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 , 2018 1 / 39
Parallel Service System ◮ Parallel servers S = { s 1 , . . . , s J } of various skills; ◮ Various customer types C = { c 1 , . . . , 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 1 α 2 α 3 α 4 c 2 c 3 c 4 c 1 µ 2,3 µ 2,2 µ 1,1 µ 1,2 µ 3,3 µ 3,4 µ 3,1 µ 1,4 s 1 s 2 s 3 Figure: A parallel skilled based service system with 3-server and 4-customer-type 2 / 39
λ α 1 α 2 α 3 α 4 c 2 c 3 c 4 c 1 µ 2,2 µ 2,3 µ 1,1 µ 1,2 µ 3,3 µ 3,4 µ 3,1 µ 1,4 s 1 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 1 , c 3 , c 4 } . • S ( c i ): the servers compatible with customers of type c i . For example, S ( c 2 ) = { s 1 , 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
◮ a ( ℓ ) denotes the arrival time of the ℓ th customer; ◮ u ( ℓ ) = a ( ℓ ) − a ( ℓ − 1) is the interarrival times, where ℓ = 0 , ± 1 , ± 2 , . . . , and a (0) ≤ 0 < a (1); ◮ { u ( ℓ ) : ℓ = 0 , ± 1 , ± 2 , . . . } is a sequence of iid with mean 1 /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
Type c i has probability α c i , i = 1 , . . . , I , let ξ ( ℓ ) be a unit vector of length I such that ξ i ( ℓ ) = 1 if customer ℓ is of type c i , for ℓ = 0 , ± 1 , ± 2 , . . . . The counts of arrivals of customers of each type are then given by A ( t ) � ξ i ( ℓ ) , t ≥ 0 , ℓ =1 A c i ( t ) = 0 � − ξ i ( ℓ ) , t < 0 . ℓ = A ( t )+1 5 / 39
◮ v s j ,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 s j ,c i (0) = 0 otherwise; ◮ v s j ,c i ( k ) , k = 1 , 2 , . . . , be the processing time of the k th customer of type c i that server s j is serving after time 0 with mean 1 /µ s j ,c i ; ◮ X s j ,c i ( t ) = max { k + 1 : � k ℓ =0 v s j ,c i ( ℓ ) ≤ t } . 6 / 39
customers Bipartite Graph 1 C 2 1 3 3 S 2 1 3 1 Key 2 Customer in queue Customer in service 1 1 Arrival Service Start 3 Departure Server Position at 0 1 Server Position at t 3 Customer type 2 Arrival time a ( ) 3 1 0 t time a (0) a (-6) a (-4) a (-2) a (2) a (4) a (6) a (8) a (10) 2 a (-5) a (-3) a (-1) a (1) a (3) a (5) a (7) a (9) 3 1 2 3 3 2 2 2 1 1 Figure: Dynamics of a 3-server and 3-customer-type system under FCFS-ALIS 7 / 39
◮ P s j ( t ) is the position of server s j at time t , where we let P s j ( t ) = ℓ if the server is serving at time t the ℓ th customer in the sequence of arrivals. If servers s j 1 , . . . , s j k are idle at time t then their positions are defined as A ( t ) + 1 , . . . , A ( t ) + k , ordered by duration of idleness, with A ( t ) + k the longest idle; ◮ Y j ( t ) is the current j th level, where we let Y 1 ( t ) < . . . < Y J ( t ) be the ordered set of the positions of J servers at time t ; ◮ T s j ,c i ( t ) is the cumulative time over (0 , t ) that server s j has served customers of type c i . Let M 1 ( t ) = arg min { P s j ( t ) : s j ∈ S} , M j ( t ) = arg min { P s ℓ ( t ) : s ℓ ∈ S \ { M 1 ( t ) , · · · , M j − 1 ( t ) }} . Clearly, P M j ( t ) ( t ) = Y j ( t ) for j = 1 , · · · , J . 8 / 39
We let Q c i ,j ( t ) denote the number of customers of type c i which are waiting between servers M j ( t ) and M j +1 ( t ) at time t . These are given by: Y j +1 ( t ) − 1 � ξ i ( ℓ ) I { c i ∈ U ( M 1 ( t ) , . . . , M j ( t )) } , ℓ = Y j ( t )+1 Q c i ,j ( t ) = j = 1 , . . . , J − 1 , (1) A ( t ) � ξ i ( ℓ ) , j = J. ℓ = Y J ( t )+1 Let Q c i ( t ) be the number of type- c i customers in the system at time t . Then, � Q c i ( t ) = Q c i (0) + A c i ( t ) − X s j ,c i ( T s j ,c i ( t )) . (2) s j ∈S ( c i ) Furthermore, the work-conserving principle gives that for t ≥ 0, � t � � + � � � A ( x )+1 − P s j ( x ) x − T s j ,c i ( x ) = 0 , s j ∈ S . (3) d 0 c i ∈C ( s j ) 9 / 39
