G-networks with synchronized partial ushing. PRi SM, Universite de Versailles, 45 av. des Etats Unis, Versailles Cedex,France

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1 G-networks with synchronized partial ushing Jean-Michel FOURNEAU ;a, Dominique VERCH ERE a;b a PRi SM, Universite de Versailles, 45 av. des Etats Unis, Versailles Cedex,France b CERMSEM, Universite Pantheon-Sorbonne, 90 rue de Tolbiac, Paris, France Abstract We study generalized networks of queues with a new type of signals which deletes customers in an ordered circular list of queues. By solving the balance equations, we prove that the steady-state distribution of the queue sizes has a product form. This model could be used to analyse assembly line systems and polling mechanisms. Keywords : Queueing networks; Product form solution; Synchronised transitions. 1 Introduction We present a new type of generalized network of queues (G-networks for short) with a steadystate product form solution. G-networks have been dened by Gelenbe in [4] and have received since this seminal paper a lot of attention (see for instance [7,, 9, 11]). G-networks consist of queues, ordinary customers and signals. Signals interact with the queues at their arrival but they are never queued and they do not receive service. At the completion of their service, a customer may join another queue as a signal. Thus G-networks exhibit much more general synchronized transitions than the Jackson networks. A lot of interactions due to signal have been studied and shown to preserve a product form solution. The rst eect, studied by Gelenbe, was the destruction of a customer [4]. Then, in [6] the product form was generalized to networks where a signal trigger a customer movement to a third queue. Several extensions have been studied; some of them are related to this work and are mentioned here. Signals which implies a batch destruction have been independently studied by Gelenbe [5] and Henderson & al. [10]. If the batch size is innite, a signal ushes out, with probability 1, the queue it enters. This eect has been studied in [1]. A generalization to networks with multiple classes of customer, one type of signal and Processor-Sharing discipline appeared in [1]. In this paper, we present a new eect of signal such that the steady-state distribution still has a product form. The signals try to ush every queue in an ordered list, deleting one customer at a time for all the queues. And it stops the deletion as soon as it visits an empty queue. This eect is denoted as a synchronized partial ushing. The paper is organized as follows. In the next section we present the model and prove the product form solution in section. Examples and links to related works are presented in section 4. Corresponding author 1

2 Markovian network with synchronized partial ushing The model consists of an open G-network with N single server queues a single class of customers and two types of signals. At its arrival to an empty queue, a signal of type 1 or will have no eect and will just disappear. The behavior of type 1 signal is usual. At its arrival in an non-empty queue i f1; ; ; N g, it triggers a customer movement from queue i to some other queue j f1; ; ; N g with probability Q ij. Type signals are associated to synchronized partial ushing which are formally described in the following. The set of queues is partitioned into A subsets. It is assumed that the queues in each subset are ordered into a rooted directed cycle denoted as C a where a is the index of the subset. These rooted directed cycles are the fundamental structures used to describe the eect of the new type of signal we study. The root of C a is a queue whose index is denoted as c(a; 1). The length of the cycle (or the size of the subset) is denoted as L a. The queues in C a are numbered from c(a; 1) (i.e. the root) to c(a; L a ) according to the ordering of the cycle. The structure of a rooted directed cycle is used to emphasis that type signals will enter at the root and will loop inside the directed cycle. If the subset is a singleton, then we consider that the cycle has length 1 and consists of a directed loop on the queue. The external arrivals to the queues of this G-network follow independent Poisson processes. The external arrivals of customers to queue i is denoted by i and? i for signals of type 1. Without loss of generality we assume that there no external arrivals of type signals. Customers at queue i request a negative exponentially distributed service time with mean 1 i. Once emitted from queue i, a customer will be transferred as a customer to queue j f1; ; ; N g with probability P? i;j, as a type 1 signal to another queue j f1; ; ; N g with probability P, will i;j leave the system with probability d i or will join as a type signal the root of a subset of queues C a with probability R i;a. These transition probabilities satisfy : P i;j N X P? i;j A X a=1 R i;a d i = 1 and Q i;j = 1 8 i (1) Note that the partition of the set of queues is only used to describe the eect of type signals. The matrices P and P?, which model the routing of customers and type 1 signals do not necessarily satisfy the same constraints. Let ~n(t) be the state vector of the G-network at time t, and ~n = (n 1 ; n ; ; n N ) be one value of queue lengths. Let p(~n) denote the stationary probability distribution i.e. p(~n) = lim t!1 P r (~n (t) = ~n). Whenever a signal of type arrives at queue c(a; 1), it deletes a customer in this queue and it goes immediately to the next queue in cycle C a, says c(a; ). If queue c(a; ) contains some customers then it deletes a customer in c(a; ) and goes to the next queue in C a. The signal loops inside the cycle C a until it reaches an empty queue where it vanishes. All these deletions and movements are instantaneous. Roughly, a type signal deletes the same number of customers in all the queues of the cycle. And this number is the size of the smallest queue in the cycle C a. More precisely, the number of deleted customers are not always equal, because the signal visits the directed cycle beginning with the root. Consider an arbitrary directed cycle C a, and let be the smallest index of queue in C a such that : n (t) = min ica n i (t). Assume that a type signal enters the root of cycle C a at time t, then at time t we have : 8 >< >: 1 i < n i (t ) = n i (t)? n (t)? 1 n (t ) = 0 n i (t ) = n i (t)? n (t) < i L a ()

