M/G/1 and Priority Queueing

Transcription

1 M/G/1 and Priority Queueing Richard T. B. Ma School of Computing National University of Singapore CS 5229: Advanced Compute Networks

2 Outline PASTA M/G/1 Workload and FIFO Delay Pollaczek Khinchine Formula M/G/1 Priority Queueing Delay M/G/1 Conservation Law

3 Steady State: A Revisit Steady state distribution π = π 0, π 1 π i is the limiting probability that the steady-state X lim t X(t) is in state i: π i = P X = i = lim t P*X t = i+ π i is also the limiting proportion of time that the system is in state i: 1 π i = lim t t 1 X s =i ds t 0

4 System State Seen by Arrivals A point process: ψ = *t n : n 0+, where t i is the arrival time of the ith customer *X t n : n 0+ defines the system states seen by the customers upon their arrivals Denote π i = lim P*X t n = i+ or π n i = lim n 1 n n 1 X tj =i j=0 Does π i = π i hold? In a D/D/1 system with T = 4, and S = 3: π 0 = 1/4 and π 0 = 1

5 PASTA Poisson Arrivals See Time Averages Explanation: If arrivals are Poisson, then the proportion of time a queueing system spends in a given state (π i ) is equal to the proportion (π i ) of arrivals who find the system in that state. Notation State process: X = *X(t): t 0+ Poisson point process: ψ = *t n : n 0+ at rate λ with counting process *N(t): t 0+

6 PASTA Assumption: Lack of Anticipation (LAA) For any t 0, the future (Poisson) increments *N t + s N t : s 0+ are independent of the joint past X u : u t, N u : u t. Theorem: Any stochastic process *X(t): t 0+ (with cadlag sample paths) jointly with a Poisson point process *t n : n 0+ that satisfies LAA, and let f be any real bounded function. Then with probability 1, 1 lim t t t f X s ds = 0 1 lim n n n j=0 f(x(t j ))

7 Intuition/Informal Proof Let π i (t) P*X t = i+ and define an arrival event E a t, dt N t + dt N t > 0 The corresponding π i (t) can be defined as π i (t) lim dt 0 P X t = i E a t, dt + π P*X t = i, E a t, dt + i t = lim dt 0 P*E a t, dt + P*E a t, dt X t = i+p*x t = i+ = lim dt 0 P*E a t, dt + P E a t, dt X t = i = P*E a t, dt + (LAA) π i t = P X t = i = π i (t)

8 M/G/1 Workload & FIFO Delay Take X(t) = V(t), check LAA & use PASTA 1 lim t t 1 lim t t 0 t 1 *V s x+ ds = 0 t P V s x ds P V x lim n 1 n n j=0 1 = lim n n = P D x 1 V tj x n j=0 P(D(t j ) x) V D E V = E D d F (FIFO delay) What is the value of E,V-?

9 Brumelle s Fomula: A Revisit With FIFO service discipline E V = λe SD + λ 2 E S2 Let ρ λe S, with G/G/1 mean workload Service time is independent of arrivals E V = λe S E D + λ 2 E S2 = ρd F + ρe R Avg. work in queue + avg. work in service M/G/1 mean workload and FIFO delay E V = d F = ρd F + ρe R (by PASTA)

10 M/G/1 Workload & FIFO Delay M/G/1 mean FIFO delay has a closed-form d F = ρd F + ρe R d F = ρ 1 ρ E R = λe,s2-2(1 ρ) So as the workload under any workconserving service discipline E V = d F = ρe R 1 ρ = λe,s2-2(1 ρ) Sanity check for M/M/1, E V = ρ μ(1 ρ).

11 Pollaczek Khinchine Mean Formula For an M/G/1 system with FIFO: Mean Queueing Delay (P-K mean formula) E D = ρe,r- 1 ρ = λe,s2-2(1 ρ) Mean Sojourn Time E W = E D + E S Mean Customers in Queue E Q = λe D Mean Customers in System E L = λe W = λ E D + E S = E Q + ρ

12 *Pollaczek Khinchine Formula D R i i=1 R i are i.i.d. distributed as R G is geometric, independent of R i, with P G = i = 1 ρ ρ i If G = 0, then D = 0, which shows P D = 0 = P G = 0 = 1 ρ E D = E G i=1 R i = E G E R = ρ G 1 ρ E,R- Can be derived by RCL & Laplace transform

13 M/G/1 Priority Queueing Settings: I service classes, indexed by i = 1,, I Lower index Higher priority Define λ i as the Poisson arrival rate, S i as the service time of a class i job Assumptions: Work-serving Non-preemptive or preemptive FIFO within each service class

14 Priority Queueing: Illustration Arrive at t = 1 S = 5 S = 3 Arrive at t = 3 S = 4 Non-preemptive: Arrive at t = Preempt-resume: t t

15 Definitions of Some Entities Define ith service class s utilization as ρ i λ i E,S i - Mean arrival rate of workload from class i System arrival rate and mean service time λ I λ i i=1 System utilization ρ λs = I i=1 I, S (λ i /λ)e,s i -. ρ i i=1 P server idle = 1 ρ

16 Mean Queueing Delay First, consider a non-preemptive system E,D i - E,waiting time for jobs from class i- E,W i - E class i sojourn time = E,D i - + E S i Average waiting time E,D i - = V R + D 1 2 i + D i V R : mean residual workload in the server 1 D i : delay due to jobs in the queue upon arrival D 2 i : delay due to jobs who arrive afterwards

