Single-Server Service-Station (G/G/1)

Transcription

1 Service Engineering July 997 Last Revised January, 006 Single-Server Service-Station (G/G/) arrivals queue departures Arrivals A = {A(t), t 0}, counting process, e.g., completely random (Poisson) vs. deterministic. Service Durations S,S,S 3,..., associated with server/customer. Work arriving up to time t: L(t) = A(t) n= S n,t 0. Netflow X(t) =L(t) t. potential outflow Theorem Given X that is RCLL, there exist unique I and U that satisfy the following: Skorohold s framework: IandUaregivenby I = X (X = lower envelope of X); U = X X. Interpretations of I? U? U = X + I, U 0, di 0, I(0) = 0, I increases { only when U =0; UdI =0. 0 U>0 di = dt U =0 Alternative derivation: I least s.t. I X, I 0 For convenience, introduce also: T,T,..., inter-arrival times; A,A,..., arrival times; (A n = T T n ).

2 X(t) S S S 3 I(t) = - X(t) Idleness (Cumulative) U(t) = X(t) + I(t) = X(t) - X(t) S S S 3 Workload (Unfinished work) G/G/ (GI/GI/): Prevalent Assumptions independent A renewal {T n } iid: interarrival times with mean = λ,cv = C a, {S n } iid: service times with mean = µ,cv = C s. lim T A(T ) T = λ arrival rate; µ service rate (potential); ρ = λ/µ traffic intensity (offered load); ρ server s utilization (Little on system = server ); since λ µ output rate (actual service rate). Assume (Meta) Theorem: Stability ρ< (λ<µ;thenλ = departure rate = throughput rate). (Meta) Proof: Fluid view λ empties eventually. µ

3 Productivity commonly defined as output input (in appropriate units) Server s productivity (capability) = customers served per time unit time served per time unit = λ = λ = µ ρ λ/µ System s productivity = unit customers of time = λ<µ. Prevalent view: Inefficiency/excess capacity, manifested through λ<µ. Typically, fundamentally flawed! (especially in service operations) (Efficiency proxy: ρ = λ/µ) MOP s in Steady-State (Equilibrium). L(t) = number of customers in system, at time t (Abuse: Q(t)). Distribution at steady-state identified as Limit distribution: L(t) t L( ), abbreviated to L. π n = P (L( ) =n) Averages calculated directly: E(L) = lim EL(t) = nπ n, or via t n=0 long-run averages, namely T E(L) = lim L(t)dt (SLLN, Ergodicity). T T 0. W q (n) = waiting-time of customer n =,,... W (n) =W q (n)+s n sojourn/cycle/response time. Steady state W q (n) W q ( ), abbreviated to W q. Averages calculated directly EW q (n) EW q,orvia N E(W q ) = lim W q (n). N N 3. Examples of other limits: lim T T Alternatively, lim T T More generally, lim T T T 0 T T 0 n= {L(t)>0} dt = ρ (utilization) (0, ) (L(t))dt = E (0, ) (L) =P (L >0) = π 0 M/G/: Prob{wait>0}, viapasta f(l(t))dt = Ef(L) = π n f(n) : Ergodicity Relations: EW = EW q + µ ; EL = λew (Little) = EL q + ρ n=0 3

4 Calculating Waiting-Times Under FIFO: Lindley s Equations A,A,... arrival times (jumps of A( )) Key observation: W q (n) = U(A n ) W q (n) = 0 A n waiting time of customer n (U is RCLL) A n + W q (n ) + S n A n departure time of customer (n ) {}}{ A n + W q (n ) + S n otherwise. In short, where W q (n) = [A n A n + W q (n ) + S n ] + ; x + = x 0 = [W q (n ) + x n ] +,n x n = S n T n Lindley s Equations W q (n) =[W q (n ) + x n ] + W q () = 0, n Recursion useful for spreadsheet calculations. Possible to obtain an explicit representation, using the corresponding expression (i.e., X X) for U. An alternative calculation via Skorohod s Map (of Lindley s Equations) W q (n) W q (n ) = U(A n ) U(A n ) = [L(A n ) L(A n )] (A n A n ) + I(A n ) I(A n ) }{{}}{{}}{{} S n T n y n (def.) y n = cumulative idleness over (A n,a n ]. W q (n) =W q (n ) + x n + y n, x n = S n T n W q (n) 0,y n 0 W q (n) y n =0 (IfW q (n) > 0, narrives to a busy server, and then y n =0.) 4

