A multiclass network with non-linear, non-convex, non-monotonic ...

A multiclass network with non-linear, non-convex, non-monotonic stability conditions Vincent Dumas

INRIA, domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, FRANCE. [email protected]

April 1995 Abstract

We consider a stochastic queueing network with xed routes and class priorities. The vector of class sizes forms a homogeneous Markov process with countable state space Z+6 . Its conditions of ergodicity, which are identi ed with the conditions of stability of the network, depend on the vector  whose components are the trac intensities of the dierent classes. In order to determine the exact stability conditions, we resort to the tools of Malyshev and Menshikov's theory of random walks in Z+N with space homogeneity and bounded jumps (see [11]). We exhibit diverging paths of the associated dynamical system, whose conditions of existence yield the conditions of transience of our network. Then, for our proofs of ergodicity and transience, we nd Lyapunov functions that satisfy the criteria given in [11]. The stability conditions thus determined have especially unusual characteristics : they have a quadratic part, the stability domain is not convex, and we may nd vectors  <  such that the network corresponding to  is ergodic, and that corresponding to  is transient (see Theorem 1.1 and section 8). 0

0

Keywords: multiclass queueing networks, preemptive resume priorities, homogeneous random walks, induced Markov processes, uid equations, dynamical system, instability cycles, domain of stability, monotonicity.

1 Introduction : a multiclass network with priorities. This paper is devoted to the stability analysis of a stochastic, multiclass network with xed customer routes and preemptive resume priorities among classes. Our network is composed of K = 3 single-server stations (or queues) and I = 2 customer types. Each type corresponds to a deterministic route visiting three stations. Type i customers at the sth stage of their route will be called class (i; s) customers; their station will be denoted kis, and then type i customers enter the network at station ki1 and leave it after station ki3. We have: (k11; k12; k13) = (1; 2; 3); (k21; k22; k23) = (3; 2; 1): It is assumed that type i customers enter the network according to a Poisson process of rate i , and that class (i; s) customers require independent, exponentially distributed services of parameter is . Dierent arrival processes and service sequences are independent. 1

-1 

11 ^ 23



12 ^ 22



13 _ 21 2

Figure 1 : Network under study. At last, the service discipline is based on class priorities: at each queue k, there are two classes, which form the set k = f(i; s)=kis = kg, and one class has preemptive resume priority over the other one, whereas inside each class customers are served in their order of arrival. The priorities are : (2; 3) > (1; 1); (2; 2) > (1; 2); (1; 3) > (2; 1); where the notation : (i; s) > (j; r) means that kis = kjr and class (i; s) has priority over class (j; r). This network is pictured in Figure 1. Let C = f(i; s); 1  i  2; 1  s  3g be the set of all the classes. We will denote by Qt (i; s) (resp. Qt (k)) the number of customers in class (i; s), (i; s) 2 C (resp. the number of customers in queue k, k = 1; 2; 3), at time t  0. Consider the process (Qt )t0 de ned by:

Qt = (Qt (i; s))(i;s)2C : It is easy to check that this is a homogeneous Markov process with countable state space ZZjCj + , where jCj = 6 is the cardinality of C . As usual we identify the stability (resp. the instability) of the network with the ergodicity (resp. the transience) of Qt. We will denote by: is =
i ; is the trac intensity for class (i; s), and: X is ; k =
(i;s)2k

the trac intensity for queue k. The vector  of trac intensities is:  = (is )(i;s)2C . It is well-known that necessary conditions of stability of such networks are that the trac intensities be smaller than one for all queues; these are the usual conditions. But these conditions are in general not sucient, as was proved by several authors at rst for some deterministic networks (see Kumar and Seidman [6], Lu and Kumar [7]), then for some stochastic networks (see Rybko and Stolyar [10], Bramson [2]). As for the exact stability conditions, they were given explicitly for only two similar models of networks with xed routes and class priorities, one studied by Rybko-Stolyar (stochastic setting, [10]) and Botvitch-Zamyatin (stochastic setting, [1]), the other one by Lu-Kumar (deterministic setting, [7]) and Dai-Weiss ( uid setting, [5]). In fact, in multiclass networks with priorities, Qt is in general not irreducible in ZZjCj + , and we will prove that this induces an additional, necessary condition of stability. This additional condition is sucient to stabilize the models which are cited above, but this is not true for our network. 2

In order to get the exact stability conditions of such a stochastic, multiclass network, there are two possible (and very close) approaches : the associated uid limit model introduced by Dai (see [4]) or the dynamical system associated to random walks in ZZN+ and de ned by Malyshev (see [11]). The rst approach derives from the ergodicity criterion introduced by Rybko and Stolyar in [10]: Dai extended the scope of this criterion, and deduced that the network is stable if the uid limit model empties in nite time. We just got acquainted with a remarkable paper of Meyn [9] where the author de nes a notion of instability of the uid model which implies the transience of the original network. However, the uid approach is still unable to distinguish between true uid limits and simple solutions of the uid equations, which creates arti cial diculties especially to check the instability of the uid model (see remarks 4.2 and 6.4). Hence we prefered applying the corresponding criteria of ergodicity and transience proved by Malyshev and Menshikov in [11] for random walks in ZZN+ . This approach provides a ner understanding of the transient behaviour of Qt (and presumably also of the behaviour of the uid limit model) via the notion of induced Markov processes. We will then be able to prove the following theorem, which describes the domain of stability of our network.

Theorem 1.1

Consider the functions:

8 < F1 () = (11 + 23 ? 1) _ (12 + 22 ? 1) _ (13 + 21 ? 1) _ (13 + 22 ? 1) (13 + 23 ? 1)(1 ? 12 ? 22 ) ? (12 + 23 ? 1)(1 ? 13 ? 21) : FF23 (()) = = (13 + 23 ? 1) ^ [(13 ? 12) _ F2 ()] and set: F () = F1() _ F3 (). Then if F () < 0, the network is stable, and if F () > 0, the network is unstable.

Here a _ b and a ^ b stand respectively for max(a; b) and min(a; b). Function F1 concentrates the usual conditions and the additional one due to the non-irreducibility of Qt. A three-dimensional projection of the stability domain, which will be studied in section 8, will clearly show that:  the quadratic condition introduced through function F2 is meaningful;  the stability domain (i.e. f=F () < 0g) is not convex;  there is no monotonicity with respect to the partial ordering

 < 0 , 8(i; s) 2 C : is < 0is : More precisely, one can nd two vectors  < 0 , with  corresponding to an unstable network and 0 corresponding to a stable network. These characteristics are especially unusual. The complexity of the stability conditions seems to leave little hope to nd a direct approach to the stability conditions of this kind of multiclass networks. The paper is organized as follows. In section 2, we introduce some essential processes: the potential loads associated to the classes, which at rst allow us to give a simple proof that F1() < 0 if the network is stable, and that the network is unstable if F1() > 0. In section 3, we expose in our special setting the notions of induced Markov processes and stationary drifts and the criteria of ergodicity and transience presented in [11]. In section 4, we show how to compute the stationary drifts via the uid equations. In section 5, we exhibit diverging paths of the associated dynamical system, whose conditions of existence are: F1()  0 and F3 () > 0, and thus yield the conditions of transience of our network. In section 6 3

(resp. in section 7), we use the criteria of [11] to prove Propositions 6.1 and 6.3 (resp. Propositions 7.2, 7.3 and 7.4) which give the conditions of transience (resp. the conditions of ergodicity) of the network, completing the proof of Theorem 1.1. The proofs of all these propositions are very similar, but they involve dierent, technical arguments, and then all of them have been completely written. In section 8 we present (and picture) a three-dimensional projection of the stability domain. We will repeatedly use the following notations : x ^ y for min(x; y), x _ y for max(x; y), x+ for x _ 0 ; when two stochastic processes f = (ft )t0 and g = (gt )t0 satisfy : ft ? gt !0 almost surely when t! + 1, t we will denote it by: f  g.

2 Structural, necessary conditions of stability.

In this section, we are going to prove that F1() < 0 (resp. F1() > 0) is a necessary condition of stability (resp. a sucient condition of instability): this is an easy result that does not require a deep insight into the behaviour of our model. Some auxiliary processes will be especially useful to us all: to each class (i; s) is associated a process W (i; s) = (W t(i; s))t0, which is the cumulated service time that will be required at stage s by all type i customers present at time t in the network, and which we will call the potential load of class (i; s); this is a random function of Qt, but by the law of large numbers we get that : Ps W (i; s)  r=1Qis (i; r) : We will also consider W t (k) (the potential load of queue k), which is de ned by: X W t(i; s): W t(k) = (i;s)2k

There is a natural decomposition of W t (i; s) as: W t(i; s) = t(i; s) ? Bt(i; s); where t (i; s) is the cumulated load brought for class (i; s) up to time t (each type i customer entering the network brings his future service time at stage s), and Bt (i; s) is the load processed (or equivalently the time spent) by server kis for class (i; s) up to time t ( t (k) and Bt (k) are derived in the obvious way). We have: 8 t(i; s) > < t !Zist almost surely when t! + 1, > : Bt (i; s) = 1IfQu (i;s)>0; Qu (j;r)=0 if (j; r) > (i; s)g du: 0 At rst, the processes W (i; s) allow us to prove the following, well-known result very easily:

Lemma 2.1

If (Qt)t0 is ergodic, then

If then (Qt)t0 is transient.

8k 2 f1; :::; K g; k < 1: 9k 2 f1; :::; K g= k > 1;

4

Proof :

!0 when t! + 1 in probability, and in consequence: 8k 2 f1; :::; K g : W tt(k) !0 when t ! +1 in probability, or equivalently: Bt (k) ! when t ! +1 in probability. k t But on the other hand: R Bt (k) = 0t 1IfQu (k)>0gdu ![Q(k) > 0] < 1; t t where  is the limit distribution (which must satisfy : [Q = 0] > 0). Conversely, if for some k we have: k > 1, since Bt (k)  t, 8t, then almost surely: W t(k)  k ? 1 > 0; lim inf t!+1 t

If (Qt)t0 is ergodic, then

Qt t

which implies the transience.

