Fluid Models of Parallel Service Systems under FCFS

Transcription

1 Fluid Models of Parallel Service Systems under FCFS Hanqin Zhang Business School, National University of Singapore Joint work with Yuval Nov and Gideon Weiss from The University of Haifa, Israel Queueing and Networks, IMA, 208 / 39

2 µ 2,3 µ 3,3 µ 3,4 Parallel Service System Parallel servers S = {s,..., s J } of various skills; Various customer types C = {c,..., c I }; A bipartite compatibility graph G where (s j, c i ) G if server s j can serve customers of type c i λ α α 2 α 3 α 4 c c 2 c 3 c 4 µ 2,2 µ, µ,2 µ,4 µ 3, s s 2 s 3 Figure: A parallel skilled based service system with 3-server and 4-customer-type 2 / 39

3 µ 2,3 µ 3,3 µ 3,4 λ α α 2 α 3 α 4 c c 2 c 3 c 4 µ 2,2 µ, µ,2 µ,4 µ 3, s s 2 s 3 Figure: A parallel skilled based service system with 3-server and 4-customer-type C(s j ): the customer types compatible with s j ; For example, C(s 3 ) = {c, c 3, c 4 }. S(c i ): the servers compatible with customers of type c i. For example, S(c 2 ) = {s, s 2 }. S(C) = c i C S(c i), C(S) = s j S C(s j), U(S) = C(S), α C = c i C α c i. 3 / 39

4 a(l) denotes the arrival time of the lth customer; u(l) = a(l) a(l ) is the interarrival times, where l = 0, ±, ±2,..., and a(0) 0 < a(); {u(l) : l = 0, ±, ±2,...} is a sequence of iid with mean /λ; A(t) = max{l : a(l) t}; A(t) is the number of customers arriving in (t, 0] for t < 0; A(t) gives the number of customers arriving in (0, t] for t > 0; A(t) A(s) counts the total number of arrivals in (s, t] with s < t. 4 / 39

5 Type c i has probability α ci, i =,..., I, let ξ(l) be a unit vector of length I such that ξ i (l) = if customer l is of type c i, for l = 0, ±, ±2,.... The counts of arrivals of customers of each type are then given by A(t) ξ i (l), t 0, l= A ci (t) = 0 ξ i (l), t < 0. l=a(t)+ 5 / 39

6 v sj,c i (0) be the remaining service time of server s j if he is serving a customer of type c i at time 0, and v sj,c i (0) = 0 otherwise; v sj,c i (k), k =, 2,..., be the processing time of the kth customer of type c i that server s j is serving after time 0 with mean /µ sj,c i ; X sj,c i (t) = max{k + : k l=0 v s j,c i (l) t}. 6 / 39

7 C S Bipartite Graph customers 3 Key Customer in queue Customer in service Arrival Service Start Departure Server Position at 0 Server Position at t Customer type Arrival time a( ) a(-6) a(-5) a(-4) a(-3) a(-2) 0 a(0) a(-) 3 t a() a(2) a(4) a(6) a(8) a(0) a(3) a(5) a(7) a(9) 2 time Figure: Dynamics of a 3-server and 3-customer-type system under FCFS-ALIS 7 / 39

8 P sj (t) is the position of server s j at time t, where we let P sj (t) = l if the server is serving at time t the lth customer in the sequence of arrivals. If servers s j,..., s jk are idle at time t then their positions are defined as A(t) +,..., A(t) + k, ordered by duration of idleness, with A(t) + k the longest idle; Let Y j (t) is the current jth level, where we let Y (t) <... < Y J (t) be the ordered set of the positions of J servers at time t; T sj,c i (t) is the cumulative time over (0, t) that server s j has served customers of type c i. M (t) = arg min{p sj (t) : s j S}, M j (t) = arg min {P sl (t) : s l S \ {M (t),, M j (t)}}. Clearly, P Mj (t)(t) = Y j (t) for j =,, J. 8 / 39

9 We let Q ci,j(t) denote the number of customers of type c i which are waiting between servers M j (t) and M j+ (t) at time t. These are given by: Q ci,j(t) = Y j+ (t) l=y j (t)+ A(t) l=y J (t)+ ξ i (l) I{c i U(M (t),..., M j (t))}, j =,..., J, ξ i (l), j = J. Let Q ci (t) be the number of type-c i customers in the system at time t. Then, Q ci (t) = Q ci (0) + A ci (t) X sj,c i (T sj,c i (t)). (2) s j S(c i ) Furthermore, the work-conserving principle gives that for t 0, t ( ) +d A(x)+ P sj (x) (x ) T sj,c i (x) = 0, s j S. (3) 0 c i C(s j ) () 9 / 39

