King Fahd University of Petroleum & Minerals Computer Engineering Dept

Transcription

1 King Fahd University of Petroleum & Minerals Computer Engineering Dept COE 543 Mobile and Wireless Networks Term 032 Dr. Ashraf S. Hasan Mahmoud Rm Ext /15/2004 Dr. Ashraf S. Hasan Mahmoud 1 Leture Contents 1. Traffi Engineering - Erlang C and Erlang B models This material is found in setion of Pahlavan s book (page 176) 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 2

2 Performane of Fixed-Assignment Aess Methods FDMA/TDMA provide a hard apaity limit (number of hannels) FDMA maximum number of arriers per ell TDMA maximum number of slots per frame X number of arriers per ell CDMA-based also has a hard apaity limit ditated by the number of Walsh odes for example, but usually pratial apaity is lower Soft-apaity figure: Near the apaity boundary, the addition of one extra user degrades the link quality for all Call admission ontrol mehanism attempt to limit maximum number of ongoing alls before link quality degrades for all If you operate a maximum no of hannels, then all bloking and all delay are the two important measures! 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 3 Erlang-B and Erlang-C Models More details to be provided in COE560 Model designed to predit bloking probability (Erlang-B) and average all delay (Erlang-C) for a given number of hannels and traffi intensity Valid for voie and traffi models onforming to the basi assumption (usually not appliable to data) Assumptions, Terminology and Parameters: Channels Servers: servers Users Calls Calls arrive aording to a Poisson proess with rate = Inter-all arrival is an exponentially distributed r.v. with mean 1/ Call duration is exponentially distributed r.v. with mean = 1/µ Traffi intensity, ρ = /µ 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 4

3 Erlang-B (M/M//) Model Call Bloking µ 2µ As was shown earlier review previous notes, the all bloking probability is given by B(, ρ) 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 5 3µ = ρ! i ρ i! i= 0 (-1)µ ρ is referred to as the offered load, while ρx[1-b(,ρ)] is referred to as the arried load Note in this model alls arriving while there are alls are bloked no buffering is employed Erlang-C (M/M/) Model Call Delay j j+1 µ 2µ 3µ (-1)µ The probability that an arriving all having to wait is given by ρ Pr( delay > 0) = 1 k ρ ρ ρ! 1 The average delay is given by D= Pr( delay > 0) k = 0 k! 3/15/2004 Dr. Ashraf S. Hasan Mahmoud µ ( ρ) The probability of the delay exeeding t time units is given by ( N ρ ) µ t ( ) ( ) Pr delay > t = Pr delay > 0 e

4 Examples An IS-136 ellular provider owns 50 ell sites and 19 traffi arriers per arrier per ell eah with bandwidth of 30 khz. Assuming eah user makes three alls per hour and the average holding time per all is 5 minutes. Determine the total number of subsribers that the servie provider an support with a bloking rate less than 2% Solution: = 19X3 = 57 per ell B(57,ρ) = 0.02 ρ = 45 Erlangs per ell (/µ) sub = 3/60*5 = 0.25 Erlangs per sub Number of subs = total traffi / traffi per user = 45 / 0.25 = 180 per ell Number of subs for all sites = 180 X 50 = 8,000 subs Note that ρ all_subs = (/µ) all_subs = all_subs /µ whereas, ρ sub = (/µ) sub = sub /µ all_subs = no of subs X sub 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 7 Bakground Slides 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 8

5 Queuing Model Consider the following system: A(t) N(t) = A(t) D(t) D(t) ith ustomer arrives at time S i Queueing System ith ustomer departs at time D i T i = D i S i A(t) number of arrivals in (0, t] D(t) number of departures in (0, t] N(t) number of ustomers in system in (0,t] T i duration of time spent in system for ith ustomer 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 9 Number of Customers in System Blue urve: A(t) Red urve: D(t) 8 7 A(t) and D(t) for a queueing system 6 Total time spent in the system for all ustomers = area in between two urves T 1 T 2 T time 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 10

6 Little s Formula ont d Little s formula: E[N] = E[T] Whih relates the average arrival rate (), the average number of ustomers in the system (E[N]), and the average time spent in the system (E[T]) Holds for many servie disiplines and for systems with arbitrary number of servers. It holds for many interpretations of the system as well 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 11 Example 1: Problem: Let Ns(t) be the number of ustomers being served at time t, and let τ denote the servie time. If we designate the set of servers to be the system m then Little s formula beomes: E[Ns] = E[τ] Where E[Ns] is the average number of busy servers for a system in the steady state. 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 12

