Stationary analysis of a queueing model with local choice Hanene MOHAMED joint work with : Christine Fricker, Plinio S. Dester Alea 2019 CIRM Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 1 / 22
Customers & Queues Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 2 / 22
The model A set of N one-server queues with (in)finite capacity arrival : independent Poisson processes P ( λ ) service time : E ( µ ) . The arrival-to service rate ratio, load , is defined by ρ = λ/µ Which Policy ? (Choice Vs No Choice) Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 3 / 22
The impact of choice in a large set of queues Choice policy is a well-known distributed way to improve load balancing. Supermarket model arriving customer chooses two queues at random joins the shortest one Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 4 / 22
The queue length process : a Markov process X i ( t ) = ♯ customers at queue i at time t , 1 ≤ i ≤ N . The queue length process ( X ( t )) t ≥ 0 = ( X i ( t )) 1 ≤ i ≤ N : Markov process on state space N N Proposition : Stability condition, Stationary distribution Markov process ( X ( t )) t ≥ 0 is ergodic iff ρ < 1 and it has a unique invariant distribution y = ( y n , n ∈ N N ) . See [Vvedenskaya 1996], [Mitzenmacher 1996] . . . Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 5 / 22
Stationary distribution No-Choice Model : geo ( ρ ) stationary distribution. Random Choice Model : Mean-field approach. (Stationary) Marginal distribution for one queue : The probability that queue i has k customers does not depend on i . This probability, denoted by π k ( ρ ) , is given by � π k ( ρ ) = y ( k,n 2 ,...,n N ) ( ρ ) ( k,n 2 ,...,n N ) Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 6 / 22
The power of choice Proposition : Limiting stationary tail probability The limit as N gets large of the stationary probability that a queue has more than m customers is : � Choice policy k ≥ m π k ∼ ρ m No-choice ∼ ρ 2 m − 1 Random choice In this table the equivalent is when ρ tends to 0. Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 7 / 22
Geographical constraints : Local choice policy Local choice policy (case with geographical constraints) : the arriving customer chooses one queue at random he compares it to its neighbor joins the shortest one Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 8 / 22
Choice policy Which Policy ? Local choice : Choice among two neighbors customer chooses randomly one queue, denoted i , among N queues. joins the least loaded between queues i and the next ( i + 1) . Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 9 / 22
Contribution function Contribution function : quantifies the amount of arrivals at the different queues, depends only on the state n i of this queue i and the two neighboring queues ( i − 1) and ( i + 1) : c i ( n ) = 1 2 ✶ { n i = n i − 1 } + ✶ { n i <n i − 1 } + 1 2 ✶ { n i = n i +1 } + ✶ { n i <n i +1 } . Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 10 / 22
Queues length process The queues length process ( X ( t )) t ≥ 0 = ( X i ( t )) 1 ≤ i ≤ N is a Markov process on state space N N with Q-matrix : Q ( n, n + e i ) = λc i ( n ) Q ( n, n − e i ) = µ ✶ n i > 0 where n = ( n 1 , . . . , n N ) and ( e i ) 1 ≤ i ≤ N is the canonical basis of R N . Proposition : Stability condition, Stationary distribution Markov process ( X ( t )) t ≥ 0 is ergodic iff ρ < 1 and it has a unique invariant distribution y = ( y n , n ∈ N N ) . Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 11 / 22
Invariant probability ; what informations about it ? The invariant probability on N N : y = ( y n , n ∈ N N ) It is solution of the global balance equations : � y ( n ′ ) Q ( n ′ , n ) = 0 , for n ∈ N N . n ′ ∈ N N Our aim is not to solve these equations. For N fixed, y = ( y n , n ∈ N N ) depends only on one parameter : the load ρ Which kind of dependency of y on ρ ? Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 12 / 22