10 U(t): the remaining time at time t until next arrival; V sj,c i (t): the remaining processing time of c i by server s j if it is processing a type c i customer at time t, and V sj,c i (t) = 0 otherwise; R sj,c i (t): the number of type-c i customers served by server s j by time t; Z(t) = ( A(t) P sj (t), Q ci,j(t), U(t), V sj,c i (t) ) are Markov processes. We say that the parallel service system is stable (ergodic) if {Z(t) : t 0} is positive Harris recurrent (ergodic). Determining condition for the stability of {Z(t) : t 0}; Finding ( condition for ) P Pr lim s (t) P t t = = lim sj (t) t t = ; Finding condition and determining constant r sj,c i such that ( ) R sj,c Pr lim i (t) t t = r sj,c i = for (s j, c i ) G. 0 / 39

11 Corresponding to ()-(3), our fluid model given by {( Tsj,c i (t), P sj (t), Ȳj(t), Q } ci,j(t)) : t 0 is: P Mj (t) (t) = Ȳj(t), j =,..., J, (4) α ci (Ȳj+(t) Q Ȳj(t)) I{c i U( M (t),..., M j (t))}, ci,j(t) = j =,..., J, (5) α ci (λt ȲJ(t)), j = J, Q ci (t) = Q ci (0) + λα ci t µ sj,c i Tsj,c i (t), (6) t 0 s j S(c i ) ( λx P ) +d sj (x) (x In addition, we assume that c i C(s j ) ) T sj,c i (x) = 0, s j S. (7) T sj,c i (x) are zero for (s j, c i ) / G, and nondecreasing with c i C(s j ) ( T sj,c i (t 2 ) T sj,c i (t )) t 2 t. (8) / 39

12 Theorem : For almost all sample paths ω and any subsequence of {( T n s j,c i (t), P n s j (t), Ȳ n j (t), Q n c i,j(t)) : n } contains a subsequence along which {( T n s j,c i (t), P n s j (t), Ȳ n j (t), Q n c i,j(t)) : n } converges u.o.c. in (0, ) and any of their limits, ( T sj,c i (t), P sj (t), Ȳj(t), Q ci,j(t)), jointly satisfies relations (4)-(8). Moreover, P sj (t), Ȳj(t), and Q ci,j(t) are Lipschitz continuous on (0, ), and T sj,c i (t) are Lipschitz continuous on [0, ) for s j S, c i C, j =,..., J, and (s j, c i ) G. 2 / 39

13 Definition : Consider the fluid model given by (( P sj (0), λα ci, µ sj,c i ) : s i S, c i C, (s j, c i ) G). Let P (0) = J P j= sj (0). (i) We say that the fluid model is stable if starting from any initial state P (0) with P (0) =, there exists t 0 such that for any (Ȳ(t),, ȲJ(t)) satisfying (4)-(8), λt Ȳ(t) = 0 for all t > t 0. (ii) We say that the fluid model has complete resource pooling if for all values of λ, starting from any initial state P (0) with P (0) =, there exists t 0 such that for any (Ȳ(t),, ȲJ(t)) satisfying (4)-(8), ȲJ(t) Ȳ(t) = 0 for all t > t 0. (iii) We say that the fluid model has complete weak resource pooling if for all values of λ, starting from any initial state P (0) with P (0) =, there exists t 0 such that for any (Ȳ(t),, ȲJ(t)) satisfying (4)-(8), Ȳ (t) = = Ȳ J (t) for all t > t 0. 3 / 39

14 Customers Arrivals 0 Server Time Server 2 C Bipartite Graph 2 3 Server 3 S 2 3 Figure: Conjectured Fluid Dynamics under FCFS-ALIS 4 / 39

15 Definition 2: (i) We say that a cumulative probability distribution H( ) of a non-negative random variable has unbounded support if H(t) < for all t. (ii) We say that a cumulative probability distribution H(t) of a non-negative random variable is spread out if there exists a function q(t) > 0 such that 0 q(t)dt > 0 and a constant k, such that H (k) (b) H (k) (a) b a q(t)dt for all 0 a < b <, where H (k) is the k-fold convolution of H( ). Theorem 2 Assume that the distributions of the interarrival times and of the service times have unbounded support and are spread out. (i) If the fluid limit model is stable, then {Z(t) : t 0} is ergodic. (ii) If the fluid limit model has complete resource pooling, then for λ large enough, the joint distribution of (Q ci,j(t), c i U(M (t),..., M j (t)), j J ) converges to a stationary distribution as t. 5 / 39