5 Lemma z = x + y z 0, y 0 z y =0 unique solution z = x+,y= x x + =max(x, 0), x =max( x, 0) Proof x = z y, z 0, y 0, z y = 0, exhibits z and y as the positive and negative parts of x, respectively. Thus, W q (n) =[W q (n ) + x n ] + W q () = 0,n Lindley s Equations Explicit representation of W q (n) Define ˆXn = X(A n ) (Recall: X(t) = netflow). Note: ˆX n = (S + + S n ) A n =(S 0 T )+ +(S n T n ) = x + + x n (Define S 0 =0.) We have: W q (n) =U(A n )=X(A n ) X(A n )= ˆX n ˆX n local minima of X are attained before jumps = ˆX n min 0 j n ( ˆX j )= ˆX n +max 0 j n ( ˆX j ) Obtained W q (n) = max 0 j n ( ˆX n ˆX j ), n 0 Explicit! Proof of Stability for GI/GI/ (Lindley, 95) Key: {x n } are iid ({ ˆX n } random walk), since x n = S n T n. W n = max{x + + x n,x + + x n,...,x n, 0} d = = max{x + + x n,x + + x n,...,x, 0} =max 0 j n ( ˆX j ) Observe: E(x j )= µ λ ; λ<µ Ex j < 0 ˆX n a.s. by SLLN. 5

6 Fact P {sup n 0 ( ˆX n ) < } = Ex < 0. Consequence: W n = max k 0 ( ˆX k ) when λ<µ (ρ <). n MOP s (Continued) Under FIFO: Workload = virtual waiting time (U V,W) (used in setting appointment times, e.g., for judges) Poisson arrivals, iid services: M/G/, then L( ) is compound Poisson. independent U = {U(t), t 0} invariant under service discipline, as long as server works at full pace when there is work to do: work-conservation. Calculating average-wait at steady-state: application of H = λg. A n = arrival time of n-th customer W n = waiting time of n-th customer, before service S n = service time f (t) n S n A n W n S n D n t Remaining (unfinished) work associated with customer n: (Cost per unit-time) T H = lim T T 0 n= U(t) = f n (t), t 0. n= f n (t)dt = lim T T T 0 U(t)dt = E(U) (ergodicity, SLLN) (Cost per customer) G = lim N N N n= 0 f n (t)dt = lim N N N n= ( S n W n + ) S n = E(SW q )+ E(S ) 6

7 Brumelle s Formula E(U) =λ [ E(S W q )+ E(S ) ] If service and wait independent: E(U) = λe(s)e(w q )+ λe(s )= = ρe(w q )+ ρe(s )/E(S) If ASTA (e.g., M/G/) then EU = EW q,whichgives E(W q )=E(S) ρ ρ +C (S) Khinchine-Pollatcheck Hall, formula (5.64) Illuminating Derivation of Khinchine-Pollatcheck (Hall 68 9) Ingredients: Little, PASTA, Biased sampling; Wald. For customer n =,,..., denote W q (n) = waiting-time of n-th customer. R(n) = residual service time, at time of n-th arrival L q (n) = # of customers in queue, at time of n-th arrival. {S n } = sequence of service-times. W q (n) = R(n)+ n k=n L q(n) S k, n. EW q (n) = ER(n)+E(S ) EL q (n), by Wald E(W q )= E(R)+E(S )E(L q ), n, assuming limit, +PASTA = E(R)+λE(S )E(W q ), by Little. E(W q )= E(R)+ρE(W q ), ρ < steady-state E(W q )= E(R)/( ρ) Prob. of arriving to a busy server. +C (S) E(R) =( ρ) 0+ρ E(S) PASTA + Biased sample E(W q )=E(S) ρ ρ +C (S) Khinchine-Pollatschek (Hall, 5.64) 7