Analyticity of stationary probabilities of a family of Markov chains depending on one parameter What does it mean ? Analyticity : Power series. There exists ε > 0 , such that for fixed N , � α k ( n ) ρ k , y n ( ρ ) = ∀ ρ ∈ [0 , ε [ , ∀ n = ( n 1 , ..., n N ) . k ≥ 0 Goal : y n ( ρ ) ∼ ρ → 0 ? Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 13 / 22
Analyticity : is it true ? Related to the explicit Lyapunov function used to prove the ergodicity. Analyticity : See the results of [Malyshev et al, 1979, Chapter IV] or [Fayolle et al, 1995, Chapter 7] Not in this framework due to the contribution function part. We look for an analytical solution for y . Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 14 / 22
Induction procedure Result : An algorithm to obtain all the coefficients α k ( n ) for k ≥ 0 . Plugging the power series into the global balance equations : for n = ( n 1 , n 2 , . . . , n N ) and k ≥ 0 : � N � N � � α k ( n ) = α k ( n + e i ) ✶ { n i > 0 } i =1 i =1 N N � � + ✶ { n i > 0 } c i ( n − e i ) α k − 1 ( n − e i ) − α k − 1 ( n ) c i ( n ) i =1 i =1 Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 15 / 22
Induction procedure Two crucial lemmas : Lemma 1 � n ∈ N N α k ( n ) = 0 , k > 0 . Lemma 2 α k ( n ) = 0 , ∀ n ∈ N N , | n | > k . where | n | = | ( n 1 , n 2 , . . . , n N ) | = n 1 + n 2 + . . . + n N . Let us derive the first coefficients : For k = 0 , state n such that | n | ≤ 0 (only one state 0 N ) α 0 ( n ) = ✶ { n =0 N } . Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 16 / 22
Induction procedure For k = 1 , states n such that | n | ≤ 1 ( 0 N or e i for 1 ≤ i ≤ N ) α 1 (0 N ) = − N α 1 ( e i ) = 1 α 1 ( n ) = 0 otherwise. For k = 2 , states n such that | n | ≤ 2 ( 0 N , e i or e i + e j , 1 ≤ i, j ≤ N ) 2 ( N 2 − Nc 1 ( e 1 )) α 2 (0 N ) = 1 α 2 ( e i ) = − N α 2 ( e i + e j ) = c i ( e j ) α 2 ( n ) = 0 otherwise. Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 17 / 22
Marginal queue length distribution π m ( ρ ) : the stationary probability that a queue has m customers. � π m ( ρ ) = y n ( ρ ) n ∈ N N ,n 1 = m Proposition 3 For N ≥ 3 and m > 2 ; π 0 ( ρ ) = 1 − ρ π 1 ( ρ ) = ρ − 3 2 ρ 3 + O ( ρ 4 ) π 2 ( ρ ) = 3 2 ρ 3 + O ( ρ 4 ) π m ( ρ ) = O ( ρ m +1 ) Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 18 / 22
Further expansions for the marginal The following expansions are obtained with help of mathematical software. Further asymptotics For N ≥ 3 and m > 3 ; π 0 ( ρ ) = 1 − ρ π 1 ( ρ ) = ρ − 3 2 ρ 3 + 11 8 ρ 4 − 17 3 ρ 5 + 10727 2880 ρ 6 + O ( ρ 7 ) π 2 ( ρ ) = 3 2 ρ 3 − 11 8 ρ 4 + 47 24 ρ 5 − 1583 320 ρ 6 + O ( ρ 7 ) π 3 ( ρ ) = 3 8 ρ 5 + 11 9 ρ 6 + O ( ρ 7 ) π m ( ρ ) = O ( ρ 7 ) Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 19 / 22
Validation by simulation 0 . 4 π 1 - simulation π 2 - simulation 0 . 3 π 3 - simulation π 1 - expansion π m ( ρ ) π 2 - expansion 0 . 2 π 3 - expansion 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 ρ Figure : For m = 1 , 2 and 3 , as a function of ρ . Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 20 / 22
The main result Proposition 4 Let N be fixed and m ≥ 2 . The stationary probability that a queue has m customers verifies � ρ � 2 m − 1 + O ( ρ 2 m ) , π m ( ρ ) = 12 2 when ρ tends to zero. idea of proof : Relation between ( α k ( n )) . Lemmas 1 and 2 . Understand the contribution of each term α k ( n ) . Find the first non vanishing term of π m ( ρ ) for each m . Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 21 / 22
Conclusion The following table illustrates where the choice local choice policy situates. u m = � k ≥ m π k Choice policy ∼ ρ m No-choice ∼ ( ρ 2 Local choice 4 ) m ∼ ρ 2 m − 1 Random choice In this table the equivalent is when ρ tends to 0. Hanene MOHAMED (Univ Paris Nanterre) Queueing model with local choice Alea 2019 CIRM 22 / 22
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