2

The above result is obviously valid for a much more general family of queueing networks. The conditions: 8k 2 f1; :::; K g; k < 1; are the usual conditions of stability. For our network they write : 8 < 11 + 23 < 1 12 + 22 < 1 : 13 + 21 < 1 However, there is a simple reason why these conditions are not sucient stability conditions for our network, and this reason is that the state process Qt is not irreducible in ZZjCj + . Let us explain this point. At rst, notice that state 0 (which corresponds to an empty network) may obviously be reached from any other state, and then the essential states are those that can be accessed from state 0. In order to describe the set of essential states, let us introduce the notion of face in IRjCj + , which will be crucial for further analysis. Let  be a subset of C ; the face F is the subset of IRjCj + de ned by: (x 2 F) () (xis > 0 if (i; s) 2 ; xis = 0 if (i; s) 62 ): This notion is related to the theory of random walks in ZZjCj + which was developped by Malyshev and Menshikov [11], and to which we will refer later. Notice that the union of all the faces is a partition of IRjCj + . When there is no ambiguity,  will also be called a face and F will simply be denoted by . We are going to prove the following lemma :

Lemma 2.2

The set of essential states (and then also the set of unessential states) is a union of faces (intersected with ZZjCj + ), and if F is an essential face, then so is any face F with 0  . More precisely, the unessential faces are the faces F with f(1; 3); (2; 2)g   (which is equivalent to say that the essential states are the 0

5

states Q 2 ZZj+C j satisfying : Q(1; 3) = 0 or Q(2; 2) = 0). In consequence, there is an additional, necessary condition of ergodicity, which is : 13 + 22 < 1; and the network is transient if 13 + 22 > 1. At last, the usual conditions ensure that the set of essential states will be reached from any state in integrable time.

Proof :

A state of the network is a set of customers positioned in dierent classes in the network. A state is essential i it can be reached from state 0, or equivalently i these customers may be brought step by step (from an empty network) to their nal positions and in some order that is compatible with the priorities (which means that a customer may not be moved forward as long as there is a customer with higher priority in his queue). It is then obvious that if a given state is essential, then any other state deduced from the rst one by suppressing one or several customers will also be essential. In consequence, if some face F possesses an essential state, then the state e where there is exactly one customer by class in  is also essential, and so is any state e with 0  . Inversely, if e is essential, it is easy to check that all the states in F are essential : you just have to keep the customers with the same nal positions grouped and to move the dierent classes in the same order as for e . Thus we proved that the essential states form a union of faces and that F is essential if F is essential and 0  . Moreover, a face  is essential i the state e can be reached from state 0. Now it is easy to check that ef(1;3);(2;2)g (that is the state with one customer in (1; 3) and one customer in (2; 2)) cannot be reached from state 0, because whichever customer is positioned rst, his priority will block the other one one stage before his nal position. Then all the faces F with f(1; 3); (2; 2)g   are unessential. We let the reader check empirically that any state e with f(1; 3); (2; 2)g 6  can be reached from state 0 (customers (1; 1) or (2; 1) may be forgotten because they can always be positioned after all the others). We de ne: 0 =
13 + 22; the trac intensity of face f(1; 3); (2; 2)g. For all t  0, if we start the network from an essential state, we have: Zt Bt (0) =
Bt (1; 3) + Bt (2; 2)  1IfQu(0)>0gdu; 0

0

0

where Qu(0) =
Qu(1; 3) + Qu(2; 2). The necessary condition of ergodicity: 0 < 1, and the sucient condition of transience : 0 > 1, are then obtained as in Lemma 2.1. At last, it is easy to check that the condition k < 1 for some k = 1; 2; 3, ensures that from any initial state queue k will empty in integrable time. It is then obvious that 2 < 1 or 3 < 1 are sucient conditions to reach the set of essential states in integrable time. 2

Remark 2.3

In fact, one can prove that in any multiclass network with xed customer routes and class priorities where there is no class (i; s) such that (i; s + 1) > (i; s), the set of essential states will be a union of faces, and any subface of an essential face will be essential. Moreover, to any unessential face  is associated an additional, necessary condition of ergodicity, which is : X  =
is < jj ? 1; (i;s)2

6

and the network is transient if  > jj ? 1. One can check that the unstable network analyzed by LuKumar [7] and Dai-Weiss [5], or that analyzed by Rybko-Stolyar [10] and Botvitch-Zamyatin [1], fall into this instability pattern.

Notice that F1() = 0max ( ? 1). Hence we proved that if the network is stable, then F1 () < 0, k3 k and if F1() > 0, then the network is unstable. However, the condition F1() < 0 is still not sucient to stabilize our network. In order to prove Theorem 1.1, we will have to exploit more deeply the structure of our markovian state process. The background will be that of Malyshev and Menshikov's work about random walks in ZZN+ (see [11]). It is presented in the following section.

3 Induced Markov processes and stationary drifts.

Let A = (axy )x;y2ZZ+ denote the matrix of transition intensities of (Qt )t0. It is easy to check that the following conditions are satis ed: Boundedness of jumps: there exists a constant d such that axy = 0 if kx ? yk > d. Space homogeneity: there exists a function a(; u),   C, u 2 ZZjCj , such that: axy = a(; y ? x), if x 2 F . The mean jump from x will be denoted by M (x) and is de ned by: X axy (y ? x): M (x) = jCj

y

2ZZ+

jCj

The space homogenenity implies that M (x) depends only on the face that x belongs to. The mean jump from face  will then be denoted by M (); it is equal to: X a(; u)u: M () = 2ZZ

u

jCj

The key to study this kind of random walks is the notion of induced Markov process. It is very intuitive in our particular setting. Consider a face   C ,  6= ;, and assume that the components of Q0 that belong to  are initially in nite. Then, these components will of course remain in nite, and the space homogeneity implies that the other components will behave as a separate random walk (Qt )t0 which is the Markov process induced by . Let us consider the example of face  = f(2; 1); (2; 3)g. In Figure 2, the saturated classes (2; 1) and (2; 3) are simply marked with a bullet. The induced Markov process Qt = (Qt (i; s))(i;s)62f(2;1);(2;3)g represents a (sub-)network with the following characteristics:  customers (1; 1) arrive at rate 1 (from outside the network) and are served at rate 0 (they are blocked in queue 1);  customers (1; 2) arrive at rate 0 (class (1; 2) is not fed), are served at rate 121IfQ(2;2)=0g, and then become class (1; 3) customers ;  customers (1; 3) arrive at rate 121IfQ(1;2)>0;Q(2;2)=0g, are served at rate 13, and then leave the network ; 7

-1 

11 ^ 23 



12 ^ 22



13 _ 21 2 

Figure 2 : Face f(2; 1); (2; 3)g.  customers (2; 2) arrive at rate 211IfQ(1;3)=0g (from outside the network), are served at rate 22 , and then leave the network.

For any face , the induced Markov process Q satis es the space homogeneity and the boundedness of jumps. We will denote M (0 ) the mean jump of Q from face 0  Cn; notice that it is the (normal) projection of M ( [ 0) on IRjCnj . We say that face  is ergodic or transient if the corresponding induced Markov process is. By convention, we say that  = C is ergodic. Now assume that face  is ergodic, and denote by  its stationary distribution, and  (0) the stationary probability to be in a face 0  Cn. The stationary drift v() 2 IRjCj + on face  is de ned by: X  0  ( )M ( [ 0 ): v() =  Cn 0

Notice that for (i; s) 62 , we have vis () = 0, because Mis ( [ 0) = Mis (0 ), and then vis () is the stationary mean jump on component (i; s) of the induced Markov process, that is 0. By a natural convention, for  = C we set v() = M (). The idea is that when the components in  are initially very high with respect to the other components, then until one of the components in  comes back to 0 (which takes a long time since the jumps are bounded), the other components will behave as (Qt )t0, and then will converge to the stationary distribution ; in consequence, the \long term" (that is until a class in  empties) drift of the components in  will be given by v(). Notice that for   C and 0  Cn, we might de ne v (0 ) the stationary drift of Q on face 0 . Since obviously: (Q) = Q[ , then v (0 ) is the (normal) projection of v( [ 0) on IRjCnj . In view of this remark, here is a simple criteria of transience of a face in terms of an outgoing face. 0

0

Lemma 3.1

Let two faces   0 be given. If 0 is ergodic and vis(0 ) > 0 for all (i; s) 2 0n, then  is transient, and 0 is called an outgoing face of .

Proof :

The complete proof was given by Malyshev in [8]. In view of our above remark, we have (vis(0 ))(i;s)2Cn = v (0 n): 8

Then a simple ergodic theorem proves that for initially high values of the components (i; s) 2 0n with respect to the components (i; s) 2 Cn, we have with positive probability:

Qt !v() when t! + 1. t

2

Remark 3.2

The name \outgoing face" must be understood in terms of the dynamical system induced by the stationary drifts (see Malyshev [8]); we will later say some words about this notion when we will present the instability cycles of our network. During the analysis of this network, we will have to cope with situations where the conditions of Lemma 3.1 will not strictly be met. We will have an ergodic face 0   such that vis (0 ) > 0 for all (i; s) 2 0n but some (i0 ; s0) 2 0 n such that vi0 s0 (0) = 0. However, in the special cases that we will meet, it will be obvious that for initially high values of the components (i; s) 2 0 n with respect to the components (i; s) 62 0 , the component (i0 ; s0 ) of Qt will not return to 0 in integrable time (it will behave as a null recurrent Markov process). In that case,  is not ergodic, and we will still say that 0 is an outgoing face of .

