Markov Processes Relat. Fields 3, 527–545 (1997)
Markov MP Processes &R F and Related Fields c Polymat, Moscow 1997
On Polling Systems where Servers Wait for Customers L.G. Afanassieva1 , F. Delcoigne2 and G. Fayolle2 1
Moscow State University, Mechanico-Mathematical Faculty, Chair of Probability, Laboratory of Large Random Systems, Vorobyevy Gori, 119899 Moscow, Russia 2 INRIA, Domaine de Voluceau, Rocquencourt, BP 105 78153 Le Chesnay Cedex, France E-mail:
[email protected],
[email protected] Received September 13, 1997
Abstract. In this paper, a particular polling system with N queues and V servers is analyzed. Whenever a server visits an empty queue, it waits for the next customer to come to this queue. A customer chooses his destination according to a routing matrix P . The model originates from specific problems arising in transportation networks. A global classification of the process describing the system is given under broad assumptions. It is shown that only transience or null recurrence can take place. A detailed classification of each node, together with limit laws after proper time-scaling, are obtained. The method of analysis relies on the Central Limit Theorem and a coupling with a reference system in which transportation times are identically zero. It is shown that a tight time-varying injection of servers can empty the system. Finally a system where servers may become impatient is briefly considered. Ergodicity conditions are given, showing that the system can stabilize, contrary to the case without impatience. Keywords: network, polling, random walk, recurrence, transience, Central Limit Theorem AMS Subject Classification: Primary 60K25; Secondary 68M20
1. Description of the model Consider an open network, consisting of N stations (nodes, parking lots) and def V cars, which circulate among the stations. Let S = {1, . . . , N }. The arrivals of customers form a simple point process, defined by a metrically transitive sequence of interarrival times {an , n ≥ 1}. Assuming the first customer arrives at Pnt1 = 0, then the (n + 1)-th customer enters the system at time tn+1 = j=1 aj and is directed to some node i with probability γi , i ∈ S, γi > 0
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L.G. Afanassieva, F. Delcoigne and G. Fayolle
PN and i=1 γi = 1. All customers choose their destination via some ergodic routing matrix P = (pij )i,j∈S . Let π = (π1 , . . . , πN ) be the invariant measure associated to P . A car arriving at a station where there are waiting customers takes one of them to some destination. Whenever the car finds a station empty, it stops and wait for the next arrival at this node. After having reached their destination, customers leave the network. Also, except in Section 5 where an impatience phenomenon will be assumed, a customer who upon arrival does not find an available car waits in a queue. Capacities of waiting rooms for clients are supposed unlimited and there are at least V available parking lots for cars at each station, i.e. empty displacements of cars are not allowed in this model. At that moment, it might be useful to explain the name “polling” appearing in the title of this paper. In the classical terminology, polling systems denote special networks with one mobile server visiting stations according to some routing mechanism. A huge literature has been devoted to these systems. In the case of Markovian routing, fresh results concerning the classification (recurrence, transience) appeared in [2,7]. However, in some applications, it is also necessary to cope with state-dependent displacements of the server. In [4], a model was analyzed, in which two routing matrices are involved, the choice being made depending on the state (empty or busy) of the last visited queue. In particular, it was remarked in [4] that, when the server always waits at an empty queue, the system is not ergodic: the purpose of the present study is to investigate in full detail the behaviour of a similar model with an arbitrary number of servers. We introduce the following quantities, for all i, j ∈ S, n ≥ 1. • τij , the time to go from node i to node j, ∀1 ≤ i, j ≤ N . These N 2 random variables do not depend on the arrival process, but can be correlated between each other. • qj (n), the number of customers at node j at time tn − 0. • xj (n), the number of cars at node j at time tn − 0. • S(n), the position of the server at time tn − 0 when V = 1. • vn , the set of nodes at which there is at least one vehicle at time tn . Assume the nth customer arrives at node i(n) and intends to go to node j(n). The pairs {i(n), j(n); n ≥ 1}, (1.1) are supposed to form a sequence of independent and identically distributed (i.i.d.) random variables. This means concretely that the choice of a destination by a customer depends only on his arrival node and on nothing else. Let also λ = (E an )−1 , τi =
N X j=1
pij E τij , τ =
N X i=1
πi τi .
(1.2)
On polling systems where servers wait for customers
529
The main goal of the study is to analyze the behaviour of the process def
Q = {Q(n), n ≥ 1}
def
=
{qj (n), xj (n); j ∈ S, n ≥ 1}
def
{~q(n), ~x(n), n ≥ 1}.