◮ U ( t ): the remaining time at time t until next arrival; ◮ V s j ,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 s j ,c i ( t ) = 0 otherwise; ◮ R s j ,c i ( t ): the number of type- c i customers served by server s j by time t ; � � ◮ Z ( t ) = A ( t ) − P s j ( t ) , Q c i ,j ( t ) , U ( t ) , V s j ,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 s 1 ( t ) P sJ ( t ) lim t →∞ = · · · = lim t →∞ = 1; Pr t t • Finding condition and determining constant r s j ,c i such that � � R sj,ci ( t ) lim t →∞ = r s j ,c i = 1 for ( s j , c i ) ∈ G . Pr t 10 / 39
Corresponding to (1)-(3), our fluid model given by �� � � T s j ,c i ( t ) , ¯ ¯ P s j ( t ) , ¯ Y j ( t ) , ¯ Q c i ,j ( t ) : t ≥ 0 is: ¯ M j ( t ) ( t ) = ¯ P ¯ Y j ( t ) , j = 1 , . . . , J, (4) α c i ( ¯ Y j +1 ( t ) − ¯ Y j ( t )) I { c i ∈ U ( ¯ M 1 ( t ) , . . . , ¯ M j ( t )) } , ¯ Q c i ,j ( t ) = j = 1 , . . . , J − 1 , (5) α c i ( λt − ¯ Y J ( t )) , j = J, � Q c i ( t ) = ¯ ¯ µ s j ,c i ¯ Q c i (0) + λα c i t − T s j ,c i ( t ) , (6) s j ∈S ( c i ) � t � � + � � � λx − ¯ ¯ P s j ( x ) d x − T s j ,c i ( x ) = 0 , s j ∈ S . (7) 0 c i ∈C ( s j ) In addition, we assume that ¯ T s j ,c i ( x ) are zero for ( s j , c i ) / ∈ G , and nondecreasing with � c i ∈C ( s j ) ( ¯ T s j ,c i ( t 2 ) − ¯ T s j ,c i ( t 1 )) ≤ t 2 − t 1 . (8) 11 / 39
Theorem 1: For almost all sample paths ω and any subsequence of { ( ¯ s j ,c i ( t ) , ¯ s j ( t ) , ¯ j ( t ) , ¯ T n P n Y n Q n c i ,j ( t )) : n ≥ 1 } contains a subsequence along which { ( ¯ s j ,c i ( t ) , ¯ s j ( t ) , ¯ j ( t ) , ¯ T n P n Y n Q n c i ,j ( t )) : n ≥ 1 } converges u.o.c. in (0 , ∞ ) and any of their limits, ( ¯ T s j ,c i ( t ) , ¯ P s j ( t ) , ¯ Y j ( t ) , ¯ Q c i ,j ( t )) , jointly satisfies relations (4)-(8) . Moreover, ¯ P s j ( t ) , ¯ Y j ( t ) , and Q c i ,j ( t ) are Lipschitz continuous on (0 , ∞ ) , and ¯ ¯ T s j ,c i ( t ) are Lipschitz continuous on [0 , ∞ ) for s j ∈ S , c i ∈ C , j = 1 , . . . , J , and ( s j , c i ) ∈ G . 12 / 39
Definition 1: Consider the fluid model given by (( ¯ P s j (0) , λα c i , µ s j ,c i ) : s i ∈ S , c i ∈ C , ( s j , c i ) ∈ G ). Let P (0) | = − � J | ¯ j =1 ¯ P s j (0). (i) We say that the fluid model is stable if starting from any initial state ¯ P (0) with | ¯ P (0) | = 1, there exists t 0 such that for any ( ¯ Y 1 ( t ) , · · · , ¯ Y J ( t )) satisfying (4)-(8), λt − ¯ Y 1 ( 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) | = 1, there exists t 0 such that for any ( ¯ Y 1 ( t ) , · · · , ¯ Y J ( t )) satisfying (4)-(8), ¯ Y J ( t ) − ¯ Y 1 ( 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) | = 1, there exists t 0 such that for any Y J ( t )) satisfying (4)-(8), ˙ Y 1 ( t ) = · · · = ˙ ( ¯ Y 1 ( t ) , · · · , ¯ ¯ ¯ Y J ( t ) for all t > t 0 . 13 / 39
Customers Arrivals 0 Time Server 1 Bipartite Graph Server 2 C 2 1 3 S 2 1 3 Server 3 Figure: Conjectured Fluid Dynamics under FCFS-ALIS 14 / 39
Definition 2: (i) We say that a cumulative probability distribution H ( · ) of a non-negative random variable has unbounded support if H ( t ) < 1 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 ) d t > 0 and a constant k , � b such that H ( k ) ( b ) − H ( k ) ( a ) ≥ a q ( t ) d t 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 c i ,j ( t ) , c i ∈ U ( M 1 ( t ) , . . . , M j ( t )) , 1 ≤ j ≤ J − 1) converges to a stationary distribution as t → ∞ . 15 / 39
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