Class 9 Stochastic Markovian Service Station in Steady State - Part I: Classical Queueing Birth & Death Models

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1 Service Engineering Class 9 Stochastic Markovian Service Station in Steady State - Part I: Classical Queueing Birth & Death Models Service Engineering: Starting to Close the Circle. Workforce Management (WFM): Hierarchical Operational View. 4CallCenters A Personal Tool for Workforce Management. Markov Jump Processes (MJP): Ergodicity; The Method of Cuts; Reversibility. The M/M/ Queue. Infinite-Server Queues (M/M/ ). The Erlang-C (M/M/n) model. Pooling. Queueing Models with Blocking.

2 Service Engineering: A Subjective View Goal (Subjective): Develop scientifically-based design principles (rules-of-thumb) and tools (software) that support the balance of service quality, process efficiency and business profitability, from the (often conflicting) views of customers, servers and managers. Contrast/Complement the traditional and prevalent Service Management (U.S. Business Schools) Industrial Engineering (European/Japanese Engineering Schools) Examples: Staffing - How many agents required for balancing servicequality with operational efficiency (or, for maximizing profit). Skills-Based Routing (SBR) - Platinum and Gold and Silver customers, all seeking Information or Purchase or Technical Support, via Telephone or IVR or e.mail of Chat. Service Process Design + Staffing + SBR. Recipe for Progress in Research, Teaching, Applications: Simple Models at the Service of Complex Realities, with a pinch of a Multidisciplinary View (Operations, HRM, Marketing, MIS) = Service Engineering. 2

3 Workforce Management (WFM): Hierarchical Operational View Workforce Management: Hierarchical Operational View Forecasting Customers: Statistics, Time-Series Agents : HRM (Hire, Train; Incentives, Careers) Staffing: Queueing Theory # FTE s (Seats) per unit of time Service Level, Costs Shifts: IP, Combinatorial Optimization; LP Shift structure Union constraints, Costs Rostering: Heuristics, AI (Complex) Agents Assignments Individual constraints Skills-based Routing: Stochastic Control 3 74

4 Software Tool: 4CallCenters (4CC) - Mathematical Engine: Technion M.Sc. thesis of Ofer Garnett. - Used in Germany, India, Brazil,..., Israel. - Important tool in our course. - Free download from 4

5 Example of 4CC Output: Congestion Curves % Abandon vs. Calls/Hour for various Number of Agents %Abandon vs. Calls per Interval for various Number of Agents (E[Service] = 3:30 min, E[(Im)Patience] = 6:00 min.) 80.0% 70.0% 60.0% %Abandon 50.0% 40.0% 30.0% 20.0% 0.0%.0% Calls per Interval 5 4

6 A Long Line for a Shorter Wait at the Supermarket - New York Times Page of 4 Pooling Queues at a NYC Supermarket A Long Line for a Shorter Wait at the Supermarket Robert Caplin for The New York Times Sam Baris directing customers at Whole Foods in Columbus Circle, where the long line moves quickly. By MICHAEL BARBARO Published: June 23, 2007 Show New Yorkers a checkout line and they ll tell you whether it s worth the wait. SIGN IN TO OR SAVE THIS PRINT REPRINTS Readers Opinions Share Your Thoughts What are some of your recent checkout experiences? Post a Comment Multimedia Starbucks at 9 a.m.? Eight minutes, head to the next one down the street. Duane Reade at 6 p.m.? Twelve minutes, come back in the morning. SHARE But now a relative newcomer to Manhattan is trying to teach the locals a new rule of living: the longer the line, the shorter the wait. Come again? For its first stores here, Whole Foods, the gourmet supermarket, directs customers to form serpentine single lines that feed into a passel of cash registers. Banks have used a similar system for decades. But supermarkets, fearing a long line will scare off shoppers, have generally favored the one-line-per-register system. By 7 p.m. on a weeknight, the lines at each of the four Whole Foods stores in Manhattan can be 50 deep, but they zip along faster than most lines with 0 shoppers. 23/06/2007 6

7 Markov Jump Processes: Brief Review Characterization: i, j S; π 0 = (π 0 i ) - initial distribution, Q = [q ij ] - generator matrix. Steady-state equations: 0 = πq i π i =, π i 0 Ergodic Theorem: Let X be irreducible (i j). Assume that there exists a solution π to its steady-state equations. Then π is its (unique) stationary and limit-distribution. Transition rates: π i q ij = long-run (steady-state) transitionrate (number of transitions per unit of time) from i to j. Balance equations: For each state j S, its long-run (steadystate) entry-rate equals its exit-rate. Formally, i j π i q ij = π j q jj = i j π j q ji, j. Cuts: B S, the long-run (steady-state) transition rate from B to B c equals that from B c to B. Formally, i B j B c π i q ij = i B c j B π i q ij. B c B 7