8 The Congestion Index (Understanding Khinchine-Pollatschek) M/G/ E(W q )=E(S) ρ ρ +C (S) G/G/: E(W q ) ρ E(S) ρ C (A)+C (S) Allen-Cunneen (Hall (5.70)) pure utilization accessibility stochastic variability (unitless) Fact: Right-hand side upper bound, becoming asymptotically exact as ρ (Heavy traffic). Kingman s Exponential Law of Congestion (Invariance principle) W q E(S) ( exp mean = ρ C (A)+C ) (S), wp ρ, 0, wp ρ, asymptotically exact as ρ. Understanding the formula: Assume C (A) =C (S) = (as in M/M/): E(W q) E(S) = ρ ρ Substitute ρ =0.5, 0.9, 0.95, Other MOP s: E(W ) = E(W q )+E(S) E(L q ) = λe(w q ) E(L) = λe(w )=E(L q )+ρ 8

9 Congestion Curves (Empirical Proof of Khinchine-Pollatcheck Formula) Service Level vs. Availability \ Accessibility Average waiting time, sec Occupancy Congestion Index: E(W q ) E(S) m = m ρ ρ C a + C s ρ ρ C (m = number of servers) 9

10 Performance vs. Availability \ Accessibility Occupancy Average waiting time, sec Queueing Science: Measurements Model Validation 0

11 % Abandonment: Theoretical Congestion Curve (Erlang-A) 70 # Servers = % Abandonment Arrival Rate

12 Justifying the Law of Congestion: Why W q exp ( mean = µ ρ ρ C a+c s )? via Strong Approximations. (Heavy Traffic Theory) S(t) = t n= S n µ t + σ sb s (t) (Donsker for partial sums) A(t) λt + λ 3/ σ a B a (t) =λt + λ / C a B a (t) (for renewals) L(t) = S[A(t)] [ λt + λ 3/ σ a B a (t) ] + σ s B s (λt) (B fluctuations) µ = λ µ t + λ/ µ C ab a (t)+ λ/ µ C sλ / B s (λt) X(t) = L(t) t ( ρ)t + λ/ µ d = ( ρ)t + λ/ [ ] C a B a (t)+c s λ B s (λt) µ (C a + Cs ) / B(t) sum of independent BM = d BM ; selfsimilarity (both by characterization) = ( ρ)t + σb(t), where σ = µ ρ(c a + C s ) Recall: V obtained from X through reflection, and reflection is Lipshitz continuous. V RBM( ( ρ),σ) with stationary distribution exp ( mean = Hence, V ( ) d exp ( ) mean = ρ Ca +C s µ ρ v. significant Generalized P K, for EW q ) σ ( ρ). Approximation improves as ρ (heavy traffic) EW µ + ρ Ca + C s ρ }{{}}{{} utilization stoch. variability cost of congestion 0 strictly convex, increasing in ρ, C a,c s.

13 Priorities (M/G/ with priorities) K customer classes, indexed by k =,...,K. Highest priorities to, then,...; lowest to K. FCFS within priority class. Nonpreemptive first (Later, preemptive-resume). Class k: Poisson arrivals, at rate λ k General service time: m k = E(S k )ande(s k)bothfinite. Steady state ρ = ρ + + ρ K <, where ρ k = λ k m k. Note: ρ k = fraction of time allocated by server to class k. ρ = idleness/availability. Calculation of E(W k q ) = average wait of class k.. E(W q )=E(R)+m E(L q)=e(r)+ρ E(W q ) E(W q )=E(R)/( ρ ), as before (K =).. E(Wq )=E(R)+ m E(L q)+m E(L q) + m λ E(Wq ) }{{}}{{} wait due to class & in queue wait due to class, arriving during wait of. E(W q )=E(R)+ρ E(W q )+ρ E(W q )+ρ E(W q ) E(W q )=[E(R)+ρ E(W q )]/( ρ ρ )= = E(R)/[( ρ )( ρ ρ )] substitute k. EW k q = ER + m EL q + + m k EL k q + λ m EW k q + + λ k m k EW k q = ER + ρ EW q + + ρ k EW k q + (ρ + + ρ k )EW k q 3

