Multimedia Communication Services Traffic Modeling and Streaming

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1 Multimedia Communication Services Medium Access Control algorithms Aloha Slotted: performance analysis with finite nodes Università degli Studi di Brescia A.A. 2014/2015 Francesco Gringoli Master of Science in Communication Technologies and Multimedia

2 Slotted Aloha: analysis Node number m finite, fresh arrivals are Poisson, time-unit rate λ Use DTMC to describe number n of Backlogged, slot duration is T Not Backlogged nodes m-n: transmit fresh packets arrived in previous slot No fresh packets at idle node: Fresh packets at idle node: ( ) = e λt / m P k = 0 q a =1 e λt / m Backlogged nodes transmit independently with probability P(trx) = q r Probability i not Backlogged nodes transmit when n are Backlogged : Q a (i, n) Probability j Backlogged nodes (re)transmit when n are Backlogged : Q r (j, n) 2

3 Slotted Aloha: analysis/2 Q a (i, n) and Q r (i, n) are binomial r.v. Let p probability event happens: on n attempts it happens k times with prob. P( k) = n p k ( 1 p) n k n, P( k) =1 k By replacing: [ ] = kp( k) E x Q a n k =0 = np ( i,n) = m n 1 q i a ( ) = n i Q r i,n 1 q r k =0 ( ) m n i q a i ( ) n i q r i 3

4 Slotted Aloha: analysis/3 State n changes when n n - 1 : no fresh arrivals, only one retransmission n n : two cases Single fresh arrival, no retransmission (fresh packet transmitted) No fresh arrival, no retransmission or at least two n n + 1 : one single fresh arrival and retransmission(s) n n + i : fresh arrivals are i 2 (don t care retransmissions) Probabilities are: P n,n 1 = Q a ( 0,n)Q r 1,n P n,n = Q a ( 1,n )Q r 0,n P n,n +1 = Q a 1,n P n,n +i = Q a ( i,n),i 2 ( ) ( ) + Q a ( 0,n) 1 Q r ( 1,n ) ( ) 1 Q r ( 0,n) [ ] [ ] Note: event fresh arrival and one retransmission does not imply transition n n 4

5 Slotted Aloha: analysis/4 One step transition matrix: from state n to State n-1, one Backlogged node turns idle State n, Backlogged nodes do not change State m > n, new nodes turned Backlogged P 00 0 P 02 P 03 NOTE: P 01 = 0, why? P 10 P 11 P 12 P 13 P =,P 0 P 21 P 22 P 23 ij = P[ n new state = j n = i] 0 0 P 32 P 33 From matrix P to asymptotic distribution n +1 p n = p i P in, p n =1 i= 0 m n= 0 5

6 Slotted Aloha: results and notes Average number of Backlogged nodes: N = Average delay (from Little s Theorem) To limit delay after collisions m n= 0 np n T = N /λ Retransmission probability q r should be high If λ low and few collisions: OK! Retransmissions are successful Average number of retransmissions depends on state n: q r n If q r n >> 1 too many collisions in future slots!! Network collapse! 6

7 Slotted Aloha: results and notes/2 New arrivals on not Backlogged nodes on average N n = ( m - n ) q a Retransmissions on Backlogged nodes on average R n = nq r Probability that one (re)transmission is successful Only one fresh arrival and no retransmissions No fresh arrivals and only one retransmission P sn = Q a ( 1,n )Q r ( 0,n) + Q a ( 0,n)Q r ( 1,n ) Why? On average, transmission rate is exactly P s n Variation/ drift D n of Backlogged nodes D n = N n P sn Increases with fresh traffic, decreases with delivered traffic = ( m n)q a P sn Attempted number of (re)transmissions Includes fresh and retransmissions G n = N n + R n = ( m n)q a + nq r 7

8 Slotted Aloha: results and notes/3 Note: P s n, D n and G n depend on n (number of Backlogged nodes) By replacing Q a and Q r into P s n : ( )( 1 q a ) m n 1 q a ( 1 q r ) n + ( 1 q a ) m n n( 1 q r ) n 1 q r = P sn = m n 1 1 = ( m n)q a + nq r 1 q a 1 q a 1 q r (( m n)q a + nq r )e q a ( m n) q r n = G( n)e G n where approximation is good if q a and q r are small ( ) m n ( 1 q r ) n ( ) Number of transmissions (attempts) well approximated by Poisson G(n) Note: number of transmission attempts increases with n ( Backlogged ) 8

9 Slotted Aloha: results and notes/4 How to plot/represent P s n, D n and G n With q a, q r and m known, compute them as a function of n ( Backlogged ) P sn ( ) ( ) ( ) = P sn [q a,q r,m] n D n = D s[qa,q r,m] n G n = G s[qa,q r,m] n G n = N n + R n = ( m n)q a + nq r By varying n in [0, m] plot m + 1 points [G n, P s n ] as a curve It s Transmission Rate P s n as a function of Attempted Rate G n Plot m + 1 points [G n, N n ] = [(m - n)q a + nq r, (m-n)q a ] on the same graph It s New arrivals rate as a function of Attempted Rate G n (this is a line) Network can deliver packets if New arrivals rate N n < Transmission Rate P s n 9

