Value functions for M/G/1 & Task Assignment Problem Esa Hyyti - PowerPoint PPT Presentation

Value functions for M/G/1 & Task Assignment Problem Esa Hyytiä Joint work with Samuli Aalto, Aleksi Penttinen, Jorma Virtamo Department of Communications and Networking Aalto University, School of Electrical Engineering, Finland YEQT-VI, November 1-3, 2012 YEQT-VI, November 1-3, 2012 1/44

Outline 1 Server systems: Performance measures 2 Value functions 3 Value functions for M/G/1 4 Task Assignment Problem 5 Summary of Results YEQT-VI Nov 2012 2/44

Server Systems: Dispatcher Servers Customers Latency E [ T ] : Sojourn time, Response Time, Delay, . . . Objective: min E [ T ] . Slowdown : “ long jobs can wait longer ” Latency T i γ i � Slowdown of job i , . Service time X i Objective: min E [ γ ] . YEQT-VI Nov 2012 3/44

Server Systems: Holding Cost Structure Holding cost: Job i accrues costs at job-specific rate b i Latency: With b i = 1, Total cost rate is the number of jobs in the system, N t Cost a job incurs is equal to the latency, b i · T i = T i . Slowdown: With b i = 1 / x i Cost a job incurs is equal to the slowdown, b i · T i = T i . x i Note: No costs associated with state transitions YEQT-VI Nov 2012 4/44

Value Function: Definition Let C z ( t ) denote the cost rate at time t for an initial state z Cumulative costs accrued during ( 0 , t ) are � t V z ( t ) � C z ( s ) ds . 0 Relative value is the expected difference in the infinite horizon cumulative costs between a) a system initially in state z , and b) a system initially in equilibrium, v z � lim t →∞ E [ V z ( t ) − r t ] . YEQT-VI Nov 2012 5/44

Value function: Latency in Server Systems For latency, the cost rate C z ( t ) is simply N z ( t ) � ”the number of jobs in the system” , Value function reads �� t � � � v z = lim N z ( s ) ds − E [ N ] t . E t →∞ 0 Similarly for the slowdown and general holding costs YEQT-VI Nov 2012 6/44

Value function: M/G/1-FCFS Example Initial state z = ( 3 , 1 ) : First job with remaining size 3 currently receiving service Second job with size 1 is waiting Also later arriving jobs have to wait (FCFS) Relative value of state z is the expected difference in infinite horizon costs: E[ N(t) ] Relative value # of Jobs 3 2 v z = blue shaded area. r = E[ N ] 1 Known jobs 1 2 3 4 Time t YEQT-VI Nov 2012 7/44

Value function: Comparison of States Given two states z 1 and z 2 , the expected difference in the infinite horizon costs is d ( z 1 , z 2 ) = lim t →∞ E [ V z 2 ( t ) − V z 1 ( t )] , which gives d ( z 1 , z 2 ) = v z 2 − v z 1 . Example : Server system Suppose state z 2 is state z 1 plus one new job Value function gives the marginal cost for accepting a new job! YEQT-VI Nov 2012 8/44

Value Function for M/G/1 Queues λ ν A. Elementary scheduling disciplines: M/G/1-FCFS M/G/1-LCFS B. Size-aware scheduling disciplines: M/G/1-SPT (shortest-processing-time) M/G/1-SRPT (shortert-remaining-processing-time) M/G/1-SPTP (shortest-processing-time-product) C. Processor sharing (PS) M/D/1-PS (fixed job sizes) M/M/1-PS YEQT-VI Nov 2012 9/44

M/G/1: Notation Basic case : Poisson arrival rate λ Service times X i i.i.d., X i ∼ X Offered load ρ = λ E [ X ] Size-aware state z = (∆ 1 ; .. ; ∆ n ) with n jobs: ∆ i is the remaining service time of job i Job n is served first (FCFS,LCFS) Backlog u z = � i ∆ i With arbitrary holding costs : State z = ((∆ 1 , b 1 ); .. ; (∆ n , b n )) b i is the holding cost of job i E [ B ] is the mean holding cost (arbitrary job) YEQT-VI Nov 2012 10/44