8 Markov Jump Processes: Time-Reversibility, Birth & Death. Reversibility: A stochastic process X = {X t, < t < } is called reversible if for any r {X t, 0 t r} d = {X r t, 0 t r}. Fact. Ergodic MJP in steady-state is reversible if and only if the detailed-balance equations hold: π i q ij = π j q ji, i, j S. Birth & Death process: MJP on S = {0,, 2,...}, with jumps only between adjacent states: q ij = 0 if i j >. λ i- λ 0 λ λ i 0 2 i- i i+ µ µ 2 µ i µ i+ Cuts: π i λ i = π i+ µ i+ (π i q i,i+ = π i+ q i+,i ). (Take B = {0,,..., i} and B c = {i +, i + 2,...}.) Corollary. Every ergodic Birth & Death process is reversible. (Follows from the cut-equations.) 8

9 Service Station: Birth & Death Animation λ i- λ 0 λ λ i 0 2 i- i i+ µ µ 2 µ i µ i+ i number-in-system; λ i arrival rate, with i customers in system; µ i service rate, with i customers in system. Cuts at i i + : π i λ i = π i+ µ i+, i 0, which yields π i+ = λ i µ i+ π i = λ iλ i µ i+ µ i π i = = λ 0λ... λ i µ µ 2... µ i+ π 0. Stability: Steady-state (Limit) distribution exists if and only if i=0 in which case it is given by π i π 0 = λ 0... λ i µ... µ i+ <, = λ 0...λ i µ...µ i π 0, i 0, [ i 0 λ 0...λ ] i µ...µ i+ 9

10 Classical Markovian Queues Assumptions (from now on): n statistically identical independent (iid) servers; FCFS discipline First Come First Served; Work conservation: a server does not go idle if there are customers in need of service; Arriving customers all join and remain till end of service (do not abandon). Queueing Notations: G a /G s /n/k, where G a : General Arrivals (M for Poisson arrivals), G s : General Services (M for Exponential service times), n = number of servers, K = maximal number in system. Next: M/M/, M/M/, M/M/n, M/M/n/k, M/M/n/n. 0

11 Measures of Performance (Steady-State, Long-Run) λ = i 0 π i λ i - arrival rate = service rate - µ = i π i µ i L - number of customers in system (sometimes L s ); L q - number of customers in queue; W - sojourn time through the system (W s ); W q - waiting time in the queue. In steady state (in the long run), E[L] = k 0 k π k = lim T T T 0 L(t)dt. E[L q ] = k=n+ (k n) π k. Little s formula yields average times: E[L] = λ E[W ]; E[L q ] = λ E[W q ]. Average service time, E[S], must satisfy: E[W ] = E[W q ] + E[S].

12 Review: MJP MJP X = {X t, t 0} on S = {i, j,...} countable. Markov property: P r {X t = j X r, r < s; X s = i} = P ij (s, t), s < t, i, j S. Time homogeneity: P r {X s+t = j X s = i} = P ij (t), s, t, i, j, transition probabilities. Characterization: π 0 = initial distribution and P (t) = [P ij (t)], t 0, stochastic. Finite-dimensional distributions: P r {X 0 = i 0, X t = i,..., X tn = i n } = π 0 (i 0 )P i0,i (t )... P in,i n (t n t n ). P (t) : stochastic ; P (s + t) = P (s)p (t), s, t (Chapman Kolmogorov); P (0) = I ; P ( (0) = Q = [q ij ], infinitesimal generator j S q ij = 0 ). Micro to Macro : P (t) = P (t)q (= QP (t)) and P (0) = I Forward (Backward) equations. Solution : P (t) = exp[tq] = n=0 t n n! Q n, t 0. Animation: i q ij j; i, j S exponential clock at rate q ij, call it (i, j). Given i, consider clocks (i, j), j S; move to the winner when rings. Thus: stay at i exp(q i = j i q ij ) and switch to j with probability P ij = q ij /q i (q ij = q i P ij, i j; q ii = q i ). Transient analysis vs. long-run/limit stability/steady-state lim t P ij (t) = π j, i; π = πp (t), t. Calculation via steady-state equations: P ( ) = P ( )Q or balance equations: i j π i q ij = π j q jj = i j π j q ji, j. { 0 = πq i π i =, π i 0 } Transition rates: π i q ij = long-run average number of switches from i to j. Cuts: i B j B π iq c ij = i B c j B π i q ij, B S. B c B 2