17 Mean Residual Workload V R Apply PASTA again V R = I ρ j E R j = ρ j where, E R j = E S j 2 2E S j I E S j 2 2E S j = I λ j E S j 2 is the mean residual time Under non-preemptive system, V R is independent of priority classes. 2

18 For Single-Class M/G/1 Average workload in queue is E Q S i i=1 = E Q E S = λe D E S = ρe D by Wald s equation E Q = λe,d- (by Little s law) Compare with Brumelle s formula: E V = λe S E D + λ 2 E S2 = ρd F + ρe R

19 D i 1 and D i 2 D i 1 can also be considered as average workload in queue upon arrival of a class i customer that needs to be finished before serving the customer i i i D i 1 = E S j E Q j = E S j λ j E,D j - = ρ j E,D j - i 1 i 1 i 1 D i 2 = E S j E M j = E S j λ j E,D i - = ρ j E,D i -

20 Triangular Equations i i 1 E,D i - = V R + ρ j E,D j - + ρ j E,D i - i i 1 = V R + ρ j E,D j - 1 ρ j Compute starting from i = 1 E,D i - = V R i i 1 (1 ρ j )(1 ρ j )

21 Priority Class i Mean Queueing Delay E,D i - = V R i i 1 (1 ρ j )(1 ρ j Mean Sojourn Time E,W i - = E,D i - + E,S i - Mean Customers in Queue E,Q i - = λ i E,D i - Mean Customers in System E,L i - = E,Q i - + ρ i )

22 How about Preemptive-Resume? Without preemption: V R E,D i - = i i 1 (1 ρ j )(1 ρ j E,W i - = E,D i - + E,S i - With preemption: E D i = i i ρ j E,R j - ) i 1 ) (1 ρ j )(1 ρ j E W i = E D i + E S i = E D i + E S i i 1 1 ρ j

23 Service Time with Preemption Consider a customer of class 2, starting service at time t 0, during service time, it might be preempted by class 1 customers Time until it finishes is S 2 S 2 (j) S 2 (1) S 2 (2) S 2 (3) t 0 t 0 + S 2

24 Deriving E S i j For any class i customer, S i S i j Condition on the length of the previous cycle E S i j+1 = E S i j+1 Si j x=0 = x f j x dx = λ 1 xe S 1 + λ i 1 xe S i 1 f j x dx x=0 = ρ ρ i 1 xf j x dx x=0 j = ρ ρ i 1 E S i

25 Deriving E S i E S i j+1 = ρ ρ i 1 j E S i 1 Let γ ρ ρ i 1 j E S i = E S i = 1 + γ + γ E S i = 1 + γ + γ 2 + E S i = 1 1 γ E S i E S i E S i = i 1 1 ρ j

26 Conservation Law You don t get something for nothing. Shorter delay for higher class longer delay for lower class What is unchanged/conserved? workload In busy periods, independent of order of service Require no work is created/destroyed in system No user leaves the system before finishing No idle time when facing non-empty queue Only consider work-conserving service disciplines

27 M/G/1 Conservation Law Under M/G/1, if the service discipline is 1. non-preemptive and work conserving 2. independent of the service times then the following must hold: I ρ ρ j E,D j - = 1 ρ V R ρ < 1 ρ 1 Weighted sum of the queueing delay E,D j - can NEVER CHANGE regardless how sophisticated the service discipline is.

28 Conservation Law: Derivation Avg. workload in queue (or D I 1 ) is conserved E V V R = E,Q j -E,S j - I I = λ j E,D j -E,S j - = I ρ j E,D j - ρ j E,D j - is the mean workload from class j in queue. From the P-K mean formula for M/G/1 E V = d F = λe,s2-2(1 ρ) = V R 1 ρ because E S 2 = λ j I = λ E,S j 2 - I 2 λ λ j E,S j 2-2 = 2 λ V R

29 Class-homogenous Service Time When E,S j - = E S, I ρ j E,D j - = ρ 1 ρ V R λ j E,D j - I = λ 1 ρ V R The average # of customers is a constant. By Little s Law, the average queueing delay ( averaged over all classes) is a constant: I (λ j /λ)e,d j - = 1 1 ρ V R

30 S = 3 S = 4 Workload in Queue Conserved S = 5 S = 3 S = 4 Workload in queue: M/G/1 FIFO t

31 S = 3 S = 4 Workload in Queue Conserved S = 5 S = 3 S = 4 Workload in queue: M/G/1 FIFO t

32 *G/G/1 Conservation Law If select customers in a way that is independent of their service time, then distribution of the number of customers in the system will be invariant of the order of service. Avg. queueing delay is also invariant. (by LL) In particular, for non-preemptive systems: I ρ j E,D j - = E V V R = E V I 2 λ j E S j 2

33 *Service-Time Dependency The mean delay with FIFO is a tight lower bound for work conserving and service time independent service disciplines Service time dependent scheduling: SPT: shortest processing time first SRPT: shortest remaining processing time first E D FIFO E D SPT E D SRPT However, uncommon in packet switching because the packet ordering will be modified and delay for large packets increases

34 References Leonard Kleinrock, Queueing Systems, Volume 2: Computer Applications, Chapter 3 and 4, John Wiley & Sons, Inc. Linus Schrage, An Alternative Proof of a Conservation Law for the Queue G/G/1, Operations Research, Vol. 18, No. 1 (Jan. - Feb., 1970), pp