=
(1.3)
Definition 1.1. The network is said to be ergodic if and only if there exists a stationary and a.s. finite sequence def b def b Q = {Q(n), n ≥ 1} = {b qj (n), x bj (n); j ∈ S, n ≥ 1},
b such that Q(n + k) converges weakly to Q(k) as n → ∞, ∀k ≥ 1. Accordingly, a node i is ergodic if and only if qi (n + k) weakly converges to a stationary sequence qbi (k) as n → ∞. Definition 1.2. The network is transient if lim
n→∞
N X
qj (n) = ∞,
a.s.
j=1
A node i is transient if lim qi (n) = ∞,
n→∞
a.s.
Definition 1.3. The process Q is said to be null recurrent if it is non-ergodic and there exists a state, say (kj , rj ; j ∈ S), such that P qj (n) = kj , xj (n) = rj , ∀j ∈ S, for infinitely many n = 1. Similar definitions hold for each separate node. General notation • Occasionally, it will be convenient to consider the system in continuous time and the same symbols will be used, e.g. Q(t), whenever no ambiguity arises. • Taboo probabilities: If A ⊂ S, i, j ∈ S, then A Nij represents the number of times the Markov chain associated to P visits node j, starting from i, under the restriction that none of the nodes in the set A is entered in-between. • {Fn , n ≥ 0} will denote the increasing sequence of σ-algebras generated by {Q(t), t ≤ n}. • Ai (n) is the event that the n-th external arrival takes place at node i. In a similar way, Aij (n) is the event that the nth external arrival takes place at node i and is directed to node j.
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• Z [resp. R] is the set of integers [resp. real numbers]. The positive parts of these sets are Z+ and R+ respectively. The components of a vector ~ ∈ Rd will be denoted by Xi , i = 1, . . . , d. X • The complementary S − A of A ∈ S will be written A and the indicator function of an arbitrary event B will be denoted by 1{B} . 2. Classification of the network The purpose of this section is to classify the process Q(n), viewed as a random walk on Z2N + , in terms of ergodicity, transience and recurrence. Throughout this paper, we suppose N > 1, λ > 0. One of the basic results is that the process Q is never ergodic. Theorem 2.1. If γ 6= π, then the network is transient. Moreover, for any k such that N X γk > γi pik , (2.1) i=1
node k is transient. Proof. When γ 6= π, there exists at least one node, say k, such that condition (2.1) holds. Define then, for all m ≥ 0, the quantity def αk (n; m) = P qk (n) ≤ m qk (1) = m . Out of the first n arrivals in the system, let νij (n) be the number of clients who arrived at node i and wanted to reach node j. Let def
νi (n) =
N X
νij (n),
def
νei (n) =
N X
νji (n).
(2.2)
j=1
j=1
In (2.2), νi (n) represents the total number of external customers arriving at node i, while νei (n) stands for the number of internal customers intending to go to station i. Now the following inequality is obvious: def αk (n; m) ≤ P νk (n) ≤ νek (n) = βk (n). (2.3) To study the asymptotic behaviour of βk (n), as n → ∞, we use the events Aij (s) defined in the general notations in Section 1. Then one can write νk (n) =
n X N X
1{Akj (s)} ,
νek (n) =
s=1 j=1
νk (n) − νek (n) =
n X N X s=1 j=1
n X s=1
δk (s),
1{Ajk (s)} ,
(2.4)
On polling systems where servers wait for customers
where δk (s) =
N X
531
(1{Akj (s)} − 1{Ajk (s)} ).
j=1
From our general assumptions, it follows immediately that the random vectors δ(s) = (δ1 (s), . . . , δN (s)), s ≥ 1, are i.i.d. Furthermore, E δk (s) = γk −
N X
γj pjk = ρk > 0.
j=1
Using well-known large deviation upper bounds (see e.g. [8]), one obtains the estimate √ (2.5) βk (n) ≤ c1 e−c2 n , where c1 and c2 are fixed constants. It follows from (2.3) and (2.5) that ∞ X
αk (n, m) < ∞.