13 Review: MJP 2 Ergodic Theorem: Let X be irreducible (i j). Assume that there exists a solution π to its steady-state equations. Then, X must be unexplosive and π must be its stationary distribution, its limit distribution and SLLN lim T T lim T T T 0 f(x t)dt = π i f(i) ( = Ef(X )) ; eg. f(x) = B (x). t Tg(X t, X t ) = i i π i q ij g(i, j), for g(x, x) = 0, x; e.g. g(x, y) = C (x, y). j Birth-and-death process: MJP on S = {0,, 2,...}, where all jumps are between adjacent states: q ij = 0 if i j >. Cuts: π i q i,i+ = π i+ q i+,i. (Take B = {0,,..., i} and B c = {i +, i + 2,...}.) Reversibility: any τ A stochastic process X = {X t, < t < } is called reversible if for {X t, < t < } d = {X τ t, < t < }. Fact. Ergodic MJP in steady-state is reversible if and only if the detailed balance equations hold: π i q ij = π j q ji, i, j S. Corollary. Every ergodic birth-and-death process is reversible. (Follows from the cut equations.) 3

14 The M/M/ Queue Poisson arrivals, at rate λ; Single exponential server, at rate µ (E[S] = /µ). λ λ λ 0 µ µ 2 i- µ i λ µ i+ Transition rates: λ i λ, i 0; µ i µ, i. Cut equations: λπ i = µπ i+, i 0. Traffic intensity ρ = λ µ < (iff stability). Steady-state distribution: L d = Geometric(p = ρ) (from 0): π i = ( ρ)ρ i, i 0. Hence, can calculate: E[L] = ρ/( ρ), then E[W ], then E[W q ], finally E[L q ]. Insightful calculations: Effective (actual) service rate = π n µ = µ( π 0 ) = µ ρ = µ λ µ = λ. n Contrast with Service Capacity (potential service rate) = µ. 4

15 M/M/: Sojourn Time Sojourn time is exponentially distributed: W d = exp mean = µ( ρ) = + µ ρ ρ. Proof: By PASTA and the memoryless-properly of Exponentials, W d = N i= N = L + d = Geom( ρ) (from ); N and {X i } are all independent. X i, X i exp(µ) i.i.d., Conclude by recalling: Geometric sum of iid Exponentials is Exponentially distributed. (The parameter is calculated via Wald.) Aside: The latter property is essentially Poisson-Splitting. A selfcontained proof can be give via Moment generating functions: φ W (t) = E [exp{tw }] = E exp{t N X i } i= = E E exp{t N X i } N i= (X i independent with E [exp{tx i }] = µ µ = E µ t µ( ρ) = µ t k=0 = φ exp(µ( ρ)) (t). N = µρ k= k µ t ( ρ)ρ k = µ t ) µ k µ t µ( ρ) µ( ρ) t 5

16 M/M/: Further Properties Delay probability (PASTA): P{W q > 0} = ρ. Waiting time in queue (given delay, it is exp): W q /µ d = 0 wp ρ exp ( mean = ) ρ wp ρ Note: E[ W q /µ ] = 0 ( ρ) + ρ ρ = Number-in-system/queue: E[L] = ρ ρ. ρ ρ ; E[L q] = ρ2 ρ. Server s utilization (occupancy) is ρ = λ/µ. Via Little s formula, applied to system = server : ρ = lim T T T 0 L Server(u) du = λ µ. Departure process in steady state is Poisson (λ) (Burke s theorem) useful in queueing networks. Support: Reversibility implies that the departure process equals (in distribution) the arrival process. Furthermore, Average inter-departure time = µ ρ + λ + ( ρ) = µ λ. 6

17 M/M/ Queue: 4CallCenters Average Waiting Times = E[S], for ρ = 50%; 4 E[S], for ρ = 80%; 9 E[S], for ρ = 90%;..., 9 E[S], for ρ = 95% (via model). 4CallCenters, performance measures: Average Speed of Answer (ASA) = E[W q ] (will be very different with abandonment); % Answer within Target = P{W q T }; (T = 0 important, as we ll learn later, but hardly used.) Average Queue Length = E[L q ]. 7

18 M/M/ Queue (Ample Servers) Poisson arrivals, rate λ; Infinite number of exponential servers, rate µ (E(S) = /µ). λ λ λ λ 0 2 i- i i+ µ 2µ iµ (i+)µ λ i λ, i 0; µ i = i µ, i. Cut equations: λ π i = (i + )µ π i+, i 0. Always stable. Steady-state distribution is Poisson: π i = e R Ri i!, i 0 ; R = λ is the offered load, (measured in units of Erlangs) µ = Average amount of work-units (time-units of service) that arrives to the system per time-unit = Average number of customers in steady-state (via Little s Law). Offered-Load Extension to a time-varying environment: Consider M t /M/ (time-inhomogeneous Poisson arrivals, at rate λ(t), t 0): R(t) = E t λ(u) du = Eλ(t S), t 0. t S 8