3 Queue is ushed while the other queues are only partially emptied. These deletions are synchronized. It may be worthy to remark that if the partition of the set of queues consists only of singletons, then type signals have exactly the same eect as the one considered by Chao in [1]. With more general partitions, the synchronized transitions are much more complex. However, we prove in the next section that these G-networks still have product form solution. The equilibrium distribution First, under the assumptions we made, ~n(t) is clearly a Markov chain. Assume that the steadystate distribution exists, the following theorem establishes that it has a product form if a xed point system satises the stability constraints. Let i be an arbitrary queue, in the following we denote by b(i) the index of the ordered cycle of queues which contains queue i. As we have made a partition of the set of queues in cycles, there is one and only one cycle which contains queue i. Let f(i) be the function which gives the index of queue i in the cycle b(i). Theorem 1 Consider the following trac equations : i = i P N j j P j;i P N (? j P N k=1 k k P? k;j ) jq j;i i? i P N j j P? j;i P N j j R j;b(i) 1? b(i) () where : 8>< >: 8a; a = YL a `=1 8i if f(i) 6= 1 = 8i if f(i) = 1 = 1 c(a;`) f (i)?1 Y `=1 c(b(i);`) If there exists a positive vector solution = ( 1 ; ; ; N ) to the system such that i < 1 then the network has a steady-state distribution with product form solution : p(~n) = NY (1? i ) n i i (4) Proof : The state space is S = f~n : n i IN 8ig and the global balance equations relating ux into and out of each state ~n of S are : P N p(~n)? P i i? i 1lfni>0g = N p(~n ~e i) i d P i N p(~n? ~e i) 1l i fn P i>0g N P N p(~n ~e j?? ~e i ) j P j;i? j Q ji P 1lfni>0g N P N P N k=1 p(~n ~e k ~e j? ~e i ) k P? kj Q ji1l P fni>0g N p(~n ~e j) j P? 1l j;i fn i=0g P N P N P N P 1 r=0 p ~n ~e j r P L b(i) `=1 ~e c(b(i);`) P f(i)?1 `=1 ~e c(b(i);`) j R j;b(i) 1l fni=0g As usual, we denote by ~e i the vector having all its components equal to zero except the i th one, which is equal to one and 1l f:g the characteristic function. First we divide by p(~n) and use the solution (4) to simplify the ratios and we group the terms with the same characteristic functions : i ( i? )1l i fn i >0g = i i d i

4 1l fni >0g 4 i i r=0 j j P? j;i 1l fn i =0g 1X j j P j;i N X 0 j j R j;b(i) 4@ L b(i) Y `=1 r c(b(i);`1 A? j Q j;i f (i)?1 Y `=1 k=1 k k P? k;j jq j;i 5 c(b(i);` 5 1lfni =0g If for all queue i the load i is smaller than 1, then we have a < 1 for all cycle C a. Therefore P 1 r=0 (QL b(i) `=1 c(b(i);`) r is well dened and is equal to 1=(1? b(i) ). We substitute b(i) and in the equations and we remark that 1l fni =0g = 1? 1l fni >0g. Thus after some permutation of terms, we get : i 4 i? i i i d i 1l fni >0g i 4 i j j P? j;i N X j j P? j;i N X j j P j;i N X j j R j;b(i) j j R j;b(i) k=1 5 1lfni >0g = k k P? k;j jq j;i? j jq j;i 5 Using denition of i, after substitution, all the characteristic functions disappear to obtain a ow equation (see next Lemma). And the proof of the lemma will conclude the proof of the theorem. Lemma 1 The following ow equation is consistent with the denition of i. i = i i d i j j P? j;i N X j j R j;b(i) (5) Proof : First consider i dened by equation, multiply both sides of equation by the denominator and sum up for all queues i : i i i j j P j;i N X? i i? j jq j;i j j R j;b(i) i k=1 k k j P? k;j Q j;i = j j i P? j;i Remember that for all i, we have : P N Q i;j = 1 and P N P i;j P N P? i;j P A a=1 R i;ad i = 1. Thus after substitution and cancellation, we obtain : i = i i d i j j P? j;i N X AX a=1 j j R j;a j j R j;b(i) i (6) 4