n=1
The Borel – Cantelli Lemma ensures that the process {qk (n), n ≥ 1} takes its values within an arbitrary bounded interval [0, A] a finite number of times only: this means that qk (n) is transient, and so is the network. The proof of the theorem is concluded. 2 Theorem 2.2. If γ = π,
(2.6)
then the network is always transient for N ≥ 4 and any arbitrary τ . It is null recurrent for N = 2, 3 and τ = 0. Proof of the case N ≥ 4. It suffices to prove the transience for τ = 0, since one can easily show that a system with τ = 0 is pathwise dominated by a system having the same arrival and routing processes, but an arbitrary τ > 0. For the sake of shortness, let us denote by V the state of Q(n) when there are no customers in the network and all cars are waiting at node 1. Since {i(n), j(n)} is a sequence of i.i.d. random vectors and τ = 0, the state V is a regeneration point for the process Q. Setting α(n) = P Q(n) = V , the following standard classification holds (see e.g. [6]): (i ) Q is transient if ∞ X
α(n) < ∞ ;
(2.7)
n=1
(ii ) Q is ergodic if lim inf α(n) = α > 0 ; n→∞
(2.8)
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L.G. Afanassieva, F. Delcoigne and G. Fayolle
(iii ) Q is null recurrent if ∞ X
α(n) = ∞
and
n=1
lim α(n) = 0.
n→∞
Assuming Q(0) = V with probability one, we have the inequality def α(n) ≤ P νi (n) = νei (n), ∀i ∈ S = β(n).
(2.9)
(2.10)
Expression (2.4), giving νi (n) − νei (n) in terms of the vectors δ(s), s ≤ n, together with condition (2.6), which yields E δi (n) = 0,
i ∈ S,
permit to use the Local Central Limit Theorem, since the vectors δ(n), n ≥ 1, are i.i.d. Consequently, C β(n) ∼ (N −1)/2 , (2.11) n so that, with (2.10), condition (2.7) holds for N > 3. 2 Proof of the case N = 2, 3. Showing the property for V = 1 ensures it holds in fact for any V . Thus take V = 1 and let ϕ ~ (n) be the vector with components ϕi (n) = νi (n) − νei (n),
∀i ∈ S, ∀n > 0.
Introduce the events def Cn = ϕ ~ (n) = ~0 and def
Li1 (n) =
n n [ o Ai1 (s) ∩ Ai (k), s < k ≤ n . s=1
Li1 (n) says that, among the n first arrivals in the network, the destination of the last client arrived at node i was node 1. Clearly Cn ⊂ {S(n) = 1}. Moreover, the event {S(n) = 1; ϕ1 (n) = 0} says in particular that, among the n first arrivals, all clients whose destination was 1 have been served right after the nth external arrival. Now the result follows from the Central Limit Theorem. N = 2. We have, for all n sufficiently large, D D √ 1 ∼ P{Cn ∩ L21 (n)} ≤ P{Q(n) = V} ≤ P{Cn } ∼ √ 2 , n n for some constants D1 and D2 , which proves the null recurrence according to (2.9). N = 3. First, the irreducibility of the routing matrix P allows to assume ad libitum, for instance p23 p31 6= 0. Then, for all n sufficiently large, there exit constants D3 and D4 such that D4 D3 ∼ P Cn ∩ L21 (n) ∩ L31 (n) ≤ P Q(n) = V ≤ P Cn ∼ . n n Again (2.9) is satisfied and the theorem is completely proved. 2
On polling systems where servers wait for customers
533
3. Local behaviour A detailed classification of the nodes will be presented in the next 5 theorems. For many quantities of interest, the reader is referred to definitions given in Section 1, in particular in (1.2) and (1.3). Theorem 3.1. If γ = π, then all the nodes are non ergodic. Proof. For the non ergodicity, it suffices to carry out the analysis in the case τ = 0 and then to prove that, for any k, the sequence {qk (n), n ≥ 1} forms a submartingale with respect to the family of σ-algebras {Fn , n ≥ 1}. We have qk (n + 1) = qk (n) = 0,
∀k ∈ vn ,
(3.1)
and qk (n + 1) − qk (n) ≥ 1{Ak (n)} −
X
vn Nik
1{Ai (n)} ,
∀k ∈ v n .
(3.2)
i∈vn
It follows from (3.2) that, for all k ∈ v n , X E qk (n + 1) − qk (n) Fn ≥ πk − πi E(vn Nik ). i∈vn
Setting Pv n v n
=
(pij , i ∈ v n , j ∈ v n ),
Pv n v n
=
(pij , i ∈ vn , j ∈ v n ),
π vn
=
(πi , i ∈ vn ),
the above relation yields, for all k ∈ v n , X X E qk (n + 1) − qk (n) Fn ≥ πk − πi pij E(vn Njk ) i∈vn
= πk −
h
t
j∈v n
π v n Pv n v n
X q≥0
(Pvn vn )q
i
= 0,
k
which, together with (3.1), shows that {qk (n), n ≥ 1} is a submartingale. Consequently qk will no reach a compact set containing 0 in an integrable time and the nodes are non ergodic. The proof is concluded. 2 In fact, there is a more precise result. Theorem 3.2. If γ=π
and τ = 0,
then each node is null recurrent and has always a positive probability of being empty.
