, since the service denials are not definitive for a given call, but only temporal

29 Conclusion Our model allows to tae into account realistic implementations of the RTS service, e.g. in the LTE networs We identify and evaluate some natural parametric class of service policies between an optimal and practical one Several numerical demonstrations are given, presenting the quality of service metrics in function of user radio conditions Future wor : non-real time streaming evaluation

30 Bibliography [1] B. B laszczyszyn, M. Jovanovic, and M. K. Karray, Quality of real-time streaming in wireless cellular networs : Stochastic modeling and analysis, To appear IEEE Trans. Wireless Commun., [2] F. Baccelli and P. Brémaud, Elements of queueing theory. Palm martingale calculus and stochastic recurrences. Springer, [3] F. Baccelli and B. B laszczyszyn, Stochastic Geometry and Wireless Networs, Volume I Theory, ser. Foundations and Trends in Networing. NoW Publishers, 2009, vol. 3, No 3 4. [4] J. Abate and W. Whitt, Numerical Inversion of Laplace Transforms of Probability Distributions, ORSA Journal on Computing, vol. 7, no. 1, [5] J. Kaufman, Blocing in a shared resource environment, IEEE Trans. Commun., vol. 29, no. 10, pp , [6] J. Roberts, A service system with heterogeneous user requirements, in Performance of Data Communications Systems and their Applications (edited by G. Pujolle), [7] 3GPP, TR V900 Further advancements for E-UTRA - Physical Layer Aspects, in 3GPP Ftp Server, 2010.