16 Proposition : Consider a fluid limit ( T sj,c i (t), P sj (t), Ȳj(t), Q ci,j(t)). If fluid server positions at levels k,..., l merge at time τ > 0, i.e., Ȳ k (τ) < Ȳk(τ) = = Ȳl(τ) < Ȳl+(τ) (or if l = J, Ȳl(τ) < λτ), for some k l, then the fluid limit must satisfy T Mj,c i (τ) =, j = k,..., l, c i C(M j )\C(M l+,...,m J ) Ȳ k (τ) = = Ȳ l (τ) = α ci l µ Mj,c i TMj,c i (τ) j=k for c i C(M k,..., M l ) \ C(M l+,..., M J ). 6 / 39

17 µ 2,3 µ 3,3 µ 3,4 Product-Form Service Rates: λ α α 2 α 3 α 4 c c 2 c 3 c 4 µ 2,2 µ, µ,2 µ,4 µ 3, s s 2 s 3 Assume µ sj,c i = µ sj ν ci, where we can think of /ν ci as the average work required by a customer of type c i and µ sj as the speed of server s j. This models the case where customer types that share the same server differ only in the amount of work that they require, and where servers that can serve the same type of customer differ only in the speed at which they perform the service. 7 / 39

18 µ 2,3 µ 3,3 µ 3,4 λ α α 2 α 3 α 4 c c 2 c 3 c 4 µ 2,2 µ, µ,2 µ,4 µ 3, s s 2 s 3 Service Rates Depend Only on Server: µ sj = µ sj,c i for all (s j, c i ) G. Service Rates Depend Only on Customer Type: ν ci = µ sj,c i for all (s j, c i ) G. 8 / 39

19 Condition for complete resource pooling in the product-form service rate (µ sj,c i = µ sj ν ci ) case: Introduce the following notation: for any c i, θ ci = αc i ν ci / c i C α ci ν ci ; for any C C, θ C = c i C θ c i ; for any s j, β sj = µ sj / s j S µ s j ; and for any S S, β S = s j S β s j. The following three conditions are equivalent (Condition A): (i) β S(C) > θ C ; (ii) θ C(S) > β S ; (iii) β S > θ U(S). 9 / 39

20 Theorem 3: Consider a parallel service fluid model with product-form service rate. (i) Assume that Condition A holds, then complete resource pooling holds, that is, for any initial conditions there exists t 0 such that for λ large enough, every fluid limit satisfies Ȳ (t) = = ȲJ(t) and Ȳ (t) = = Ȳ J (t) = µ for all t > t 0, where µ = / α ci µ sj. ν ci s j S c i C (ii) Assume that Condition A holds only as weak inequalities. Then complete weak resource pooling holds, and furthermore, for any initial conditions there exists t 0 such that for λ large enough, every fluid limit satisfies Ȳ (t) = = Ȳ J (t) = µ. 20 / 39

21 Network with Complete Bipartite Compatibility Graph: Every server can serve all types of customers, i.e., the compatibility graph is a complete bipartite graph. m sj = c i C α ci m sj,c i, µ sj = m s j, µ = J µ sj. (9) Theorem 4: Consider the FCFS service discipline fluid model with a complete bipartite compatibility graph. We have (i) It is always complete resource pooling; (ii) The fluid server-level process is given by j= Ȳ (t) = = ȲJ(t) = min ( Ȳ J (0) + µt, λt ), t > 0, where µ is given in (9); (iii) When ȲJ(t) < λt, the matching rates are given r sj,c i = µ sj α ci /(µ c k C α ck ). 2 / 39

22 Network with Tree Bipartite Compatibility Graph: A tree graph is a connected graph with no loops. With K nodes it will have exactly (K ) edges, and it will always have at least two leaves c i C(s k ) T sk,c i (τ) =, Psk (τ) = P s (τ) = = PsJ (τ). s j S(c m) This means that the set of linear equations { c i C(s k ) η s k,c i = s k S, s j S(c m) µ sj,c m α cm µ sj,cm α cm η sj,c m = µ, c m C Tsj,c m (τ), (0) with the (I + J ) unknowns η sj,c i, (s j, c i ) G and an additional unknown µ has a positive solution. Thus complete resource pooling will relate to the set of linear equations (0). 22 / 39

23 Theorem 5: Consider a fluid model with tree bipartite compatibility graph. (i) The system will have complete resource pooling if and only if (0) has a positive solution, and it will have complete weak resource pooling if the solution is non-negative. (ii) If complete resource pooling holds then µ given by (0) is the pooled service rate, and the matching rates are given by r sj,c i = µ sj,c i η sj,c i /µ. 23 / 39