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Proof. Let αi (n) be the probability that node i is empty at the instant of the (n + 1)th arrival in the system. We shall prove that lim inf αi (n) = αi > 0, n→∞
and this will be sufficient for node i to be null recurrent. Here, there is no loss of generality in assuming V = 1 and qi (0) = 0, ∀i ∈ S. Then, with the notation of Theorem 2.1, we have \ νk (n) > νek (n) ⊂ S(n) = i ⊂ qi (n) = 0 . k6=i
Thus
\ n Pn o t=1 δk (t) √ P qi (n) = 0 ≥ P >0 . n k6=i
Hence, using γ = π and the Central Limit Theorem, we get lim inf αi (n) = αi > 0. n→∞
2 Theorem 3.3. Here we refer to Section 1, especially (1.1), (1.2), and denote by τij (n) the time to go from node i to node j at the nth arrival instant. The sequence {tn , τij (n); n ≥ 1} is assumed to be stationary ergodic. Then, when γ = π, the nodes are (i ) transient if λτ > V ; (ii ) recurrent if λτ < V , under the additional assumption E(τij τkl ) < ∞,
∀i, j, k, l ∈ S,
(3.3)
and in this case all conclusions of Theorem 2.2 hold. Proof of (i ). One can use standard queueing theory arguments. Consider an arbitrary node, i say. It can be viewed as a single queue with V servers working in parallel and an equivalent average service time which is larger than X πk τk τ + τi = . πi πi k6=i
The corresponding traffic intensity ρi satisfies the inequalities ρi ≥
λτ γi = λτ > V, πi
which means that each node is transient.
2
On polling systems where servers wait for customers
535
e which is obtained from Q by Proof of (ii ). Let us introduce the system Q, e work under the same taking τ = 0. This means in particular that Q and Q e arrival and routing processes. Assuming Q(0) = Q(0), the idea is to couple Q e and Q pathwise, at properly chosen instants. Step 1. e to Q.
In the next lemma, an important coupling is constructed, relating Q
Lemma 3.1. If at some instant T all cars are blocked (that is waiting for new e ). arrivals) in system Q, then Q(T ) = Q(T Proof. It will be assumed all cars are blocked at the same node, say i. Let eij (t)] be the number of clients who arrived at node i and have Uij (t) [resp. U e been transported to node j on the time interval [0, t] in Q [resp. Q]. The machinery of the two systems ensures that eij (t), Uij (t) ≤ U
∀i, j ∈ S, ∀t ≥ 0.
(3.4)
Now the definition of T yields eii (T ) Uii (T ) = U
and
N X
Uji (T ) =
j=1
N X
eji (T ), U
j=1
whence, by (3.4), eji (T ), Uji (T ) = U
∀j ∈ S.
(3.5)
Suppose there ∃j 6= i, such that qj (T ) > qej (T ) and let r be the destination e before time T . But node of the last customer served at node j in system Q, e then Ujr (T ) < Ujr (T ). For r = i, this immediately contradicts (3.5). Otherwise, qr (T ) > qer (T ), since there were more visits of cars at node i, before time T , in e than in system Q, remembering that at time T all cars are blocked system Q esi (T ), at node i. Now, by induction, there would exist s, such that Usi (T ) < U contradicting again (3.5). The reader will check that similar arguments hold in the general situation, when cabs are blocked at different nodes. The proof is concluded. 2 Step 2. Now we will construct an increasing sequence of stopping times {Tk , k ≥ 1} satisfying E Tk+1 − Tk FTk < C < ∞ and
N X
xi (Tk ) = V, ∀k ≥ 0.
i=1 def
Setting ϕ(t) = inf i qi (t), the basic point to prove is the existence of a random variable L, such that, for some fixed but arbitrary t, L = inf s > t, ϕ(s) = 0 ϕ(t) = 1 , with E L < D < ∞ uniformly.