7 Example 1: ont d Note: for a single server Ns(t) an be either 0 or 1 E[Ns] represents the portion of time the server is busy. If p 0 = Prob[Ns(t) = 0], then we have 1 - p 0 = E[Ns] = E[τ], Or p 0 = 1 - E[τ] The quantity E[τ] is defined as the utilization for a single server. Usually, it is given the symbol ρ ρ = E[τ] For a -server system, we define the utilization (the fration of busy servers) to be ρ = E[τ] / 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 13 Multi-Server Systems: M/M/ The transition rate diagram for a multiserver M/M/ queue is as follows: Departure rate = kµ when k servers are busy j j+1 µ 2µ (-1)µ 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 14

8 Multi-Server Systems: M/M/ ont d When k servers are busy, the time until the next departure is given by: X = min(τ 1, τ 2,, τ k ) where τ i are iid exponential r.v. with mean 1/µ The CDF for X is given by (refer to definition) Prob[X > t] = Prob[min(τ 1, τ 2,, τ k ) > t] = Prob[τ 1 >t, τ 2 >t,, τ k >t] = Prob[τ 1 >t] Prob[τ 2 >t] Prob[τ k >t] = e -µt e -µt e -µt = e -kµt Therefore, the time till the next departure (X) is an exponentially distributed r.v. with mean 1/(kµ) 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 15 Multi-Server Systems: M/M/ ont d Writing the global balane equations: p 0 = µp 1 jµ p j = p j-1 for j=1, 2,, p j = p j-1 for j=, +1, p j = a j /j! p 0 (for j=1, 2,, ) and p j = ρ j- /! a p 0 (for j=, +1, ) where a = /µ and ρ = a/ 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 16

9 Multi-Server Systems: M/M/ ont d To find p 0, we resort to the fat that p j = 1 p 0 1 j a a 1 = + ρ = 0!! 1 j j 1 The probability that an arriving ustomer has to wait Prob[W > 0] = Prob[N ] = p + p +1 + p +2 + = p /(1-ρ) Erlang-C formula 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 17 Multi-Server Systems: M/M/ ont d The mean number of ustomers in queue (waiting): j= ( j ) E [ N ] = Pr[ N( t) = j] q = = j= ( j ) ρ ( 1 ρ ) 2 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 18 ρ p = ρ Pr[ 1 ρ W j p > 0]

10 Multi-Server Systems: M/M/ ont d The mean waiting time in queue: E [ W ] = E[ Nq ]/ The mean total delay in system: E [ T ] = E[ W ] + E[ τ ] = E[ W ] + 1/ µ The mean number of ustomers in system: E[ N] = E[ T ] = E N q ] + a [ Why? 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 19 Example: A ompany has a system with four private telephone lines onneting two of its sites. Suppose that requests for these lines arrive aording to a Poisson proess at rate of one all every 2 minutes, and suppose that all durations are exponentially distributed with mean 4 minutes. When all lines are busy, the system delays (i.e. queues) all requests until a line beomes available. Find the probability of having to wait for a line. 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 20

11 Example: ont d Solution: = ½, 1/µ = 4, = 4 a = /µ = 2 ρ = a/ = ½ p 0 = { /2!+2 3 /3!+2 4 /4! (1/(1-ρ))} -1 = 3/23 p = a /! p0 = 2 4 /4! X 3/23 Prob[W > 0] = p /(1- r ) = 2 4 /4! X 3/23 X 1/(1-1/2) = 4/ /15/2004 Dr. Ashraf S. Hasan Mahmoud 21 Multi-Server Systems: M/M// The transition rate diagram for a multiserver with no waiting room (M/M//) queue is as follows: Departure rate = kµ when k servers are busy µ 2µ (-1)µ 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 22

12 PMF for Number of Customers for M/M// Writing the global balane equations, one an show: p j = a j /j! p 0 (for j=0, 1,, ) where a = /µ (the offered load) To find p 0, we resort to the fat that p j = = j a p = 0! j j 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 23 Erlang-B Formula Erlang-B formula is defined as the probability that all servers are busy: Pr[ N = ] = p a / j! = 2 1+ a + a / 2! a /! 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 24

13 Expeted Number of ustomers in M/M// The atual arrival rate into the system: = 1 p ) ( Average total delay figure: Average number of ustomers: a E [ T ] = E[ τ ] E[ N] = E[ τ ] a Why? 3/15/2004 Dr. Ashraf S. Hasan Mahmoud 25