M/G/1-FCFS Proposition: The size-aware relative value of state z with respect to delay in an M/G/1-FCFS queue is 12 n λ u 2 � z v z − v 0 = i ∆ i + 2 ( 1 − ρ ) . (1) i = 1 With respect to arbitrary job specific holding costs b i ,   n i λ u 2 � � z  + v z − v 0 =  ∆ i b j 2 ( 1 − ρ ) E [ B ] . (2) i = 1 j = 1 Note: Insensitive to service time distribution. 1 Hyytiä et al., Eur. J. Oper. Research (2012) 2 Hyytiä et al., J. Applied Probability (2012). YEQT-VI Nov 2012 11/44

M/G/1-LCFS (preemptive) Proposition: The size-aware relative value of state z with respect to delay in an M/G/1-LCFS queue is 34 n 1 � v z − v 0 = i · ∆ i . (3) 1 − ρ i = 1 With respect to arbitrary job specific holding costs b i ,   n i 1 � �  . v z − v 0 =  ∆ i b j (4) 1 − ρ i = 1 j = 1 Note: Later arrivals immune to state z . Insensitivity: v z − v 0 depends only on ρ . 3 Hyytiä et al., Eur. J. Oper. Research (2012) 4 Hyytiä et al., J. Applied Probability (2012). YEQT-VI Nov 2012 12/44

Value Function for M/G/1 Queues λ ν A. Elementary scheduling disciplines: M/G/1-FCFS M/G/1-LCFS B. Size-aware scheduling disciplines: M/G/1-SPT (shortest-processing-time) M/G/1-SRPT (shortert-remaining-processing-time) M/G/1-SPTP (shortest-processing-time-product) C. Processor sharing (PS) M/D/1-PS (fixed job sizes) M/M/1-PS YEQT-VI Nov 2012 13/44

Size-aware M/G/1: Scheduling Notation: (∆ i , ∆ ∗ i ) = remaining and initial service time of job i . Index policy α serves first the job with the lowest index. Scheduling Index Optimality ∆ ∗ SPT optimal non-preemptive / delay & slowdown i SRPT ∆ i optimal preemptive / delay i optimal preemptive / slowdown 5 ∆ i · ∆ ∗ SPTP 5 Hyytiä, Aalto, Penttinen, SIGMETRICS’12. YEQT-VI Nov 2012 14/44

Size-aware M/G/1: Scheduling non-preemptive preemptive class-aware size-aware non-anticipating anticipating size-aware SEPT SPT FB, FIFO,. . . SRPT delay ( c µ -rule) (depends on f ( x ) ) slowdown -”- -”- FB, FIFO, . . . SPTP (depends on f ( x ) ) (M/G/1) YEQT-VI Nov 2012 15/44

Size-aware M/G/1: Additional notation Notation: Jobs are numbered so that (without new arrivals) job 1 is served first and job n last. f ( x ) denotes the service time pdf. ρ ( x ) denotes the load due to jobs shorter than x , � x ρ ( x ) = λ x f ( x ) dx . 0 Define f ( x ) b ( x ) h ( x ) � ( 1 − ρ ( x )) 2 , where b ( x ) is the mean holding cost of a job with size x , b ( x ) = E [ B | X = x ] YEQT-VI Nov 2012 16/44

M/G/1-SPT (Non-preemptive) Proposition: The size-aware relative value of state z with respect to arbitrary holding costs in an M/G/1-SPT queue is 6  � n � i − 1 1 � �  + v z − v 0 = b i  ∆ i + ∆ j 1 − ρ (∆ i ) i = 1 j = 1 (5)     � ˜ n n i � 2 ∆ i + 1 � λ � � � ∆ 2 j + ∆ j h ( x ) dx     2 ˜ ∆ i i = 1 j = i + 1 j = 1 where job 1 receives service and ∆ 2 < . . . < ∆ n  0 , i = 1 ,  ˜ ∆ i = ∆ i , i = 2 , . . . , n ∞ i = n + 1 .  6 Hyytiä et al., Eur. J. Oper. Research (2012) YEQT-VI Nov 2012 17/44