19 M/M/ Queue: Continued Very useful: -server models provide bounds (ideal): Recall, DS-PERT s with Infinite-Servers. Later, queues with abandonment. Average number-in-system in steady-state (via Little s formula): E[L] = E(# Busy Servers) = λ µ = R. All the steady-state results are valid, as is, also for M/G/ generally distributed service times. (Insensitivity of performance to the service-time distribution.) Time-Varying Arrivals: The observation R = E[L] paves the way to the definition of offered-load in a time-varying environment: In M t /G/, with a time-inhomogeneous Poisson arrival process at rate {λ(t), t 0}), the average number-in-system at time t is: R(t) = E t t S λ(u) du = Eλ(t S e), t 0. Here S denotes a service-time and S e a residual service-time (as in biased-sampling). In the special case of S exponential, S e has the same exponential distribution as S, due to the memoryless property. 9

20 The Erlang-C (M/M/n) Queue Poisson arrivals, rate λ; n exponential servers, each at rate µ. Erlang-C Widely used in call centers - the workhorse of WFM. agents arrivals ACD queue Transition-rate diagram: λ λ λ 0 µ 2µ 2 n- nµ n λ λ n+ n+2 nµ nµ Erlang-B λ j λ, j 0, Lost Calls agents µ j = (j n)µ, j. Agents utilization arrivals ρ = λ nµ. Assume ρ < (R < n) to ensure stability (in analogy to M/M/). 20

21 Example of Instability, via 4CallCenters Steady-state distribution: π i π 0 = = Ri π 0, i n, i! = nn ρ i π 0, i n, n! n j=0 R j j! + where R = λ/µ is the offered load. R n n!( ρ), 2

22 M/M/n Queue: Properties Erlang-C Formula (97) for the Delay probability: P{W q > 0} = E 2,n = π i = Rn n! i n ρ π 0. Erlang-C computation: via recursion, see Erlang-B below. Erlang-C approximations: important later in course. Number-in-queue: P{L q = i} = E 2,n ( ρ)ρ i, i > 0, or L q = 0 wp E 2,n Geom 0 ( ρ) wp E 2,n Waiting time distribution: W q /µ = Compare with M/M/! 0 wp E 2,n Exp ( mean = n ρ) wp E2,n Departure process: Poisson(λ) in steady-state. Proof via reversibility, as with M/M/. 22

23 M/M/n Queue: Waiting-Time Distribution In analogy to M/M/, Formally: P E[S] W q W q > 0 d = exp(n( ρ)), E[S] W q > t W q > 0 = e n( ρ)t. P{W q > t} = k= P{L q = k } P{E k > t} (where E k d = Erlang(k, nµ)) = E 2,n k= ( ρ)ρ k t = E 2,n nµ( ρ) t = E 2,n nµ( ρ) t = E 2,n e nµ( ρ)t. e nµx nµ(nµx) k (k )! k= e nµ( ρ)x dx e nµx dx (nµρx) k dx (k )! 23

24 Pooling: Economies of Scale Kleinrock, L. Vol. II, Chapter 5 (976) (Pelephone s Call Center) Example: Kleinrock, L. Vol.II, Resource Chapter Sharing 5 (976) (a) m m (d) C m C m C C (b) m (e) m m C m C m C C (c) m (f) m m C mc Simplest is Best! Do not model complicated undesirable scenarios! 4CallCenters output m M/M/ scale-up M/M/m technology M/M/ λ, µ mλ, µ mλ, mµ Combine: queues servers Saved inefficiency idleness lost capacity ( long queue, 2 idle) (rate mµ at all times) ( ) Remark EW q m, λ, µ m EWq (, λ, µ) ( ) while EW s m, λ, µ m EWs (, λ, µ) individual server s capacity (Explain, via P m (Wait > 0), noting W q W q > 0.) Summary (pg. 287) Large systems (scaling up input rate and system capacity) yield improvements (in average response-time) that are proportional to the scaling factor. For a given scale factor, the single-server (fast) system is superior to the multiple-server (slow) system, as far as total time a system in concerned. The opposite is true, however, when restricting to only waiting time. (See Homework)