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L.G. Afanassieva, F. Delcoigne and G. Fayolle
Clearly one must have L < inf Li , i
where Li is the time to empty an arbitrary component, say qi , of the vector ~q, under the condition ~q(s) > 0, s ≥ t, and ϕ(t) = 1. (3.6) Let Mvi (a, b), 1 ≤ v ≤ V , be the number of visits at node i made by an arbitrary vehicle v, on the time interval (a, b] and denote by us , s ≥ 0, the arrival times at node i, from t onwards. Then, under the conditioning (3.6) and using the ergodic theorem for Markov chains, we have V
θ
V πi V 1 XX E Mvi (us−1 , us ) = = , θ→∞ θ λγ τ λτ i v=1 s=1 lim
a.s.
(3.7)
Thus, as in a standard G/G/V queueing system, (3.7) shows that qi (us ) behaves in the positive orthant as a stopped supermartingale, which will reach the boundary 0 in an integrable time (see e.g. [5]), provided that V > 1. λτ Since this condition does not depend on the coordinate index i and ϕ(t) = 1, the existence of L is proved. To proceed further with the proof of step 2, it suffices to note that, as soon as a coordinate has reached the value 0, it takes an integrable time ∆, either to block all cars at various nodes or to jump again into the positive orthant. We have exactly ∆ = inf(∆1 , ∆2 ), where • ∆1 is the first time to block all cars, which is integrable by (3.3); • ∆2 is the time necessary to jump again into the positive orthant. Keeping in mind that ϕ(L + ∆2 ) = 1, the above procedure can be repeated a number of times bounded by a random variable W , geometrically distributed with some parameter, say ε > 0, which does not need fuller specification. This establishes the existence of the sequence {Tn , n ≥ 1} satisfying Tn+1 − Tn ≤ D1 + D2 + · · · + DW +1 , sup E(Tn+1 − Tn ) < C < ∞ , n
where the random variables Di , i ≥ 1, have the same distribution as L + ∆.
On polling systems where servers wait for customers
Step 3.
537
Since e n ), n ≥ 1, Q(Tn ) = Q(T
the conclusion of part (ii) of the theorem follows immediately from Theorems 2.2 and 3.2, using the uniform boundedness of E(Tn+1 − Tn ). Theorem 3.3 is completely proved. 2 In the next two theorems, pathwise comparisons are needed between the e introduced in Theorem 3.3, and its restriction to a polling system process Q, where only a subset of the nodes is visited. def Set Λ = {i1 , . . . , ik }, for all k-tuple of integers 1 ≤ i1 < · · · < ik · · · ≤ N , and let PeΛ denote the transition matrix of the restriction to Λ of a Markov chain with state space S and transition matrix P . Introducing the two vectors π eΛ and Λ γ e , with components γj πj , γ ejΛ = P , ∀j ∈ Λ, (3.8) π ejΛ = P i∈Λ πi i∈Λ γi it is well known (see e.g. [6]) that the invariant measure of PeΛ is indeed given by π eΛ . The exact form of PeΛ can be found in [6] but is not needed there. To avoid uninteresting technicalities, we take xi (0) = 0, ∀i ∈ Λ. Let τ = 0 Λ e Λ on ZΛ and consider the process Q + × Z constructed as follows: • it corresponds to a new polling system with zero transfer times; • its sample paths are obtained by assuming the server never stops at the nodes in Λ and that the arrival and routing processes are the same as for Q. Then, writing qekΛ (n) for the number of customers at the n-th arrival instant, e Λ , we have waiting at node k ∈ Λ, in the system Q qeiΛ (n) ≤ qi (n),
∀i ∈ Λ,
∀n ≥ 0.
Theorem 3.4. Let ∂B = {(x1 , . . . , xN ) ∈ ZN + ; xi = 0, ∀i ∈ Λ}, for any arbitrary Λ, and assume γ = π. Then P q1 (n), . . . , qN (n) ∈ ∂B, for infinitely many n = 0, ∀ |Λ| ≥ 4. In other words, the boundaries of codimension ≥ 4 are transient. Proof. As already remarked, it suffices to prove the statement for τ = 0, since e Λ , with the case τ 6= 0 follows by direct sample path comparison. Consider Q Λ Λ Λ = {i1 , . . . , ik }. Here γ e =π e (see (3.8)) and Theorem 2.2 implies P qeiΛ1 (n) = . . . = qeiΛk (n) = 0 ∼
CΛ n(k−1)/2
,
for some constant C Λ . Since |Λ| ≥ 4, it follows that X P qi1 (n) = · · · = qik (n) = 0 < ∞ n≥0
and the proof is concluded by using the Borel – Cantelli lemma.