10 Slotted Aloha: results and notes/5 Blue curve from simple analysis (infinite nodes, m = ): rate = Ge -G Note: q a must be chosen so that m q a < P s max Here we have m = 20, q a = 0.01, q r = 0.4 Transmission rate P s New arrival rate Attempted rate G n 10

11 Slotted Aloha: instability Drift D n can be easily evaluated : D n = N n - P s = (m - n) q a - P s Transmission rate P s Where Drift is zero, steady state: how many? Depends on q r Steady states: D n = 0 s 1 s 2 q r = 0.4 Transmission rate P s s 1 q r = 0.2 s 3 Attempted rate G n Attempted rate G n If they are three, the one in the middle is unstable! (why?) 11

12 Slotted Aloha: instability/2 Threshold T for q r : one or three steady states If q r < T only one steady state s 1 on the left If q r > T two more states on the right s 2 and s 3 (that at center unstable) 1. Start with low q r : only state s 1 on the left If network works (regime) at its left Drift is positive (in fact, D n = N n P s > 0 ), fresh traffic not delivered More nodes get Backlogged, G n increases, regime moves to the right If regime at its right Drift is negative (in fact, D n = N n P s > 0 ), retransmissions successful Less nodes Backlogged (turn idle), G n decreases, regime moves to the left Steady state s 1 is stable: If regime perturbed, soon it returns to s 1 12

13 Slotted Aloha: instability/3 Threshold T for q r : one or three steady states If q r < T only one steady state s 1 on the left If q r > T two more states on the right s 2 and s 3 (that at center unstable) 2. Increase q r : three steady states s 1, s 2 and s 3 State s 1 as before State s 2, if regime perturbed to the left, drift negative, G n tend to decrease, regime moves to s 1 to the right, drift positive, G n tend to increase, regime moves to s 3 State s 2 is unstable State s 3, if regime perturbed to the left, drift > 0, regime returns to s 3 State s 3 is stable 13

14 Slotted Aloha: instability/4 Below threshold, too few retransmissions When collision occurs, some nodes get Backlogged (A few) nodes remain Backlogged for a long time (low q r ) Nodes tend to be idle (unless they have some fresh arrival) Network works with low P s Above threshold, too many retransmissions When collision occurs, Backlogged nodes retransmit immediately Soon, all nodes are Backlogged Network prone to work on the rightmost state with low P s Close to threshold, network oscillates between n = 0 and n = m 14

15 Slotted Aloha: instability/5 When above threshold Increasing q r moves s 2 to the left Behavior gets worse, even less Backlogged nodes to escape from s 3 In general, average transmission delay decreases with q r 15

16 Slotted Aloha: instability/6 Example: m = 25 λ max = 1 e 0.36 Choosing q a greater than that is useless, maximum throughput can not deliver traffic q a < λ max m q a = 0.01 Choose q r = 0.2 Only one steady state (stable, s 1 ). P s is high Transmission rate P s Probability # Backlogged # Backlogged 16

17 Slotted Aloha: instability/7 By increasing q r, system tends to oscillate: # Backlogged Probability Hundreds of steps Oscillates between high P s and Congestion Never stops at unstable state Transmission rate P s # Backlogged Both sides highly probable # Backlogged 17

18 Slotted Aloha: instability/8 Above threshold, if unstable state s 2 exceeded, works in s 3 with P s = 0 # Backlogged Probability Hundreds of steps Transmission rate P s # Backlogged # Backlogged 18

19 Slotted Aloha: how to make it stable When n is small, choose high q r P s is high (good!), average delay low (very good!) Problem: what if system moves to s 3? When n is big, choose low q r Drift turns negative, n decreases system escapes from congestion! It s a sort of BackOff Exponential There is no agreement between nodes, fix q r small for as long as the (single) node is Backlogged. # Backlogged Hundreds of steps 19

20 Slotted Aloha: with Infinite Nodes By increasing the number of nodes Fresh traffic q a must be decreased (remember about maximum throughput!) It is already a limiting hypothesis Threshold for q r decreases To have stability, delay must be increased Network tends to become unusable! Change hypothesis: from no-buffering to infinite nodes Attempted rate is G(n) = λ + nq r and Drift is D n = λ - P s Fresh traffic (red line, straight) becomes horizontal (check slide 13) There is no asymptotic distribution, number of Backlogged nodes is huge 20