24 Foss and Chernova s Example (stability depends on distributions rather than the first moments): (On the stability of a partially accessible multistation queue with state-dependent routing. Queueing Systems, 29(998), 55-73) 3-server, 3-customer-type and an almost complete bipartite compatibility graph as illustrated in the following Figure. C λ λ λ 2 3 F L S 2 3 F R α c = α c2 = α c3 = /3, and the service time distributions are v s,c 2 F L, v s,c 3 F R ; v s2,c F R, v s2,c 3 F L ; v s3,c F L, v s3,c 2 F R with means a L and a R respectively. f L (t) = M e t/m, f R (t) = + A A e t/a + A + A Ae At 24 / 39

25 Let L : service time = exp( M ); R : service time = exp(a); R 2 : service time = exp( A ). Then the state-space can be written as for n = 0,,, (n, L, L, L), (n, L, L, R ), (n, L, L, R 2 ), (n, L, R, L), (n, L, R, R ), (n, L, R, R 2 ), (n, L, R 2, L), (n, L, R 2, R ), (n, L, R 2, R 2 ); (n, R, L, L), (n, R, L, R ), (n, R, L, R 2 ), (n, R, R, L), (n, R, R, R ), (n, R, R, R 2 ), (n, R, R 2, L), (n, R, R 2, R ), (n, R, R 2, R 2 ); (n, R 2, L, L), (n, R 2, L, R ), (n, R 2, L, R 2 ), (n, R 2, R, L), (n, R 2, R, R ), (n, R 2, R, R 2 ), (n, R 2, R 2, L), (n, R 2, R 2, R ), (n, R 2, R 2, R 2 ). 25 / 39

26 The generator is given by D 0 D D 2 D 3 C 0 C C 2 C 3 0 C 0 C C C 0 C where the first block-row corresponds to state n = 0, the second block-row corresponds to state n =, and so on. 26 / 39

27 D k = C k = D k D2 k D3 k D2 k D22 k D23 k D3 k D32 k D33 k C k C2 k C3 k C2 k C22 k C23 k C3 k C32 k C33 k, k = 0,, 2, ;, k = 0,, 2,. First consider the transition rate matrix D 0. More specifically, D 0 is the transition rates between (0, L, L, L), (0, L, L, R ), (0, L, L, R 2 ), (0, L, R, L), (0, L, R, R ), (0, L, R, R 2 ), (0, L, R 2, L), (0, L, R 2, R ), and (0, L, R 2, R 2 ). 27 / 39

28 D 0 = δ a δ a 0 δ a 3 M 0 δ a 0 0 δ a 3 a 0 0 δ a 0 3 M δ a 3 a δ a δ / 39

29 Now define G to be the minimal nonnegative solution to C k G k = 0, k=0 D k = D k+i G i, Ck = i=0 C k+i G i, k =, 2,. i=0 Then π n = 0 = ( π 0 D n n + π j C ) ( ), n+ j C n = 2, 3,, (π 0, π ) j= [ D0 j=0 D +jg j C 0 j=0 C +jg j ]. 29 / 39

30 M = alpha hat A Figure: The fraction of customers served with service time distribution F L, as a function of A. 30 / 39

31 M = 5 alpha hat Figure: The fraction of customers served with service time distribution F L, as a function of A. A 3 / 39

32 Matching Rates Depends on Service Distribution: C 2 3 C S 2 3 () S (2) C S (3) Figure: Topologies of the systems for the simulation study. For each system, we used four service time distributions: System : α = (.2,.6,.2) and µ = (.4,.2,.4); System 2: α = (.,.4,.4,.) and µ = (.4,.3,.2,.); System 3: α = (.,.2,.2,.,.2,.2) and µ = (.05,.,.5,.2,.2,.3); 32 / 39

33 Exponential; Pareto (denoted by the subscript p ), density f(x) = 3γ(γx + ) 4, x 0 with mean /2γ; Uniform : U(0, 2/µ) with µ = 2γ; Uniform 2: U(.9/µ,./µ) with µ = 2γ. System Exponential Pareto Uniform Uniform *.0078 * * * * Table: Resulting p-values of Hotelling s T 2 test. * denote p-value < / 39

34 System r =.3 0.3, r p = r u = , r u2 = (see Adan and Weiss, Exact FCFS matching rates for two infinite multi-type sequences. Operations Research, 60(20), ) 34 / 39

35 System 2 r = , r p = r u = , r u2 = / 39

36 System r = r p = / 39

37 r u = r u2 = / 39

38 In summary, by Hotelling s T 2 -test, the matching rates of systems 2 and 3 under the Pareto or uniform service time assumption are different from the exponential service time assumption in a statistically significant manner. 38 / 39

39 Summary: Stability for the parallel service system; Completely resource pooling; Matching rate; The first moment is not sufficient to determine the stability, the completely resource pooling, and matching rate. 39 / 39