More Queueing Theory Carey Williamson Department of Computer - PowerPoint PPT Presentation

More Queueing Theory Carey Williamson Department of Computer Science University of Calgary

Motivating Quote for Queueing Models “Good things come to those who wait” - poet/writer Violet Fane, 1892 - song lyrics by Nayobe, 1984 - motto for Heinz Ketchup, USA, 1980’s - slogan for Guinness stout, UK, 1990’s 2

M/M/1 Queue ▪ M/M/1 queue is the most commonly used type of queueing model ▪ Used to model single processor systems or to model individual devices in a computer system ▪ Need to know only the mean arrival rate λ and the mean service rate μ ▪ State = number of jobs in the system l l l l l l … 0 1 2 j-1 j j+1 m m m m m m 3

Results for M/M/1 Queue (cont’d) ▪ Mean number of jobs in the system: ▪ Mean number of jobs in the queue: 4

Results for M/M/1 Queue (cont’d) ▪ Probability of n or more jobs in the system:      = = −   =  n k P ( n k ) p ( 1 ) n = = n k n k ▪ Mean response time (using Little’s Law): — Mean number in the system = Arrival rate × Mean response time — That is: 5

M/M/1/K – Single Server, Finite Queuing Space l K K 6

Analytic Results ▪ State-transition diagram: l l l l … 0 1 K-1 K m m m m l ▪ Solution =   = n p p , w here m n 0 −   1     1   + − −   K 1 1 1 K =  = n  p = 0 n 0 1   = 1   + K 1 7

M/M/m - Multiple Servers 8

Analytic Results ▪ State-transition diagram: l l l l l l … 0 1 m-1 m m+1 2 m (m-1) m m m m m m m m ▪ Solution  1   n p n m l  − n 1 0  = = n !  j p p m n 0 1    = n j 0 + p n m j 1  − 0 n m m ! m 9

M/M/  - Infinite Servers ▪ Infinite number of servers - no queueing 10

Analytic Results ▪ State-transition diagram: l l l l l l … 0 1 j-1 j j+1 m 2 m (j-1) m j m (j+1) m (j+2) m ▪ Solution 1 =  n p p n 0 n ! − 1   1   =  = −  n p e   =   0 n 0 n ! ▪ Thus the number of customers in the system follows a Poisson distribution with rate 𝜍 11

M/G/1 Queue ▪ Single-server queue with Poisson arrivals, general service time distribution, and unlimited capacity 1 𝜈 and variance 𝜏 2 ▪ Suppose service times have mean ▪ For 𝜍 < 1 , the steady-state results for 𝑁/𝐻/1 are:  = l m = −  / , p 1 0  +  m  +  m 2 2 2 2 2 2 ( 1 ) ( 1 ) =  + = E [ n ] , E [ n ] −  −  q 2 ( 1 ) 2 ( 1 ) l m +  l m +  2 2 2 2 1 ( 1 / ) ( 1 / ) = + = E [ r ] , E [ w ] m −  −  2 ( 1 ) 2 ( 1 ) 12

M/G/1 Queue — No simple expression for the steady-state probabilities — Mean number of customers in service: 𝜍 = 𝐹 𝑜 − 𝐹 𝑜 𝑟 — Mean number of customers in queue, 𝐹[𝑜 𝑟 ] , can be rewritten as: 𝜍 2 𝜇 2 𝜏 2 𝐹[𝑜 𝑟 ] = 2 1 − 𝜍 + 2 1 − 𝜍 ▪ If 𝜇 and 𝜈 are held constant, 𝐹[𝑜 𝑟 ] depends on the variability, 𝜏 2 , of the service times. 13

Effect of Utilization and Service Variability ▪ For almost all queues, if lines are too long, they can be reduced by decreasing server utilization ( 𝜍 ) or by decreasing the service time variability ( 𝜏 2 ) ▪ Coefficient of Variation: a measure of the variability of a distribution 𝑊𝑏𝑠 𝑌 𝐷𝑊 = 𝐹[𝑌] — The larger CV is, the more variable is the distribution relative to its expected value. ▪ Pollaczek-Khinchin (PK) mean value formula:  + 2 2 ( 1 ( CV ) ) =  + E [ n ] −  2 ( 1 ) 14

Effect of Utilization and Service Variability ▪ Consider 𝐹[𝑜 𝑟 ] for M/G/1 queue:  +  m 2 2 2 ( 1 ) = E [ n ] −  q 2 ( 1 ) Mean no. of customers in queue    +   2 2 1 ( CV ) =         −      1 2 Same as for Adjusts the M/M/1 M/M/1 formula to account for queue a non-exponential service time Traffic Intensity ( 𝜍 ) distribution 15