Now let us present the two fundamental criteria obtained by Malyshev and Menshikov (Theorem 2.1 of [11]). The notation df denotes the derivative of a dierentiable function f .

Theorem 3.3  If there exists a Lipschitz (and then almost everywhere dierentiable) function f : IRjCj + !IR+ such that for some  > 0:

8  C ;  ergodic; 8q 2 F : df (q; v())  ?; then the Markov process (Qt )t0 is ergodic.

 If there exists a Lipschitz (and then almost everywhere dierentiable) function f : IRjCj + !IR such that for some  > 0:

8  C ;  ergodic; 8q 2 F : df (q; v())  ; then the Markov process (Qt )t0 is transient. Remark 3.4

These criteria were actually proven for discrete time Markov chains. However, there is a classic way of coping with this problem; it consists in introducing the discrete time random walk (Q~ n )n2IN whose transition matrix is

A~ = I + A; 9

where I denotes the identity matrix and  satis es:

P1 : 0 <   min x a y

xy

For   C , if Qt admits A as intensity matrix, then A~ = I + A will be the transition matrix of Q~ n ; in consequence, Q~ n is ergodic if and only if Qt is ergodic, and then their stationary distributions ~  and  are equal; moreover one easily checks that:

v~() = v(): All these relations justify the passage from discrete time to continuous time.

In the following section, we will explain how to calculate the stationary drifts as solutions of some limit equations.

4 Stationary ows and limit equations. At rst we must introduce a few new notations. For a given type of customers i = 1; 2, we will denote by Nt (i) the number of type i arrivals into the network up to time t. By convention, we set: N0 (i) = 0, which means that Nt (i) does not include the customers already present in the network at time 0; then it is simply the counting function of a Poisson process of parameter i. For a given class (i; s) 2 C , Dt (i; s) is the number of departures from class (i; s) up to time t (with the convention: D0 (i; s) = 0). By laws of large numbers, we get that almost surely for any type i: Nt (i) ! when t! + 1, i t and for any class (i; s): B (i; s) ' D(i; s) : is

Now take an ergodic face , and consider the induced Markov process Qt jointly with the in nite components in . If Qt is ergodic, then there is a limit distribution  and limit probabilities  (0 ) to be in a face 0 2 Cn, or rather (if we consider all the components in C ) limit probabilities  (0) to be in a face 0  . In consequence, for any class (i; s) 2 C , we have almost surely: X  0 Bt (i; s) = 1 Z t 1I  ( ); f Qu (i;s)>0; Qu (j;r )=0 if (j; r ) > (i; s)g du ?! t t 0  2Lis 0

where Lis is the set of all the faces that satisfy: 0  ; 0 3 (i; s); 0 \ f(j; r)=(j; r) > (i; s)g = ;: Set:

is() = is

X  0  ( ):

 2Lis 0

10

-1 23

11 -11 12 -12 13 -13 ^ ^ _ 23 22 22 21 21 2

Figure 3 : Flow vector. Then almost surely for any class (i; s): Dt (i; s) ! () when t! + 1. is t It is easy to check that for any class (i; s):

vis () = is?1() ? is(); with the convention: i0 () = i.

(1)

You just have to notice that the (i; s) component of M () may be written:

Mis () = is?1() ? is() with i0() = i, and for s  1:  if (i; s) 2  and (j; r) 62  for (j; r) > (i; s) is () = 0is otherwise Relation (1) is the limit version of the following relation:

Qt(i; s) = Q0(i; s) + Dt (i; s ? 1) ? Dt (i; s); with the convention Dt (i; 0) = Nt (i). Then for (i; s) 2 , vis () may be interpreted as the limit, average \drift" of Qt(i; s) when all the classes in  (initially) possess an in nite number of customers. The vector () = (is ())(i;s)2C will be called the stationary ow on face . Figure 3 pictures the

ows through our network on an unspeci ed, ergodic face (from now on, the stationary ow will simply be denoted by ). It will be simpler (but obviously equivalent) to compute the stationary ow than the stationary drift. There are indeed some limit equations that must be satis ed by such ow vectors.

Lemma 4.1 Consider a queue k 2 f1; 2; 3g with k = f(i1 ; s1); (i2 ; s2 )g and (i1 ; s1) > (i2 ; s2) (see Figure 4). Suppose that   C is an ergodic face, and let  be the stationary ow on . Rule 1. If (i1; s1) 2 , then: i1 s1 = i1 s1 ; and: i2 s2 = 0:

Rule 2. If (i1; s1) 2  and (i2; s2) 62 , then we must have i2s2?1 = i2s2 = 0. 11

i1 s1 i1 s1 ?1 i s 1 1 i2 s2 i_s i2 s2 ?1 2 2

Figure 4 : Single queue with two classes. Rule 3. If (i1; s1) 62 , then i1s1 = i1 s1 ?1, and we must have: i1 s1?1  1, or i1s1 ?1  i1s1 . i1 s 1

Rule 4. If (i1; s1) 62  and (i2; s2) 2 , then:

i2 s2 = i2 s2 (1 ? i1 s1 ?1 ): i1 s 1

Rule 5. If (i1; s1) 62  and (i2; s2) 62 , then i2 s2 = i2 s2?1 , and we must have: i1 s1?1 + i2 s2 ?1  1: i1 s1 i2 s2

Proof :

At rst, as we already noticed, if (i; s) 62  we must have vis() = 0, or is() = is?1(), which in fact comes directly from Qt (i; s) !0: t Now x a class (i; s) 2 C . Assume that  \ f(j; r)=(j; r)  (i; s)g = 6 ;, and then take initially Q0 (j; r) = +1 for (j; r) 2 , (j; r)  (i; s). Then the server of queue kis will never be idle, and moreover its activity will be monopolized by the classes (j; r)  (i; s) (due to the priority in the queue). Then at any time t > 0 we will have: X Bt (j; r) = t; (j;r)(i;s)

which implies that:

X jr = 1: (j;r)(i;s) jr

If on the contrary  \ f(j; r)=(j; r)  (i; s)g = ;, then me must have: P  (j;r)(i;s) Qt (j; r) !0 t by ergodicity of Qt . Since the classes (j; r)  (i; s) are served in priority, one can check that this will happen i the limit, average service demand at the queue does not exceed the capacity of the server 12

(which is equal to one); or equivalently i

X jr?1  1: (j;r)(i;s) jr

All the rules enounced in the above lemma may be deduced from these three fundamental results. 2 Lemma 4.1 leads us to make the following de nition: a vector  2 IRjCj + is an admissible ow on face  if it satis es all the conditions enounced in Lemma 4.1, regardless of the ergodicity of .

Remarks 4.2  Notice that we always have: is  is .  Any admissible ow  in IRjCj + must satisfy a system of jCj ane equations (plus additional constraints) which will usually have a unique solution except for special values of the parameters i, i = 1; 2, and is , (i; s) 2 C . Then in general, each face admits at most one admissible drift or ow.  In terms of uid models, one can easily check that an admissible drift is only a solution of the dierential, uid equations. But transient faces may have admissible ows. However, in view of Theorem 3.3, only stationary drifts, that is admissible ows on ergodic faces, have to be considered. The underlying idea is that admissible ows on transient faces must correspond to pathological solutions of uid equations.

At this point, we are supposed to be able to identify all the ergodic faces (which in fact we will not have to do) and to calculate their stationary drifts. Now the natural way to analyze the behaviour of our stochastic network consists in studying the associated dynamical system, which will be de ned in the next section. It is very close to the usual uid model. In particular, any path of the associated dynamical system satis es the uid equations (in view of the above, last remark and the de nition to be given below). But the dynamical system excludes paths that would spend a non-null time on transient faces: this restriction is crucial in order to avoid pathological paths. The following section is more a practical guide to nd the instability conditions of the network as conditions of existence of diverging paths in the dynamical system, than the beginning of the proof of Theorem 1.1 (this will be the object of sections 6 and 7). This proof will however exploit some results of the next section, namely the identi cation of some ergodic (or non-ergodic) faces and the calculation of their stationary drifts; these results are located at the beginning of the proofs of Propositions 5.1 and 5.2.

5 Dynamical system and instability cycles.

For a general introduction to the dynamical system associated to the stationary drifts v(), see Malyshev [8]. Here we just rewrite the very rst lines of his work. It is assumed that all the stationary drifts v() satisfy : vis () 6= 0 if (i; s) 2 . A path Q of the dynamical system is a continuous mapping from some interval [0; T ] (T  +1) in IRjCj + such that: 13

(i) Qt belongs to the union of ergodic faces for almost every t;

(ii) If Qt 2 F for some t  0 and some ergodic face , then dQ dt = v().

Notice that there may be several (if not in nitely many) ways to go out of  if  is a non-ergodic face (this phenomena is called scattering). The simplest way to go out of a non-ergodic face  is to follow the drift v(0 ) of an outgoing face 0 of  (if there is one). Roughly speaking, the idea is that the original Markov process is transient if the dynamical system admits a diverging path. In some special cases, it is possible to prove that, from some initial state, the stochastic model stays in the neighbourhood of a diverging path with positive probability, which accounts for transience. In this section, we will simply exhibit some diverging paths of the dynamical system associated to our network. Our intention is:  to give the intuition of the modes of instability of the network;  to show how the conditions of stability may be found out. In section 6, we will indeed prove that the diverging paths correspond to zones of transience (with respect to the vector of trac intensities), and in section 7 we will prove that the network is stable outside these zones. For this we will apply the criteria of Theorem 3.3, which may be expressed in terms of the dynamical system since 8  C ;  ergodic; 8q 2 F : df (q; v()) = d fdt(Q) for a path Q in state q at time t. For this reason we will systematically use the notation f_ rather than df (q; v()). Here we are going to work under the condition F1 ()  0, which is equivalent to the following set of conditions : Extended, usual conditions : 11 + 23  1 (2) 12 + 22  1 (3) 13 + 21  1 (4)

Extended, additional condition : 13 + 22  1

(5)

since the network is already known to be transient if one of these conditions is not satis ed. We are now going to present dierent diverging paths corresponding to dierent values of the parameters (that is the trac intensities). All have the same form: they start at time t = 0 from face f(1; 1)g, which is transient ; then they consecutively run through several ergodic faces (the rst one being an outgoing face of f(1; 1)g) and nally reach face f(2; 1)g, which is transient ; then they consecutively run through several ergodic faces (the rst one being an outgoing face of f(2; 1)g) and nally reach face f(1; 1)g again ; this pattern is then repeated in nitely often. We call this kind of path a \cycle" ; these cycles are diverging because when the path comes back to face f(1; 1)g (at some time that we will denote T ), we have : QT (1; 1) = Q0 (1; 1) for some constant  > 1. 14

-1 0

&

% %   11 12 11 - 12 - 13 -13 ^ ^ _ 23 0 22 0 21 2 ! ! %

-1 )

0

)

-1 0

!