2
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L.G. Afanassieva, F. Delcoigne and G. Fayolle
The next theorem deals with the complete classification when γ 6= π. In this case, it is convenient to introduce a numbering of the nodes, according to the following inequalities: γ1 γl−1 γl γN = ··· = < ≤ ··· ≤ . (3.9) π1 πl−1 πl πN Note that, since l ≤ N , (3.9) excludes γ = π. Theorem 3.5. The ordering (3.9) being assumed, the following properties hold. (i ) For any i ≥ l, limn→∞ qi (n) = ∞ a.s. (ii ) Suppose l = 2 and put ρ1 =
λτ γ1 . Then, π1
– for ρ1 < V , node 1 is ergodic; – for ρ1 > V , node 1 is transient. def
(iii ) Assume l ≥ 3. Then the set of nodes Λ = {1, . . . , l−1} behaves as a polling system, with parameters γ Λ , π Λ and PΛ , which have been formally defined in the preamble before Theorem 3.4. mutatis mutandis. This theorem claims, among other things, that the original polling system has at most one ergodic node when γ 6= π. eΛ Proof. (i ) Choosing a node i with i ≥ l, we consider the system P Q corresponding to Λ = {1, i}. Here γ eΛ 6= π eΛ and, more precisely, γ eiΛ > j∈Λ γ ejΛ peΛ ji . Using Λ Λ now Theorem 2.1, it follows that qei is transient and so is qi , since qei (n) ≤ qi (n), for all n ≥ 1. (ii ) It follows from part (i ) that lim qi (n) = ∞,
n→∞
∀i ≥ 2, a.s.
Hence, the traffic intensity at node 1 is exactly ρ1 =
λτ γ1 π1
and the result follows easily. (iii ) Using again part (i ), lim qi (n) = ∞,
n→∞
∀i ≥ l, a.s.
So, for n sufficiently large, the cars will not stop anymore at the nodes belonging to Λ. In fact, omitting the details, one sees that the original network decomposes into two subsystems: • a system of N − l + 1 transient nodes; • a polling system QΛ of l − 1 nodes. The proof of the theorem is concluded.
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On polling systems where servers wait for customers
539
4. Time scaling and limit laws When γ = π, it has been shown above that each node is null-recurrent and the system is either null recurrent or transient. In this section, limit laws are obtained for the joint distribution of the position of the server and the number in the queues. Notation. Here vectors will often be written with arrows (e.g. ~x, ~q(n), ~k, etc.). Setting → − def ∀i ∈ S, i N = (i Ni1 , . . . , i NiN ), → − the quantities Y (s; i), s ≥ 1 will stand for a sequence of vectors which, for → − each fixed i, are i.i.d., independent of the arrival process and distributed as i N . Let → − def ∀i ∈ S, i H j = (i Hj1 , . . . , i HjN ), where i Hjk , in the Markov chain with matrix P, is the number of visits to k, starting from j, without hitting i. In addition, define N n X X → − → − def 1{Aj (s)}~ej , ∀i ∈ S, 1{Ai (s)} Y (s; i) − Z (n; i) = s=1
j=1 N
where ~ej is the jth unit vector of R and, by convention, i Nii = 1, ∀i ∈ S. → − − → − def → Letting ∆ Z (n; i) = Z (n + 1; i) − Z (n; i), it follows (since γ = π) that → − E[∆ Z (n; i)] = ~0, ∀i ∈ S. Introduce also the following covariance matrices, which do not depend on n: def Γi = E ∆Zj (n; i) ∆Zk (n; i) , ∀i ∈ S. j,k∈S
→ − In the sequel N (Γi ) will represent a random vector in RN , normally distributed, centered at the origin, having its i-th coordinate identically 0 and Γi as covariance matrix. Using the symbol “ * ” for convolution, we are in a position to state the main theorem. Theorem 4.1. If γ = π, τ = 0, ~x(0) = ~x, ~q(0) = ~0, then for any ~k ∈ RN + and for any i, j ∈ S, lim P ∃i 6= j, {i, j} ⊂ vn
n→∞
=
o n ~q(n) = P vn = {i}; √ ≤ ~k n o n ~q(n) = lim P vn = {i}; √ ≤ ~k n→∞ n
0,
(4.1)
→ − n ~ (∗)xj o Z (n; i) ~ X i H j √ P ~0 ≤ √ ≤k+ , (4.2) n n j6=i
→ − P N (Γi ) ∈ [~0, ~k] .
(4.3)
It is worth remarking that equation (4.2) gives an other proof of Theorem 3.4.