M/G/1-SRPT Proposition: The size-aware relative value of state z with respect to arbitrary holding costs in an M/G/1-SRPT queue is 7   � i − 1 � ∆ i n � 1 1 � � v z − v 0 = b i ∆ j + 1 − ρ ( x ) dx   1 − ρ (∆ i ) 0 i = 1 j = 1 (6) � 2 ∆ i + 1 ∆ i + 1   n i + λ � � � x 2 h ( x ) dx � � ∆ j h ( x ) dx + ( n − i )   2 i = 0 j = 1 ∆ i ∆ i where job 1 receives currently service and ∆ 1 < . . . < ∆ n , ∆ 0 = 0 and ∆ n + 1 = ∞ 7 Hyytiä et al., Eur. J. Oper. Research (2012) YEQT-VI Nov 2012 18/44

M/G/1-SPTP Proposition: The size-aware relative value of state z with respect to arbitrary holding costs in an M/G/1-SPTP queue is 8 � i − 1 n ˜ ∆ i � � � 1 + 2 � x dx � � v z − v 0 = b i ∆ j 1 − ρ ( ˜ ∆ ∗ 1 − ρ ( x ) ∆ i ) 0 i i = 1 j = 1   ˜ � ˜ ∆ i + 1 ∆ i + 1 n i n � 2 � � + λ � � x 4 h ( x ) dx � � � j ) − 2 ∆ j h ( x ) dx + (∆ ∗   2   i = 0 j = 1 j = i + 1 ˜ ˜ ∆ i ∆ i where � Job 1 receives service and � ∆ 1 ∆ ∗ 1 < . . . < ∆ n ∆ ∗ (SPTP) n  0 , i = 0  ˜ � ∆ i ∆ ∗ ∆ i = i , i = 1 , . . . , n ∞ , i = n + 1 .  8 Hyytiä, Aalto, Penttinen, SIGMETRICS’12. YEQT-VI Nov 2012 19/44

Value Function for M/G/1 Queues λ ν A. Elementary scheduling disciplines: M/G/1-FCFS M/G/1-LCFS B. Size-aware scheduling disciplines: M/G/1-SPT (shortest-processing-time) M/G/1-SRPT (shortert-remaining-processing-time) M/G/1-SPTP (shortest-processing-time-product) C. Processor sharing (PS) M/D/1-PS (fixed job sizes) M/M/1-PS YEQT-VI Nov 2012 20/44

M/G/1-PS: (Processor sharing) Basics : PS serves the existing n jobs at equal rates 1 / n . Mean delay in M/G/1-PS is insensitive to job size distribution, E [ T ] = E [ X ] 1 − ρ. Unfortunately, the size-aware value function is not! (∆ 1 ; .. ; ∆ n ) denotes the remaining service times, ∆ 1 ≥ . . . ≥ ∆ n . Without new arrivals: Job n leaves the system first and job 1 last Cumulative delay (myopic cost) is given by V z = ∆ n n 2 + (∆ n − 1 − ∆ n )( n − 1 ) 2 + . . . + (∆ 1 − ∆ 2 ) n (7) � = ( 2 i − 1 )∆ i . i = 1 YEQT-VI Nov 2012 21/44

M/D/1-PS PS λ ν Proposition: The size-aware relative value of state z with respect to the delay in an M/D/1-PS queue is given by 9 n λ 1 − ρ u 2 � v (∆ 1 ; .. ;∆ n ) − v 0 = z − u z + 2 i ∆ i . (8) i = 1 Note: Compact form as a new job will always depart last. Converges to (7) when λ → 0 9 Hyytiä et al., ITC’11. YEQT-VI Nov 2012 22/44