-1

11 ^ 23 0 !

!

11 ^ 23 !

-1 0

&

%  13 12 12 - 13 ^ _ 22 0 21 2 ! % !

12 ^ 22 !

-1 0

&

13 13 _ 21 2 %

Figure 5 : From f(1; 1)g to f(2; 1)g: path (1:1).

-1 0

&

! %   13 11 11 11 - 12 - 13 ^ ^ _ 23 0 22 0 21 2 ! ! %

-1 )

0

!

!

&

13 11 12 13 ^ ^ _ 23 0 22 0 21 2 ! ! %

-1

-1

Figure 6 : From f(1; 1)g to f(2; 1)g: path (1:2). It is then easy to check that the whole path satis es :

8t  0 : QT +t = Q t ; or : 8n 2 IN; 8t 2 [0; n+1T ] : QT +T +:::+n T +t = n+1 Q nt+1 : Three dierent intermediate paths from face f(1; 1)g to face f(2; 1)g (resp. two dierent paths from face f(2; 1)g to face f(1; 1)g), which will be denoted as paths (1:1), (1:2), (1:3) (resp. as paths (2:1) and (2:2)), will be involved in our dierent cycles. Figures 5 to 9 picture the sequences of ergodic faces corresponding to these dierent paths. For an ergodic face , the notation: % (resp. &) next to a class (i; s) means that (i; s) 2  and vis() > 0 (resp. vis () < 0); the notation ! means that (i; s) 62  and then vis() = 0. The exact ows are also written in these pictures (except for path (1:3), because the formulas for 11, 12 and 21 are a bit complex).

It is easy to check that when these faces are actually ergodic and their drifts have the signs pictured on the gures, they actually provide paths from f(1; 1)g to f(2; 1)g or from f(2; 1)g to f(1; 1)g. We are now going to present the conditions under which diverging cycles may be built from these intermediate paths.

Proposition 5.1

Assume that the following, supplementary conditions are satis ed:

15

-1 21

&

11 ^ 23 !

-11 21

%

12 ^ 22 !

-12 21

!

13 _ 21 %

-12 2

)

-1 21

!

11 ^ 23 !

-1 21

&

12 ^ 22 !

-12 21

!

13 -12 _ 21 2 %

Figure 7 : From f(1; 1)g to f(2; 1)g: path (1:3).

-1 23

%

11 ^ 23 %

-0 22

!

12 ^ 22 %

-0 21

!

13 _ 21 &

-0 2

)

-1 23 -1

)

%

11 ^ 23 %

-0 22

%

!

12 ^ 22 &

!

11 -0 12 ^ 23 ^  2  22 23

&

!

-0  2 -0  2

!

13 -0 _ 2 21  !

!

13 -0 _ 21 2 !

Figure 8 : From f(2; 1)g to f(1; 1)g: path (2:1).

-1 23

% ! ! 11 -0 12 -0 13 -0 ^ ^ _ 23 21 22 21 21 2 ! % &

)

-1 23

% ! ! 11 -0 12 -0 13 -0 ^ 2 _ 2 ^ 2   21  23  22 ! ! &

Figure 9 : From f(2; 1)g to f(1; 1)g: path (2:2).

16

13 + 23 > 1 12 < 13

(6) (7)

Then 11 ^ 12 > 13 and 21 ^ 22 > 23 ; moreover, the paths (1:1) (if 11 > 12 ) or (1:2) (if 11  12) and (2:1) (if 21 > 22) or (2:2) (if 21  22) form a diverging cycle.

Proof :

Condition (6), compared with conditions (2), (4) and (5), yields 13 > 11 , 23 > 21 and 23 > 22 , which in view of condition (7) is equivalent to: 11 ^ 12 > 13; and: 21 ^ 22 > 23 : It is easy to check that the faces of paths (1:1) and (2:1) are ergodic in any case, and that the rst face of path (1:2) (resp. the rst face of path (2:2)) is ergodic i 11 < 12 (resp. i 21 < 22): the induced Markov process always has some components that empty in integrable time, while the other components form a stable queue or a stable tandem. The ows are also easy to compute: the corresponding drifts have the signs pictured in the dierent gures (under condition 11 > 12 for the rst face of path (1:1), under condition 21 > 22 for the rst face of path (2:1)). In the limit case 11 = 12 (resp. 21 = 22), we have v12() = 0 (resp. v22() = 0) on the rst face  = f(1; 1), (1; 2), (1; 3), (2; 1)g of path (1:1) (resp. on the rst face  = f(2; 1), (2; 2), (2; 3), (1; 1)g of path (2:1)). It is not dicult to check that any subface of  including class (1; 1) (resp. including class (2; 1)) admits  as an outgoing face in the sense of remark 3.2, and then is non-ergodic. Notice also that this kind of limit case was not taken into account in the de nition of the dynamical system; it is clear that by following the drift v() from f(1; 1)g (resp. from f(2; 1)g), the dynamical system will in fact run through face f(1; 1); (1; 3); (2; 1)g (resp. f(2; 1); (2; 3); (1; 1)g), even if here this face is not ergodic (in fact it could be proven to be null recurrent), and then our diverging cycle will in fact follow path (1:2) (resp. path (2:2)). The condition under which these cycles are diverging might be obtained by an explicit calculation of QT (1; 1) with respect to Q0 (1; 1) (T is the length of the rst cycle). A simpler way to get it consists in considering the following function which will later be useful to prove the transience of the network in this case: f (Q) = Q(1; 1) + Q(1; 2) + Q(1; 3) + Q(2; 1) + Q(2; 2) + Q(2; 3) 13 23  W (1; 3) + W (2; 3): We obviously have: f_ = 13 + 23 ? 13 ? 23 ; 13 23 and it is easy to check that on any face which our cycles run through we have: 13 23 13 + 23 = 1; or equivalently: f_ = 13 + 23 ? 1: Then under condition (6), our cycles are actually diverging. 2 The following proposition describes a second type of diverging cycles. 17

Proposition 5.2

Assume that the following, supplementary conditions are satis ed:

13 + 23 > 1 12  13 (12 + 23 ? 1)(1 ? 13 ? 21) < (13 + 23 ? 1)(1 ? 12 ? 22 ):

(6) (8) (9)

Then 11 > 13  12 and 22 > 21 > 23. Moreover, the paths (1:3) and (2:2) form a diverging cycle.

Proof :

Condition (8) is equivalent to 12  13. When confronted to conditions (6) and (9), it yields 21 > 22 , or 21 < 22. The other conditions are obtained as for the cycles of the rst type. The faces of path (2:2) have already been proved to be ergodic. Let us analyze the Markov processes induced by the two faces of path (1:3) ; we assume that 12 < 13. At rst notice that the components (1; 3) and (2; 2) form a sub-Markov process of the induced Markov process, whose essential states are the vectors q = (q13; q22) 2 ZZ2+ such that q13q22 = 0. If q13 > 0 and q22 = 0, the mean jump is (12 ? 13; 0), with 12 ? 13 < 0. If q13 = 0 and q22 > 0, the mean jump is (0; 21 ? 22), with 21 ? 22 < 0. This subprocess is then ergodic. The stationary ows 12 and 21 are common to both faces. They satisfy (cf. Lemma 4.1): 8 > < 12 = 12(1 ? 21 ) 22 > : 21 = 21(1 ? 12 ):  We get that:

13

8 > < > :

12 = 21 ? 22   21 ? 13 22 211 = 12 12 ? 13 2 12 21 ? 13 22 We will later prove that condition (9) is equivalent to : 12 + 23 ? 1 21 < 2 22(12 + 23 ? 1) + 12 (23 ? 22 ) ;

(10)

and we let the reader check that

12 + 23 ? 1 22 (12 + 23 ? 1) + 12 (23 ? 22 )  1 , 12 + 22  1; which is condition (3). Then we get that : 21 < 2:

(11)

In consequence, queue 1 will be fed by two arrival processes of rates 1 and 21 < 2 , and then, in view of condition (2), the Markov processes induced by the faces of path (1:3) are ergodic (a complete proof could be given in terms of the criterion of Theorem 3.3). 18