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Proof. Equation (4.1) is a straightforward consequence of Theorem 3.4 and the limit law in (4.3) is obtained by letting n → ∞ in (4.2). It is also worth noting − in (4.2), which gives the time-dependent probability of having the choice ~k = → ∞ at least one vehicle at a given node. Thus we are left with the proof of (4.2). The argument will be essentially developed for V = 1. It relies on a counting (and in some sense combinatorial) argument, where the time does not play any role, the crucial fact being that every car displacement corresponds to the service of exactly one customer. Consider the system at the nth arrival instant and condition its state on def ~ν (n) = ~b = (b1 , b2 , . . . , bN ), where ~ν (n) is the vector of the number of arrivals at each queue, introduced in Section 2. Let X(t; n) denote the position of the server after t effective visits (or services) up to time n. Clearly, t ≤ n and the evolution of X(t; n) is Markovian, with transition matrix P . Thus one can write, for some fixed i, using the notation S(n) instead of vn since V = 1, o n (∗)ν (n) P S(n) = i ~ν (n); S(0) = i = P i Nj i ≤ νj (n), ∀j 6= i ~ν (n); S(0) = i , which in turn yields − n→ Z (n; i) ~ o √ ≥0 . P S(n) = i S(0) = i = P n Similarly, remarking that the event {~q(n) = ~k} can occur only if X(t; n) takes the value i exactly bi times and the value j exactly bj − kj times, for all j 6= i, we get → − n o n Z (n; i) ~ o ~q(n) P S(n) = i; √ ≤ ~k S(0) = i = P ~0 ≤ √ ≤k , n n and the result follows from the Central Limit Theorem. The extension to V > 1 is not difficult, noting simply that when all cars are blocked at some node i, the procedure for V = 1 can be reproduced. Details are omitted. 2 The last theorem given hereafter does in some sense justify, beyond its thee in which τ = 0. oretical interest, the detailed analysis made for the system Q Theorem 4.2. When γ = π and λτ < V , the distribution of Q(t), as t → ∞, satisfies − N n→ o X n o Q (t) → − lim P √ ≤ ~k = P N (Γi ) ∈ [~0, λ−1/2~k] , ∀~k ∈ RN (4.4) +. t→∞ t i=1 √ The noticeable fact is that, after a scaling in 1/ t, neither transfer times nor the number of vehicles do appear in the explicit form of the limiting distribution. This is indeed a phase transition phenomenon. Obviously, this is not the case as far as speed of convergence is concerned.
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PN Proof. Let ν(a, b) = i=1 νi (b)−νi (a) be the total number of customers arrived in the time interval (a, b). Pathwise, we have e ≤ Q(t) ≤ Q(t) e + ν(t − Tl(t) ), Q(t)
(4.5)
where {Tn , n ≥ 1} is the sequence introduced in Theorem 3.3 and l(t) = inf{n; Tn ≤ t < Tn+1 }. But the constructive procedure of this sequence shows that the random variables Tn+1 − Tn , n ≥ 1 can be bounded by the increments of a renewal point process, having a proper distribution for its stationary residual time, whence o n ν(t − T ) l(t) √ > a = 0, ∀a ≥ 0, lim P t→∞ t and the relation (4.4) follows directly. The theorem is proved.
2
Remark. The fact that τ does not take place in (4.4) can be explained by coupling. The non-influence of V follows from Theorem 3.4 and equation (4.4): asymptotically, all cars are blocked at some node (not always the same) and only one customer can be transported, no matter the value of V may be. 4.1. Time-dependent number of vehicles It has been shown previously that the system is never ergodic, for any arbitrary but fixed V . Here we permit the number of vehicles in the system to be an increasing function of time: they are injected at each node independently of the arrival process. Let • ri (t), the number of vehicles injected at node i on the interval [0, t]; def PN • r(t) = i=1 ri (t). Without loss of generality, we assume the system to be empty at time t = 0. Then the next theorem gives necessary conditions for the system to empty, when t → ∞. Theorem 4.3. When γ = π, for the family of random variables {qi (t), t ≥ 0} √ to be tight, it is necessary to have limt→∞ ri (t)/ t = +∞ in probability and then, in fact, lim P qi (t) = 0 = 1. t→∞
Proof. Remarking that qi (t) ≤ k ⊂ νi (t) ≤ νei (t) + ri (t) + k , one can write n ν (t) − νe (t) ri (t) + k o i √ i √ P qi (t) ≤ k ≤ P ≤ . t t
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Thus the Central Limit Theorem implies 1 lim inf P qi (t) ≤ k ≤ lim inf √ t→∞ t→∞ 2πσ
+∞ Z 2 n r (t) + k o − x i √ e 2σ2 P > x dx, t
−∞
√ def where σ 2 = 2λγi . If ri (t)/ t does not tend to infinity in probability, then there exist ε > 0, M > 0, and a sequence {tj , j ≥ 0} such that n r (t ) o i j P √ > M ≤ 1 − ε. tj Thus +∞ Z 2 − x e 2σ2 dx,
1 lim inf P qi (tj ) ≤ k ≤ 1 − ε √ j→∞ 2πσ
∀k > 0,
M
which contradicts the tightness of the family {qi (t), t ≥ 0}. Let Q denote the process corresponding to a network which the same arrival and routing sequences as before, but in which r(t) = 0, ∀t. Then, pathwise, we have qi (t) ≤ [q i (t) − ri (t)]+ , ∀t ≥ 0. It suffices to use Theorem 4.2 to conclude the proof.