25 Pooling M/M/ to M/M/n Pooling Queues (one vs. many) and Servers (slow vs. fast) Erlang-C n M M λ, μ Multiple queues Multiple Servers 2 M M n nλ, μ Single queue Multiple Servers 3 M M nλ, nμ Single queue Single server λ λ. n μ μ nλ. n μ μ nλ nμ Tradeoff Enabler Gain Utilization Separate vs. single queue Process design Load balancing t (avoid a long queue & idle servers) λ μ = ρ ( nλ ) ( n) μ = ρ Slow vs. fast server Technology Maximal capacity nμ vs. nμ ( nλ ) ( n ) t μ = ρ PW { q > 0} ρ 2 E2, n( ρ) 2 ρ E W q W > 0 q μ ρ nμ ρ = nμ ρ EW q = [ ] E W W > 0 P{ W > 0} q q q ρ μ ρ 4 E2, n μ n( ρ) 3 ρ μ n( ρ) E [ S] μ = μ > nμ E W sys = = E[ W ] + E[ S] q μ ρ 5 E μ n( ρ) 2, n + 6 nμ ρ 25

26 Basic Relations: E2, n E2, n ( ρ) ρ n ρ Intuition? Proof: E = P( at least oneserver idle) = 2, n n n P{ server i idle} = n( ρ) = P { server iidle} i= i= P{ server i idle} = ρ qed... Corollary: ρ E2, n E2, n ρ 2 3 Corollary: E E n ρ n( ρ) n( ρ) + 2, n 2, n ( ) EW ( n, nλ, μ) EW (, nλ, nμ) EW ( n, nλ, μ) EW (, nλ, nμ) q q sys sys Multiple-slow Single-fast Multiple-slow Single-fast Intuition: E2, ( ρ) ρ n nλ. n μ μ vs. random nλ λ λ. n μ μ E2, n ( ρ ) ρ since have an earlier commitment 26

27 Pooling M/M/ to M/M/n: Conclusions 2 : Pooling yields E[W q ] decrease by more than factor n; 3 : Fast server yields E[W ] and E[W q ] decrease by factor n; 2 3 : Fast server better for E[W ]; Pooling better for E[W q ]. 27

28 M/M/n/K Queue Poisson arrivals, rate λ; n exponential servers, rate µ; K trunks (K n); If all trunks busy, arriving customer blocked (busy signal). λ λ λ λ 0 n- n n+ K- K µ nµ nµ nµ λ j = λ, 0 j K, µ j = (j n)µ, j K. Formulae straightforward but cumbersome (simply truncate M/M/n). Always reaches steady state. 28

29 M/M/n/K Queue: 4CallCenters Use Change Settings = Features = Trunks. Note new indicators: Average Trunks Utilized and %Blocked. 4CallCenters: Advanced Profiling Arrival rate varied from 900 to 07 per hour, in step 9. Excel interface: graphs and spreadsheets. 29

30 M/M/n/K vs. Erlang-C Average service time = 6 min, 00 agents, 50 trunks Average Speed of Answer (secs) Calls per Interval M/M/00/50 M/M/00 Similar performance for light loads. Erlang-C explodes as ρ = λ nµ. 30

31 The Erlang-B (M/M/n/n) Queue λ λ λ 0 µ 2µ 2 n- nµ n λ i λ, 0 i n, µ i = i µ, i n. Steady-State: π i = Ri i! / n j=0 R j j!, 0 i n. Note: The above applies to M/G/n/n - again, insensitivity. 3

32 M/M/n/K vs. Erlang-B l Average service time = 6 min, 00 agents 40% 35% 30% 25% %Block 20% 5% 0% 5% 0% Calls per Interval M/M/00/50 M/M/00/00 Moderate load: additional trunks prevent blocking. Heavy load: % blocking /ρ ( fluid limit ). 32

33 Erlang-B Formula (97) Loss probability: E,n = π n = Rn n! / n j=0 Follows from PASTA. Recall: Erlang-B valid for M/G/n/n (General services.) λπ n rate of lost customers, λ( π n ) effective throughput. Erlang-B Computation via recursion: R j j!. E,n = RE,n n + RE,n = ρe,n + ρe,n E,0 =. Erlang-B Computation: E,n = (n R)E 2,n n RE 2,n ; E 2,n = E,n ( ρ) + ρe,n ; E 2,n > E,n, as expected: why? 33

34 Telephone Call Center arrivals retrials lost calls busy ACD 23 queue k agents 2 retrials abandonment n lost calls returns Two customer-centric (subjective) operational measures of performance: Abandonment due to (im)patience, or need; Retrials/Redials, which can often be absorbed into the Poisson arrival process). How to model Abandonment? The Palm / Erlang-A model. 34