The stationary ows on these faces are then easy to compute (see Figure 7). On the rst face of path (1:3), we have by Lemma 4.1 : 11 = 11(1 ? 21 ) 23 > 1 by conditions (2) and (11). Since this face is supposed to be the outgoing face of f(1; 1)g, we must check that 11 > 12. In view of the expressions of 11 and 12 with respect to 21 , we get that: 11 : 11 > 12 , 21 <  12 ? 2 12 23 ? 1122 Thus in view of relation (10), we just have to check that : 12 ? 11 12 + 23 ? 1 22 (12 + 23 ? 1) + 12 (23 ? 22 )  1223 ? 11 22 : This is equivalent to 11 + 23  1; which is condition (2). In consequence, we actually have 11 > 12 . In the limit case 12 = 13 , the faces of path (1:3) could be proven to be null recurrent ; they admit the corresponding faces of path (1:1) as outgoing faces in the sense of remark 3.2 (on these faces we have v13() = 0) ; notice that the admissible ows on these null recurrent faces of path (1:3) are equal to the ows on their corresponding outgoing faces of path (1:1). Let us now make the explicit calculation of QT (1; 1) with respect to Q0(1; 1) after one cycle. Thus we start from face f(1; 1)g with Q0 (1; 1) = q > 0. Let T1 denote the time when the path reaches face f(2; 1)g. For 0  t  T1 , we have: 8 d(Q(1; 1) + Q(1; 2) + Q(1; 3)) > < = 1 ? 12; dt > : dQ(2; 1) = 2 ? 21 dt (with 12 and 21 already caculated), and then: 8 q > < T1 =  ?  ; 12 1 > QT1 (2; 1) = (2 ? 21)T1 = 2 ? 21 q: : 12 ? 1 Let now T2 denote the time when the path comes back to face f(1; 1)g. For T1  t  T2 , we have: 8 d(Q(2; 1) + Q(2; 2) + Q(2; 3)) > < = 2 ? 23; dt > : dQ(1; 1) = 1 ; dt and then: 8 2 ? 21 QT1 (2; 1) > > < T2 ? T1 = 23 ? 2 = (12 ? 1)(23 ? 2 ) q; 1(2 ? 21) q = 23 (1 ? 212 ) q > (1 ; 1) =  ( T ? T ) = Q > T2 1 2 1 ( ?  )( ?  ) : (1 ?  )( 12 ? 1) 12

1

19

23

2

23

1

Thus the path is diverging i

23 (1 ? 212 ) > 1: (1 ? 23 )( 121 ? 1) Let us express this condition in terms of 21 :

! 1 ? 22 212    12 21 21 23(1 ?  ) > (1 ? 23)(  ? 1) , 23 (1 ?  ) > (1 ? 23) 12 ? 1 2 1 2 , 21 <  ( + 12?+1)23+ ? 1( ?  ) 2

22 12

23

12 23

22

which is condition (10). More explicitly :

12 + 23 ? 1 21 < 2 22(12 + 23 ? 1) + 12 (23 ? 22 ) ,  12 ?? 13  <  ( + 12?+1)23+ ? 1( ?  ) 12 21 22 13 22 12 23 12 23 22 , (12 + 23 ? 1)(1 ? 13 ? 21 ) < (13 + 23 ? 1)(1 ? 12 + 22 ); which is condition (9) of the proposition. The proof is now complete.

2

Notice that these diverging paths correspond to the condition: F1 ()  0, F3() > 0. By use of the transience criterion of Theorem 3.3, we are now going to prove that the network is transient under this condition. The ergodicity criterion of the same theorem will later allow us to prove that the network is stable if F3 () < 0 and F1() < 0.

6 Instability conditions. We keep on working under conditions (2), (3), (4) and (5). In this section, we are going to give the proofs of transience of our network in the zones delimited in the previous section. For this we will apply the criterion of transience in Theorem 3.3. Our Lyapunov functions will be based on functions W (i; s), (i; s) 2 C .

Proposition 6.1

If the following conditions are satis ed :

22 < 23 13 + 23 > 1 12 < 13

(12) (6) (7)

then the network is unstable (transient).

Remark 6.2

Condition (12), which in fact arises as a consequence of conditions (5) and (6), was added because it plays a speci c role in the stability conditions that we will analyze in the next section.

20

Proof :

Let us recall that the above conditions imply that : 11 ^ 12 > 13 and 21 ^ 22 > 23. Consider the following functions: 8 > ) = [Q(1; 2) ^ Q(2; 1) ? Q(1; 3)]+ > < ff12 ((Q Q) = [Q(1; 2) + Q(1; 3)] ^ [Q(2; 2) + Q(2; 3)] ; 2) + Q(1; 3) + Q(2; 1) + Q(2; 2) + Q(2; 3)  W (1; 3) + W (2; 3) > : f3 (Q) = Q(1; 1) + Q(1   13

23

and:

f (Q) = ? ( f1 (q) + f2 (q)) + f3 (Q); where , and are \suciently large", positive constants. We already checked that f_3 = 13 + 23 ? 1 > 0 along the corresponding cycle. Our idea is that from any initial state the dynamical system eventually joins this diverging path. Functions f1 and f2 are arbitrary means to translate this idea in terms of Lyapunov functions. We have indeed : f1 (Q) = f2 (Q) = 0 on any face of the cycle, and we will prove that:  for suciently large values of , on any part of an ergodic face where f1 > 0, we have f_1  0, and either f_1  ?1 for some positive constant 1 , or f_1 = 0 and f_3 = 13 + 23 ? 1 > 0 ;  on any ergodic face where f1 = 0 and f2 > 0, we have f_1 = 0 (which is obvious in view of the above assertion) and f_2  0, and either f_2  ?2 for some positive constant 2 , or f_2 = 0 and f_3 = 13 + 23 ? 1 > 0;  on any ergodic face where f1 = f2 = 0 and f3 > 0, we have f_1 = f_2 = 0 (which is obvious) and f_3 = 13 + 23 ? 1 > 0. It will then be obvious that for suciently large values of , and , there exists a constant  > 0 such that on any ergodic face, we have f_  . Since f is obviously Lipschitz, transience will follow from Theorem 3.3. Case f1 > 0 : At rst consider an ergodic face where f1 > 0, and then f(1; 2); (2; 1)g  . We have : f_1  (11 ? 12) _ (2 ? 21 ) ? (12 ? 13 ):

 If (1; 3) 2 , then (2; 2) 62  (otherwise the face would be unessential), and then we have: 13 = 13 , 22 = 21 = 0, and 12 = 12 (see Figure 10). In consequence: 12 ? 13 = 12 ? 13 > 0, and then f_1 < 0 for suciently high values of .  If (1; 3) 62 , we must have (2; 2) 2 , because otherwise there would be an outgoing face on the model of Figure 10. We thus have 22 = 22 > 23 , and then we must have (2; 3) 2  and (1; 1) 2 ; that is:

 = f(1; 1); (1; 2); (2; 1); (2; 2); (2; 3)g (see Figure 11). In consequence, we have 23 = 23 and 11 = 0; 12 = 0; 13 = 0 and 21 = 21 . So we get that: f_1  0 _ (2 ? 21) = 0, and f_3 = 13 + 23 ? 1 > 0, which was the expected result. 21

-1 

 %  13 12  11 11 - 12 - 13 ^ ^ _ 23 0 22 0 21 2 ! %

Figure 10 : f(1; 2); (1; 3); (2; 1)g  .

-1 23

%

!

!

%



&

11 -0 12 -0 13 -0 ^ ^ _ 23 22 22 21 21 2

Figure 11 : Face f(1; 1); (1; 2); (2; 1); (2; 2); (2; 3)g. Case f1 = 0, f2 > 0 : Now assume that f2 > 0, which means that f(1; 2); (1; 3)g \  6= ;, and f(2; 2); (2; 3)g \  6= ; (or (2; 3) 2  as we already noticed). Then 23 = 23 and 11 = 0, and f_2  (11 ? 13) _ (21 ? 23) = (?13 ) _ (21 ? 23):

 If (1; 3) 2 , then 13 = 13 and 21 = 0, and we get that: f_2  (?13) _ (?23 ) < 0:  Then assume that (1; 3) 62 , (1; 2) 2  and still (2; 3) 2 . If f1 = 0, we cannot have (2; 1) 2 . Moreover, we must have (2; 2) 2 , otherwise f(1; 1); (1; 2); (1; 3); (2; 1); (2; 3)g would be an outgoing face (see Figure 12). Then obviously  = f(1; 1); (1; 2); (2; 2); (2; 3)g and: 21 = 2 and 13 = 12 = 0 (see Figure 13). In consequence: f_2  0 _ (2 ? 23) = 0; but in any case we have f_3 = 13 + 23 ? 1 > 0. Thus we proved that on ergodic face where f1 = 0 and f2 > 0, we have f_2  ?2 for some positive constant 2, or f_2 = 0 but f_3 = 13 + 23 ? 1 > 0. Case f1 = f2 = 0, f3 > 0 : 22

-1

%

11 -0

23 ^ 0 23 &

&

%  13 12 12 - 13 ^ _ 22 0 21 2 ! %

Figure 12 : Face f(1; 1); (1; 2); (1; 3); (2; 1); (2; 3)g.

-1 23

%

!

11 -0 12 ^ ^ 23 22 22

%

&

-0  2

!

13 -0 _ 21 2 !

Figure 13 : Face f(1; 1); (1; 2); (2; 2); (2; 3)g. At last, consider an ergodic face where f2 = 0, that is :

f(1; 2); (1; 3)g \  = ;; or: f(2; 2); (2; 3)g \  = ;:  If  \ f(1; 2); (1; 3); (2; 2); (2; 3)g = ;, then   f(1; 1); (2; 1)g. If (1; 1) 2  (resp. (2; 1) 2 ), faces f(1; 1); (1; 2); (1; 3); (2; 1)g or f(1; 1); (1; 3); (2; 1)g (resp. faces f(2; 1); (2; 2); (2; 3); (1; 1)g or f(2; 1), (2; 3), (1; 1)g) are outgoing faces of  according that 11 > 12 or 11  12 (resp. 21 > 22 or 21  22), as was proved in the previous section (see Figures 5, 6, 8 and 9).  Then assume that either  \f(1; 2); (1; 3)g = ; and  \f(2; 2); (2; 3)g =6 ;, or  \f(1; 2); (1; 3)g =6 ; and  \ f(2; 2); (2; 3)g = ;. { If  \ f(1; 2); (1; 3)g = ; and  \ f(2; 2); (2; 3)g =6 ;, then we must have (2; 3) 2  (see Figure 14). Since (2; 3) 2 , we have 23 = 23 and 13 = 12 = 11 = 0, and consequently: f_3 = 13 + 23 ? 1 > 0. { If on the contrary f(2; 2); (2; 3)g\  = ; and f(1; 2); (1; 3)g\  =6 ;, then we must have (1; 3) 2 , because if (1; 3) 62  and (1; 2) 2 , faces f(1; 2); (1; 3); (2; 1)g or f(1; 1); (1; 2); (1; 3); (2; 1)g are outgoing faces of  (see Figure 15). In consequence, we have 13 = 13 and 23 = 22 = 21 = 0, and f_3 = 13 + 23 ? 1 > 0. Thus we proved that on any ergodic face where f2 = 0 and f3 > 0, we have f_3 = 13 + 23 ? 1 > 0. The proof is then complete.