2
5. Model with impatience of the server In this section there is only one server (or cab). The model is similar to the preceding one, except that the server might become impatient. In other words, when he comes to an empty node, say i, he waits for the next arrival at most Ci units of time. If no arrival takes place within time Ci , the cab chooses a destination according to the routing matrix P ; otherwise he starts serving, as before. The Ci0 s, i = 1, . . . , N are exponentially distributed random variables with parameter µi . The total input process is assumed to be a Poisson process with parameter λ. Let Q(n) = q1 (n), . . . , qN (n), S(n) be the state of the process at polling instants, i.e. the instants when the server reaches a node, so that Q is a Markov chain. We need to introduce the following quantities π=
N X πi , µ i=1 i
γ=
N X γi . µ i=1 i
Theorem 5.1. Q is ergodic if and only if πk τ +π > , λγk 1 + λγ
∀k ∈ S.
(5.1)
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Proof. To show the necessity of the conditions, we suppose the system ergodic. Then writing the equality of the ingoing and outgoing flows at an arbitrary station k, yields N X µk P S = k, qk = 0 P S = i, qi = 0 = λγk . πk − πi τi + λγk + µk λγi + µi i=1 It follows that N X P S = i, qi = 0 π − λτ γ F = = , λγi + µi λγ + 1 i=1 def
(5.2)
whence µk P S = k, qk = 0 πk (1 + λγ) − λγk (τ + π) = , ∀k, 1 ≤ k ≤ N. λγk + µk 1 + λγ But, when the network is ergodic, P S = k, qk = 0 must be strictly positive ∀k, 1 ≤ k ≤ N : this shows that conditions (5.1) are necessary. The sufficiency of (5.1) for the system to be ergodic can be shown by constructing a linear Lyapounov function, defined on the dynamical system associated to the random walk (see e.g. [4, 5]). We refrain from presenting the complete proof, since it would require too many additional definitions. 2 By way of conclusion, let us just mention the possibility of carrying out the analysis of the preceding system (and also of polling systems) in thermodynamical limit, i.e. when V and N simultaneously increase to infinity under suitable conditions. This is the subject of ongoing works. Acknowledgement The authors are grateful to M. Parent (program director in the PRAXITELE project now developed at INRIA) and to J.-M. Lasgouttes for many fruitful discussions, which helped to the define model analyzed in this paper. References [1] L.G. Afanassieva, G. Fayolle and S.Yu. Popov (1996) Models for transportation networks. Preprint 2837, INRIA. [2] A.A. Borovkov and R. Schassberger (1994) Ergodicity of a polling network. Stochastic Process. Appl. 50, 253–262. [3] K.L. Chung (1967) Springer, Berlin.
Markov Chains with Stationary Transition Probabilities.
[4] G. Fayolle and J.M. Lasgouttes (1995) A state-dependent polling model with Markovian routing. In: Stochastic Networks, F.P. Kelly and R. Williams (eds.), IMA Volumes in Math. and its Applications 71. Springer Verlag, 283–301.
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[5] G. Fayolle, V.A. Malyshev and M.V. Menshikov (1995) Topics in the Constructive Theory of Markov Chains. Cambridge Univ. Press. [6] W. Feller (1971) An Introduction to Probability Theory and Applications. 2nd edn. Wiley. [7] Ch. Fricker and M.R. Jaibi (1994) Stability of a polling model with a Markovian scheme. Preprint 2278, INRIA. [8] I.A. Ibragimov and Yu.V. Linnik (1971) Independent and Stationary Sequences of Random Variables. Wolters Noordhoff Publishing, Gronigen. [9] J. Jacod and A.N. Shiryaev (1987) Limit Theorems for Stochastic Processes. Springer.