21 Slotted Aloha: Comparison DTMC vs m = m = 10 nodes m = 50 nodes 21

22 Multimedia Communication Services Medium Access Control algorithms Simulation of AlohaNet (slotted) with finite number of nodes Università degli Studi di Brescia A.A. 2014/2015 Francesco Gringoli Master of Science in Communication Technologies and Multimedia

23 Aloha-slotted, simulation of finite network {method 1} Easy to do: all events are synchronous Do not need discrete event simulator: at every t = jδt compute evolution Method based on DTMC: simulate ONLY state n evolution Keep as less state as possible: n number of Backlogged nodes Once statistics of n available, compute network efficiency and throughput Fix number of nodes m, q a and q r and start with n = 0 Compute matrix P of the DTMC that rules the evolution (it s m + 1 * m + 1) Repeat N times the following experiment to extract the new state: Choose X, r.v. uniform in [0,1] Partition [0,1] with probabilities from row n + 1 of P {P n,0, P n,1 } 0 P n,0 P n,1 P n,2 P n,3 1 X next n=2 23

24 Aloha-slotted, simulation of finite network {method 1}/2 Some implementation notes for MATLAB To compute DTMC matrix use a function like this one function M = slalohamatrix(m,qa,qr)! % create empty matrix! M = zeros([m+1 m+1]);! % compute all elements in the matrix! for id = 1:m+1,! for jd = id-1:m+1,! if ( jd == 0 )! continue;! end;! n = id-1; % this is state n = i-1! % four cases! if ( id == jd ) % from n stay in n! M(id,jd) = q(m-n,1,qa)*q(n,0,qr) + q(m-n,0,qa)*(1-q(n,1,qr));! 24

25 Aloha-slotted, simulation of finite network {method 1}/3 Some implementation notes for MATLAB Use separated function to compute binomial probabilities Q a e Q r function prob = q(n,i,p);! if( n < i )! prob = 0;! else! prob = nchoosek(n,i) * p^i * (1-p)^(n-i);! end;! 25

26 Aloha-slotted, simulation of finite network {method 1}/4 Some implementation notes for MATLAB Main loop:! % DTMC matrix in M! steps = ;! n = 0;! ntrace = zeros([1 steps]);! for step = 1:steps,! levels = cumsum(m(n+1, :));! n = sum(rand > levels);! ntrace(step) = n;! end;! 26

27 Aloha-slotted, simulation of finite network {method 1}/5 Some implementation notes for MATLAB For stable network, two matrixes M 1 (high q r1 ) and M 2 (low q r2 ) If n m/2 use matrix M 1 to extract next state If n > m/2 use matrix M 2 Precompute M 1 and M 2! Otherwise very slow code About fresh traffic q a : choose it to have a working network Base on maximum throughput for an infinite network Choose q r1 freely Choose q r2 according to stationary state analysis Only state s 1 should appear 27

28 Aloha-slotted, simulation of finite network {method 2} Simulate independent stations State: for each station just maintain Backlog status (yes/no) Fix m, q a and q r + initialize all stations to not Backlogged Repeat N times this experiment For a station not backlogged: Partition [0,1] in S={1 arrival, 0 arrivals} S={[0,q a ),[q a,1]} NOTE P[n = 1] = q a!! Choose X U[0,1], verify if there is a new arrival For a backlogged station: Partition [0,1] in S={retransmit, skip} S={[0,q r ),[q r,1]} NOTE P[retransmit] = q r!! } Choose X U[0,1], verify if there is a retransmission attempt or not q a 1-q a 1 q r 1-q r 1 0 X new arrival 0 X no retransmission 28

29 Aloha-slotted, simulation of finite network {method 2}/2 Repeat N times this experiment ( continue from previous slide) If n arr > 1: Fresh arrivals > 1, Put corresponding stations to backlog If n arr == 1 & n retr > 0: where n retr is the total number of retransmissions Collision for the only fresh arrival Put corresponding station to backlog If n arr == 0 & n retr > 1: Backlogged stations collided If n arr == 1 & n retr == 0: Fresh arrival delivered! If n arr == 0 & n retr == 1: Retransmission delivered Put corresponding station to not backlog For every loop keep trace (statistics) of Backlogged nodes n bklgd Status of transmission = f(n bklgd ) How many fresh arrival and retransmissions= g,h(n bklgd ) As functions of n bklgd 29

30 Aloha-slotted, simulation of finite network {method 2}/3 E.g. simulation, m = 3 (nodes) Backlogging statistics 30

31 Aloha-slotted, simulation of finite network {method 2}/4 At the end compute statistics of B Outcome is a m + 1 cells vector: Cell 2+1 = 3 tells how many times there were 2 backlogged nodes Divide elements from last row of P s by elements in statistics of B Outcome: statistics of delivered traffic as function of backlogged nodes Divide elements from last row of G by elements in statistics of B Outcome: statistics of offered traffic as function of backlogged nodes Plot statistics of θ(n) as a function of G(n) 31