2

The following proposition corresponds to second-type cycles. 23

!

-1

!

11 -0 12 -0 13 -0 23 ^ 22 ^ 21 _ 2 23 22 21



Figure 14 :  \ f(1; 2); (1; 3)g = ; and  \ f(2; 2); (2; 3)g =6 ;.

-1 0

11 ^ 23 !

-11 0



12 ^ 22 !

-12 0

%

13 13 _ 21 2 %

Figure 15 :  \ f(2; 2); (2; 3)g = ; and (1; 2) 2 . Proposition 6.3

If the following conditions are satis ed :

22 < 23 13 + 23 > 1 12  13 (12 + 23 ? 1)(1 ? 13 ? 21 ) < (13 + 23 ? 1)(1 ? 12 ? 22 )

(12) (6) (8) (9)

then the network is unstable (transient).

Proof :

We recall that the above conditions imply that 11 > 13  12 and 22 > 21 > 23 . The following functions will allow us to prove the transience of our network: 8 ) = Q(2; 2) > < ff12 ((Q Q) = [Q(1; 2) + Q(1; 3)] ^ [Q(2; 2) + Q(2; 3)] ; 2) + Q(2; 3) + (1 ?  +  ) Q(1; 1) + Q(1; 2) + Q(1; 3) > > : f3 (Q) = 13 Q(2; 1) + Q(2 23 23 13  13 W (2; 3) + (1 ? 23 +  )W (1; 3) and: f (Q) = ? (f1 (Q) + f2 (Q)) + f3 (Q); where  and are \suciently large", positive constants, and  is a \suciently small", positive constant. We could have checked that for suciently small, positive  , there exists a constant 3 > 0 such that f_3  3 along the second-type cycles. Again functions f1 and f2 are arbitrary means to express the 24

idea that the dynamical system eventually joins such a cycle. We have indeed f1 = f2 = 0 on any face of the cycle, and we will prove that:  on any ergodic face where f1 > 0, we have f_1  ?1 for some constant 1 > 0 ;  on any ergodic face where f1 = 0 and f2 > 0, we have f_1 = 0 (obviously) and f_2  ?2 for some constant 2 > 0;  for suciently small  , on any ergodic face where f1 = f2 = 0 and f3 > 0, we have f_1 = f_2 = 0 (obvious) and f_3  3 > 0. It will then be obvious that for suciently large values of  and , there exists a constant  > 0 such that on any ergodic face, we have f_  . Case f1 > 0 : So consider a face  where f1 > 0, that is (2; 2) 2 . Then we have: f_1 = 21 ? 22 = 21 ? 22  21 ? 22 < 0; which was to be proved. Case f1 = 0, f2 > 0 : Now assume that f2 > 0 and f1 = 0, that is  \ f(1; 2); (1; 3)g = 6 ;, (2; 2) 62  and (2; 3) 2 . We have 23 = 23 and 11 = 0 (and then (1; 1)) 2 ), and then f_2  (11 ? 13) _ (21 ? 23) = (?13 ) _ (21 ? 23):  If (1; 3) 2 , then 13 = 13 and 21 = 0, and thus: f_2  (?13) _ (?23 ) < 0:

 If (1; 3) 62 , (1; 2) 2 , we must be on faces f(1; 1); (1; 2); (2; 3)g or f(1; 1); (1; 2); (2; 1); (2; 3)g; but the analysis of path (1:3) in the previous section proves that the latter is an outgoing face of the former (see Figure 16) ; moreover, we have 21 < 2 and 13 = 12 = 12(1 ? 2122 ) > 1 (by condition (3)), and then:

f_2  (?1 ) _ (2 ? 23) < 0: Thus we proved that on any ergodic face where f2 > 0 and f1 = 0, we have f_2  ?2 for some constant 2 > 0. Case f1 = f2 = 0, f3 > 0 : At last consider an ergodic face  where f1 = 0 and f2 = 0, that is (2; 2) 62  and f(1; 2); (1; 3)g\  = ; or (2; 3) 62 . We have: f_3 = 13 23 (1 ? 23 ) + (1 ? 23 +  )13 (1 ? 13 ) 2 1   23 13 = 13 [(1 +  ) ? 23  ? (1 ? 23 +  )  ]: 2

25

1

-1

%

&

!

&

!

%

12 11 -0 12 -12 13 23 ^ 21 ^ 21 _ 2 23 22 21

Figure 16 : Face f(1; 1); (1; 2); (2; 1); (2; 3)g.  !   12 11 12 11 - 12 - 13 21 ^23 21 ^22 21 _21 2 ! ! %

-1

Figure 17 : Face f(1; 1); (1; 2); (2; 1)g or f(1; 2); (2; 1)g.  At rst, consider the case when f(1; 2); (1; 3)g \  = ; (and then 13 = 12 = 11 ), and (2; 3) 2 . In consequence, 23 = 23 and 11 = 0, and then 13 = 0. So we get that: f_3 = 13 [(1 +  ) ? 1 ? 0] = 13  > 0:

 Now suppose that f(1; 2); (1; 3)g\  6= ; and f(2; 2); (2; 3)g\  = ; (and then 23 = 22 = 21).

{ If (1; 3) 2 , we get that: 13 = 13, 21 = 0, and then 23 = 0. In consequence: f_ =  [(1 +  ) ? 0 ? 1 ? 23 +  ] = ( +  ? 1) ?  (1 ?  ); 3

13

13

13

23

13

which is positive for suciently small values of  . { If (1; 3) 62 , then (1; 2) 2 , and even (2; 1) 2  because otherwise faces f(1; 1); (1; 2); (2; 1)g or f(1; 2); (2; 1)g would be outgoing faces of the current face (see the analysis of path (1:3)). So  is one of these two faces (see Figure 17). We already got (see the proof of Proposition 5.2): 8 ? 22  > < 12 =  21 ? 1 12 21 ? 13 22 12 13 > : 21 = 12 21 ? 13 22 2 26

-1

%

!

!

%

!

&

11 -0 12 -0 13 -0 23 ^ 21 ^ 21 _ 2 23 22 21

-1 21

&

11 ^ 23 !

-11 21

%

12 ^ 22 !

-12 21

!

13 -12 _ 21 2 %

Figure 18 : Faces f(1; 1); (2; 1); (2; 3)g and f(1; 1); (1; 2); (2; 1)g. In consequence :

? 13 ? (1 ?  +  ) 21 ? 22 > 0 f_3 > 0 , (1 +  ) ? 23  12 ? 23 12 21 ? 13 22 12 21 1322 , (1 +  )(12 21 ? 1322 ) > 23 (12 ? 13) + (1 ? 23 +  )(21 ? 22) , (12 + 23 ? 1)(1 ? 13 ? 21 ) < (13 + 23 ? 1)(1 ? 12 ? 22 ) ? [21 (1 ? 12 ) ? 22 (1 ? 13 )] which is valid under condition (9) for suciently small values of  .  At last assume that  \ f(1; 2); (1; 3); (2; 2); (2; 3)g = ;. Then either f(1; 1); (2; 1); (2; 3)g or f(1; 1), (1; 2), (2; 1)g is an outgoing face of  (see Figure 18) in view of the analysis of second-type cycles, and then  is transient.

Remark 6.4 For  = f(1; 1)g, there may be an admissible drift v with v11 < 0 (and of course vis = 0 if (i; s) = 6 (1; 1)), and then f_3 < 0 on the path of drift v. It is then crucial to make the remark about the non-ergodicity of .

So we proved that for some constant 3 > 0, we have f_3  3 > 0 on any ergodic face where f1 = f2 = 0. The proof is now complete. 2 Thus we proved that the network is unstable if F1()  0 and F3 () > 0. Since we already knew that it was unstable if F1() > 0, we actually proved that F () > 0 implies the transience of the model. The next section is devoted to the proof that the network is stable when F () < 0.

7 Stability conditions. We will assume that the following, necessary conditions of stability are satis ed (see Section 2): 27

Usual conditions : 11 + 23 < 1 12 + 22 < 1 13 + 21 < 1

(13) (14) (15)

Additional condition : 13 + 22 < 1:

(16)

At rst, we will prove a technical lemma that will be useful for our calculations.