14 E(W k q )= E(R)+ρ E(W q )+ + ρ k E(W k q ) ( ρ ρ ρ k ) Induction = E(R) ( ρ ρ k )( ρ ρ k ), k We now show E(R) = K λ k E(Sk) k= E(R) =( ρ) 0+ k ρ k m k +C k(s) = λ k E(Sk) k E(W k q )= Kj= λ j E(Sj ) ( ρ ρ k )( ρ ρ k ), k K. Convenient notation: ρ k = ρ + + ρ k, k K. Priorities: Preemptive Resume (e.g., Face-to-Face and Telephone) Now, Class k does not see classes k +,...,K. Recall: for M/G/-like queues, E(U) = E(R) ρ = E(W q) E(W k q )= E(R k k ) (ρ + + ρ k ) + λ j m j [E(Wq k )+m k ] j= j k preempts k = E(Rk ) ρ k + ρ k [E(W k q )+m k ] E(W k q ) = E(R k ) ( ρ k )( ρ k ) + ρ k ρ k m k where E(R k )= k j= ρ j m j +C (S j ) = k λ j E(Sj ) j= E(W k q )= k λ j E(Sj ) ( ρ k )( ρ k ) + ρ k E(S k ) ρ k 4

15 A Conservation Law (Kleinrock II, pg. 4) M/G/: For any work-conservation, non-preemptive strategy, k ρ k E(Wq k )= ρ E(R) ρ<, ρ = ρ. Proof E(U) =E(R)+ k m k E(L k q)=e(r)+ k ρ k E(W k q ) Recall: Unfinished work independent of strategy. M/G/-like: E(U) =E(W q )= E(R) ρ when ρ<. K ρ k E(Wq k )= k= ρ ρ K λ k E(Sk) k= When ρ>, at least one of the classes will have EW k q. Classic Application Suppose cost C k for one unit wait of class k. min k C k λ k E(W k q ) s.t. ρ k E(Wq k )=constant k Optimal Highest priority to largest C kλ k ρ k = C k m k = C k µ k. Interpretation: Equal m s costliest first Equal C s SPT (shortest processing time). 5

16 Generalizations G/G/S : steady-state when ρ< (subtle; e.g., Asmussen, S.) G/G/ : Conservation Laws and Performance Space Georgiadis, L. and Viniotis, I. On the conservation law and the performance space of single server systems, Op. Res 4, , 994. Priorities Simon, B. Priority queues with feedback J. Assoc. Computing Machinary 3, 34 49, 984. Cµ. With feedback: Klimov (long-run average) Tcha & Pliska (discounted). Costs that are convex increasing, say C k (t), k=,...,k. (C k (t) = cumulative cost for waiting t units of time, for class k.) (Van Mieghem, Jan A., Dynamic scheduling with convex delay costs: The generalized cµ rule, Annals of Applied Prob., 5, , 994.) Denote c k (t) =C k(t) marginal cost. Optimal (asymptotically): at time t, serve class k for which µ k c k (a(t)) is max; here a(t) = age of class k customer, at head of line. 3. In fact, the above Gcµ is asymptotically optimal under much broader circumstances. e.g., skills-based routing of an efficiency-driven multi-server multi-queue system, under sufficient skills - overlap. (Mandelbaum A. and Stolyar A., Scheduling Flexible Servers with Convex Delay Costs: Heavy-Traffic Optimality of the Generalized cµ Rule, submitted to Operations Research, January 00 ; Downloadable in our web site). 6