Lemma 7.1

Assume that x0 and y0 are some non-negative solutions of the system:



where the constants rij satisfy:

Then we have:

Proof :

r11x + r22y = 1 r21y + r12x = 1

8 < r11 + r22 < 1 + r21 < 1 : rr12 12 + r22 < 1 r12(1 ? x0 ) + r22(1 ? y0 ) < 0:

If r11r21 ? r12r22 = 0, then the two equations must be equivalent, and then r11 = r12 and r21 = r22. In consequence: r12x + r22y = 1; and then: r12(1 ? x0) + r22(1 ? y0 ) = r12 + r22 ? 1 < 0: If r11r21 ? r12r22 6= 0, the system has a single solution: 8 ? r22 > < x = r rr21 ? 11 21 r12r22 > y = r11 ? r12 : r11r21 ? r12r22 Let  = 1 if r11r21 ? r12r22 > 0,  = ?1 if r11r21 ? r12r22 < 0. Since x  0 and y  0, we must have (r11 ? r12)  0 and (r21 ? r22)  0, and: (r11 ? r12) + (r21 ? r22) > 0: In consequence:

r12(1 ? x0) + r22(1 ? y0 ) < 0 () r12 + r22 ? r rr21 ?? rr22r r12 ? r rr11 ?? rr12r r22 < 0 11 21 12 22 11 21 12 22 () (r21 ? r22)r12 + (r11 ? r12)r22 ? (r12 + r22)(r11r21 ? r12r22) > 0 () (r21 ? r22)r12(1 ? r22 ? r11) + (r11 ? r12)r22(1 ? r12 ? r21) > 0 28

which is valid under the assumptions of the lemma.

2

We will now explore the stability domain, which will be divided into three dierent parts. The following proposition deals with the rst part.

Proposition 7.2

If the following conditions are satis ed:

22 < 23 13 + 23  1 12 > 13 (12 + 23 ? 1)(1 ? 13 ? 21 ) > (13 + 23 ? 1)(1 ? 12 ? 22 )

(12) (17) (18) (19)

then the network is stable (ergodic).

Proof :

Notice that conditions (15) and (17) on one hand, condition (12) on the other hand, imply that 21 ^ 22 > 23: We de ne: 8 < f1 (Q) = Q(1; 3) ; 2) + Q(2; 3) + (1 ?  ?  ) Q(1; 1) + Q(1; 2) + Q(1; 3) : f2 (Q) = 13 Q(2; 1) + Q(2 23 23 13  13 W (2; 3) + (1 ? 23 ?  )W (1; 3) and: f (Q) = f1 (Q) + f2 (Q); for some positive constants  and  ,  \suciently small" (in particular  < 1 ? 23 ) and  \suciently large". Case f1 > 0 : Condition (18) ensures that if f1 > 0 (or: (1; 3) 2 ), then f_1 = 12 ? 13  12 ? 13 < 0; and so for suciently large , f_ < 0 if f1 > 0. Case f1 = 0, f2 > 0 : Consider now an ergodic face  where f1 = 0 (or: (1; 3) 62 ) and f2 > 0. We have: f_ = f_2 = 13 23 (1 ? 23 ) + (1 ? 23 ?  )13(1 ? 12 ) 2 1 = 13 [(1 ?  ) ? 23 ? (1 ? 23 ?  ) 12 ]: 23

29

1

!   11 12 11 - 12 - 13 -12 21 ^23 21 ^22 21 _21 2 ! !

-1

Figure 19 :  \ f(1; 3); (2; 2); (2; 3)g = ;.  If (2; 3) 2 , then 23 = 23 and f_  ?13  < 0.  If (2; 3) 62 , then (2; 2) 62  since 22 > 23 (condition (12)); in consequence, we have  \ f(1; 3), (2; 2), (2; 3)g = ; and f_ = 13 [(1 ?  ) ? 21 ? (1 ? 23 ?  ) 12 ] 23

(see Figure 19).

{ If (2; 1) 62 , then

1

21 = 20 = 2; f_ = 13 (1 ? 23 ?  )(1 ? 12 ); 1

and   f(1; 1); (1; 2)g. But either (1; 2) 2  and 12 = 12(1 ? 22) > 1 (by condition (14)), or  = f(1; 1)g and 12 = 11 = 11(1 ? 23 ) > 1 (by condition (13)), and in both cases f_ < 0. { If (2; 1) 2  and (1; 2) 62 ,  is not ergodic: since 21 ^ 22 > 23, it admits face f(2; 1), (2; 2), (2; 3), (1; 1)g or face f(2; 1); (2; 3); (1; 1)g as an outgoing face according that 21  22 or 21  22 (see the analysis of paths (2:1) and (2:2) in section 5). { Then assume that (2; 1) 2  and (1; 2) 2 . If 21 > 22, then again  is not ergodic (this time it admits f(1; 1), (1; 2), (2; 1), (2; 2), (2; 3)g as an outgoing face, see Figure 11), and thus we proved that f_ < 0 on all the ergodic faces. If (2; 1) 2 , (1; 2) 2 , and 21  22, we know that: 8 22  > < 12 =  21 ? ? 13 22 1 12 21 ? > : 21 =  12 ? 13  2 12 21 13 22 In consequence, we get that: f_ < 0 , (12 + 23 ? 1)(1 ? 13 ? 21 ) > (13 + 23 ? 1)(1 ? 12 ? 22 ) + [21(1 ? 12) ? 22 (1 ? 13 )] (we already made the caculation in the proof of Proposition 6.3), which is valid under condition (19) for suciently small values of  . 30

Thus we proved that f_ < 0 on all the ergodic faces: since f is obviously Lipschitz, Theorem 3.3 allows us to conclude that the network is ergodic. 2 Let us now explore the second part of the stability domain.

Proposition 7.3

If the following conditions are satis ed:

22 < 23 13 + 23 < 1

(12) (13)

then the network is stable (ergodic).

Proof :

Notice that condition (13) is stronger than condition (16) under condition (12). We set: f (Q) = Q(1; 1) + Q(1; 2) + Q(1; 3) + Q(2; 1) + Q(2; 2) + Q(2; 3)  W (1; 3) + W (2; 3): 13 23 The arguments to prove that f_ < 0 everywhere are the same as those already encountered. Notice that : f_ = 13 (1 ? 13 ) + 23 (1 ? 23 ): 1

2

 If (1; 3) 2  or (2; 3) 2 , then 13 = 13 or 23 = 23 and: f_  13 + 23 ? 1 < 0:  We now assume that (1; 3) 62  and (2; 3) 62 , and then again (2; 2) 62  in view of condition (12)

(otherwise there is no admissible ow); see Figure 19 to visualize these characteristics. { If (1; 2) 62 , then   f(1; 1); (2; 1)g, and : f_ = 13 (1 ? 11 ) + 23 (1 ? 21 ): 1 2 If  = f(1; 1); (2; 1)g, we may apply the result of Lemma 7.1 with x0 = 11=1, y0 = 21 =2, r11 = 11, r12 = 13 , r21 = 21 , and r22 = 23 ; the hypothesis of this lemma are indeed satis ed in view of Lemma 4.1, condition (13), and the usual conditions (13) and (14). In consequence, we get that f_ < 0. If  = f(1; 1)g or  = f(2; 1)g, the result is even more immediate in view of the usual conditions. { So we assume that (1; 2) 2 .  If (2; 1) 62 , then 23 = 22 = 21 = 2 and: 13 = 12 = 12 (1 ? 2 ) > 1 (by (14)); 22 which means that f_ < 0. 31

 !   12 11 12 11 - 12 - 13 21 ^23 21 ^22 21 _21 2 ! ! 

-1

Figure 20 :  = f(1; 1); (1; 2); (2; 1)g or  = f(1; 2); (2; 1)g.  At last, assume that (2; 1) 2  (see Figure 20), which means that  = f(1; 1); (1; 2); (2; 1)g or  = f(1; 2); (2; 1)g. This time, x0 = 12=1 and y0 = 21=2 must be non-negative solutions of the following system:  12x + 22 y = 1 13x + 21 y = 1 If 1221 ? 22 13 = 0, the two equations must be equivalent and then: 12 = 13 and 21 = 22 , and: by (12) 13 12 + 23 21  13 12 + 22 21 = 1; which implies that:

1

2

1

2

f_  13 + 23 ? 1 < 0 by (13): Now assume that 12 21 ? 22 13 6= 0. Let  = 1 if 1221 ? 1322 > 0,  = ?1 if 12 21 ? 1322 < 0. In view of the above equations, we have (12 ? 13 )  0 and (21 ? 22 )  0. Moreover, by solving the above system (see the proof of Lemma 7.1), we get: 22  ? 12 ? 13  ; f_ = 13 + 23 ?  21 ? 13   ?   23 12 21 ? 13 22 12 21 13 22 which may be considered as a linear function of 23 . By (12) and (13), we have 23 2 ]22; 1 ? 13 [. But for 23 = 22, we get f_ < 0 as a direct consequence of lemma 7.1, since 13 + 22 < 1 (condition (16)). So we just have to prove that f_  0 for 23 = 1 ? 13 . This amounts to: ? 22  ? 12 ? 13 (1 ?  )  0 f_  0 () 13 + (1 ? 13 ) ?  21 ? 13   ?   13 12 21 1322 12 21 13 22 () (21 ? 22)13 + (12 ? 13 )(1 ? 13 ) ? (12 21 ? 13 22)  0 () (12 ? 13)(1 ? 13 ? 21 )  0; which is valid under the usual condition (15). All the conditions of Theorem 3.3 have thus been checked, and in consequence we may conclude that the network is stable. 2 The third and last part of the stability domain is presented in the following proposition. 32

Proposition 7.4

If the following condition is satis ed:

then the network is stable (ergodic).

22  23

(14)

Proof :

Consider the following functions : 8 < f1 (Q) = Q(2; 3); : f2 (Q) = Q(1; 1) + Q(1; 2) + Q(1; 3) + Q(2; 1)+ Q(2; 2)  W (1; 3) + W (2; 2); 13 22 and: f (Q) = f1 (Q) + f2 (Q) for some positive constant . Case f1 > 0 : Consider an ergodic face  where f1 (Q) = Q(2; 3) > 0 (that is (2; 3) 2 ). Then, in view of Lemma 4.1: f_1 = 22 ? 23  22 ? 23  0; and f_1 = 0 i 22 = 22 = 23, which implies that: f_2 = 13 + 22 ? 13 ? 22  13 + 22 ? 1 < 0: 13 22 Thus if f1 > 0 we have f_ < 0 for any suciently large . Case f1 = 0, f2 > 0 : Now we just have to prove that f_2 < 0 on any (ergodic) face  such that f2 > 0, or (2; 3) 62 .  If (1; 3) 2  (resp. (2; 2) 2 ), then 13 = 13 (resp. 22 = 22), and : f_2  13 + 22 ? 1 < 0:

 So assume that f(1; 3); (2; 2); (2; 3)g \  = ; (see Figure 21). We have now :

f_2 = 13 (1 ? 12 ) + 22(1 ? 21 ): 1

2

{ If (2; 1) 62 , then 21 = 2 and we already checked that in this situation 12 > 1 due to the

usual conditions; in consequence: f_2 < 0. { If (2; 1) 2  and  \ f(1; 1); (1; 2)g = ;, then 12 = 11 = 1 and 21 > 2 by the same argument, and so f_2 < 0. { Now assume that (2; 1) 2  and (1; 2) 2 . This time we may apply Lemma 7.1 with x0 = 12=1, y0 = 21=2, r11 = 12, r11 = 12 , r12 = 13 , r21 = 21 and r22 = 22 , which directly yiels that f_2 < 0. 33

!   11 12 11 - 12 - 13 -12 21 ^23 21 ^22 21 _21 2 ! !

-1

Figure 21 :  \ f(2; 3); (1; 3); (2; 2)g = ;.

-1 21



11 ^ 23 !

!

-11 21

12 ^ 22 !

-11 21

!

13 -11 _ 21 2 

Figure 22 :  = f(1; 1); (2; 1)g. { So only one case remains, which is  = f(1; 1); (2; 1)g (see Figure 22). We set x0 = 11 =1,

y0 = 21=2. In view of Lemma 4.1, x0 and y0 are non-negative solutions of the following system:  11 x0 + 23y0 = 1 13 x0 + 21y0 = 1 If 11 21 ? 13 23 = 0, then the two equations must be equivalent, which implies that 11 = 13 and 21 = 23 . In consequence: by

(14)

13x0 + 22 y0  13x0 + 23 y0 = 1; and then: f_2  13 + 22 ? 1 < 0: If 1121 ? 13 23 6= 0, let  = 1 if 11 21 ? 13 23 > 0,  = ?1 if 1121 ? 13 23 < 0. In view of the above equations, we have (11 ? 13 )  0 and (21 ? 23 )  0. Moreover, by solving the system we get: ? 23  ? 11 ? 13  ; f_2 = 13 + 22 ?  21 ? 13   ?   22 11 21 13 23 11 21 13 23 which may be considered as a linear function of 22. By (14) and (16), we have 22 2 [23; 1 ? 13 [. But for 22 = 23, f_2 < 0 in view of Lemma 7.1 (since 13 + 23  13 + 22 < 1). So we just have to prove that f_2  0 for 22 = 1 ? 13. 34

In this case, we get that:

? 23  ? 11 ? 13 (1 ?  )  0 f_2  0 () 13 + (1 ? 13 ) ?  21 ? 13   ?   13 11 21 13 23 11 21 13 23 () (21 ? 23)13 + (11 ? 13 )(1 ? 13) ? (11 21 ? 13 23)  0 () (11 ? 13)(1 ? 13 ? 21)  0; which is valid under the usual condition (15). Thus we proved that there exists a positive constant  such that on any ergodic face, f_  ? < 0. The network is then stable (by Theorem 3.3). 2

8 Three-dimensional projection of the stability domain. In order to visualize the stability domain, let us consider its projection on the following, three-dimensional subspace of IR6: 11 = 22; 12 = 23 ; 13 = 21: The three variables above will be denoted respectively x, y and z . Then the usual conditions and the additional one write: x + y < 1; x + z < 1; z < 12 ; which delimits a simplex in IR3+ . We let the reader check that the projection of the stability domain is this simplex amputated of the following part: ( x < 12 ; 21 < y < 1 ? x; 1 ? y < z < 12 (2y ? 1)(1 ? 2z )  (y + z ? 1)(1 ? x ? y) It is more convenient to make the following change of variables: X = x ? 12 ; Y = y ? 21 ; Z = z ? 12 : Then the simplex becomes: X + Y < 0; X + Z < 0; Z < 0 (with the implicit constraint: (X; Y; Z )  (?1=2; ?1=2; ?1=2)), and it is amputated of :  X < 0; 0 < Y < ?X; ?Y < Z < 0 4Y Z  (Y + Z )(X + Y ) In this form we see that the stability domain is the intersection of a cone of vertex (0; 0; 0) with the cone (X; Y; Z )  (?1=2; ?1=2; ?1=2), or (in terms of (x; y; z )) the intersection of a cone of vertex (1=2; 1=2; 1=2) with the orthant (x; y; z )  (0; 0; 0) (see Figure 23 to have a global view of the stability domain). The form of the cut at x = 0 of the stability domain is then representative of the cuts at x = x0 for x0 2 [0; 1=2]. It is the zone delimited by full lines in Figure 24. There it is obvious that the previously announced properties are true:  the quadratic condition is meaningful; 35

x

6

1

0.8

0.6

0.4

PPPP P

0.2

y

0 0 

PPPP P Pq 0 P 0.2

0.2 0.4

0.4 0.6

0.6 0.8

0.8 1 1

Figure 23 : Three-dimensional projection of the stability domain.

36

z

 the stability domain is not convex;  in any cut x = x0 2]0; 1=2[, we can nd (y; z ) in the stability domain and (y0 ; z 0) in the instability domain such that (y; z ) > (y0 ; z 0). By continuity of the cuts, we can even nd (x; y; z ) > (x0 ; y0 ; z 0) such that (x; y; z ) is a stability point and (x0; y0 ; z 0) is an instability point.

9 Conclusion. By a repeated use of the potential loads, we gave simple proofs of the usual conditions and the additional condition of stability, and then we formed Lyapunov functions (based on simple, linear combinations of potential loads) that satis ed the criteria of Malyshev and Menshikov. We took care of determining connected stability and instability domains, and only the frontier between these two domains was not explored. For the moment, there is no practical tool to determine which part of the frontier corresponds to a zone of ergodicity, null recurrence or transience of the model. We emphasize the original form of the stability domain : to the best of our knowledge, the few models whose stability was completely analyzed (regardless those which are stable under the usual conditions) exhibited linear, convex and monotonic conditions on  (the vector of trac intensities). Very little is known about general properties of the stability domain. In fact, it is not even obvious that the conditions of stability could always be expressed only in terms of the trac intensities, and not in terms of the primitive parameters  = (i)i2I and  = (is)(i;s)2C . However, trivially sucient conditions of stability (for example the sum of all the trac intensities being smaller than one) show that any network is stable when  is suciently close to zero. In [3], Chen proved a monotone property for the stability of uid networks: if a uid network with parameters (; ) is stable under all the work-conserving disciplines, then so is it with parameters ( 0; ),  0   . One can check that this property is satis ed by our stochastic network (whose service discipline is xed). If it were valid for all networks, this would at least show that the domains of stability or instability are always connected subsets of IRjCj +. Then, notice that one of the mains reasons why our model was tractable is that each zone of instability corresponds to a unique way to go to in nity: from any initial state, the dynamical system eventually joins the same type of diverging path (which was shown implicitely through Lyapunov functions), but it would have been much more dicult to apply the criterion of transience if there had been several modes of instability with separate attraction sets. This enlights the need for a \local" criterion of transience in terms of a \locally attractive" or (using the language of dynamical systems) a \stable" diverging path. At last, concerning uid limit models of networks represented by random walks in ZZN+ , we should be able to characterize them better if we understood more deeply how they are tied to the associated dynamical systems. Anyway, we made simulations of our network for trac intensities around the quadratic part of the frontier : on the unstable side, it seems that the model asymptotically follows the corresponding, diverging path of the dynamical system with probability one (on the stable side, the queue lengths keep \bounded" as expected), which corroborates the signi cance of the dynamical system.

References [1] Botvitch, D., and Zamyatin, A. Ergodicity of conservative communication networks. Rapport de recherche 1772, INRIA, Domaine de Voluceau, Rocquencourt B.P. 105 78153 Le Chesnay Cedex France, October 1992. 37

z=

z

y2 3y-1

1/2 4/9

z=1-y

0

1/3

1/2

2/3

Figure 24 : Cut of the three-dimensional stability domain at x = 0.

38

1

y

[2] Bramson, M. Instability of FIFO queueing networks. Preprint, 1993. [3] Chen, H. Fluid approximations and stability of multiclass queueing networks I: work-conserving disciplines. Preprint, 1993. [4] Dai, J. On the positive Harris recurrence for multiclass queueing networks: a uni ed approach via

uid limit models. Preprint, 1993. [5] Dai, J., and Weiss, G. Stability and instability of uid models for certain re-entrant lines. Preprint, February 1994. [6] Kumar, P., and Seidman, T. I. Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems. IEEE Transactions on Automatic Control 35, 3 (March 1990), 289{298. [7] Lu, S., and Kumar, P. Distributed scheduling based on due dates and buer priorities. IEEE Transactions on Automatic Control 36, 12 (December 1991), 1406{1416. [8] Malyshev, V. Networks and dynamical systems. Rapport de recherche 1468, INRIA, May 1991. [9] Meyn, S. Transience of multiclass queueing networks via uid limit models. Submitted, November 1994. [10] Rybko, A., and Stolyar, A. Ergodicity of stochastic processes describing the operation of open queueing networks. Problemy Peredachi Informatsii 28, 3 (July-September 1992), 3{26. [11] V.A.Malyshev, and M.V.Menshikov. Ergodicity, continuity, and analyticity of countable Markov chains. Trans. Moskow Math. Soc. 1 (1981), 1{47.

39