5 First, we slightly change the index in the last term of the RHS : Then, as for all cycle C a, we have : = 1 X ic a AX X a=1 ic a? a 1? a P? a ic i = a? a (7) 1? a 1? a we nd again equation (5). Therefore, this equation is a ow equation and is consistent with the denition of i. This concludes the proof of the lemma and the theorem. It remains to prove that indeed equation (5) have positive solutions which satisfy the stationarity constraints. It is quite clear now that G-networks exhibits non linear ow equations. In [7], Gelenbe and Shassberger have presented a method to prove that, under some assumptions, these ow equations have a positive solution. But the stationarity constraints (i.e. i < 1) have to be veried after numerical resolution. Applying this method to G-networks with ushing is not straightforward because of some continuity arguments. Furthermore, we conjecture that the stationarity constraints are always satised for G-networks with ushing if the routing matrices implies that type signals may enter all queues. To conclude, the existence of positive solutions to ow equation which satisfy stationarity constraints is a dicult problem which will be investigated in a forthcoming paper. 4 Examples and related works 4.1 A small example Consider the following example, one of the smallest with non trivial synchronized transitions. The network consists of 4 queues (depicted in gure 1). To focus on type signals, we assume that there is no type 1 signals in the network : 8i? i = 0, matrix P? is a null matrix, matrix Q is not dened and matrix P is an arbitrary irreducible sub-stochastic matrix. The queues are divided into a two subsets partition : ff1; ; g; f4gg. The roots of cycle C 1 and C are respectively queue 1 and 4. In gure 1 the cycles are depicted as plain black lines. Type signals are represented by dotted lines joining queue 1 and queue 4, while the routing of customers is omitted to simplify the presentation. Note that a queue may send a type signal to the root of a cycle of which it belongs. To put more emphasis on the non linearity of the ow equations, we write them with more details. 8>< 1 = 1 P 4 j j P j;1 1 P 4 j j R j;1 1 1? 1 = = P 4 j j P j; P 4 j j R j;1 P 4 j j P j; P 4 j j R j;1 1 1? 1 1 1? 1 (8) >: 4 = 4 P 4 j j P j;4 4 P 4 j j R j; 1 1? 4 5

6 1 4 cycle signal Figure 1: A small network with partial synchronized ushing on queues Such a model may represent a production system and more specically an assembly subsystem where items queued in several lines are packed together to produce only one item. It is assumed that one item from every queue in a cycle is necessary to produce one object. The production is delaied until an order arrives (i.e. a signal in our model). The order asks to produce as much as possible and the production takes place until an empty queue stops the production process. Others application include the join operation in parallel systems where tasks in several queues wait to leave together a set of queues. One could also remark that type signals poll the cycle of queues and take advantage of this remark to model systems based on polling mechanisms. 4. Model with synchronized batch deletion In [14], Henderson, Northcote and Taylor have introduced a model of G-networks with batch arrivals and services such that any event occurring in the network can trigger a batch of customers to be routed through the network. They proved, that under some assumptions, their networks have product form solution. The key idea is that the service rate and the probability of trigger success can be described with the same set of functions. More precisely, they have to verify the following assumptions : When the system is in state ~n, a positive vector ~a is released with rate q(~n; ~a). Component a i of ~a is the number of customers served in queue i. The queue labeled 0 represents the outside of the G-network. The customers that compose vector ~a attempt to trigger the service completion of the customers distributed in vector ~a 0, if they succeed, the customers of ~a 0 join the queues according to vector ~a 00 with the probability P (~a; ~a 0 ; ~a 00 ). Denote A to be the set of all possible vectors ~a, ~a 0 and ~a 00 which are transferred in the G-network. It is assumed that q(~n; ~a) has the form : q(~n; ~a) = (~n? ~a)(~a) (~n) (9) where functions, and are non-negative and dened on IZ N. Furthermore, functions is positive and functions must satisfy : 8~n S 8~a A (~n) (~n? ~a) (10) 6

7 When ~a is served from state ~n, with probability P (~a; ~a 0 ; ~a 00 ) it succeed to trigger the vector ~a 0 with state dependent probability (~n?~a?~a0 ) ; thus state ~n? ~a? ~a 0 ~a 00 is reached. With (~n?~a) probability 1? (~n?~a?~a0 ), the trigger fails and and state ~n? ~a is reached. (~n?~a) Consider the trac equations for this class of G-network for all ~a 6= ~ 0 : X X (~a)f(~a) = ~a 0 A ~a 00 A (~a 0 )f(~a 0 ) f(~a 00 )P (~a 0 ; ~a 00 ; ~a)? f(~a)p (~a 0 ; ~a; ~a 00 ) (11) If the solution function f() of the equation (11) is geometric (i.e. f(~n) = Q N n i i with i < 1) then the G-network has a product form steady-state distribution : p(~n) = (~n)f(~n) 8~n S. The function related to the service in the G-network may put some restriction on the state space S. Clearly, the behavior of G-networks with synchronized partial ushing may be captured by Henderson's model, even if such a complex example is not presented. However, to prove product form, one have to exhibit functions, and in accordance with the dynamic of our model. This is quite complex. First, remark that the set of transfer vectors A is innite and the number of customers that constitute a batch vector is state-dependent. To nd the solutions ff(~a) : ~a A g of \trac equations", we have to express the sets of served/released, triggered and deposited vectors. On state space IZ N without truncations, these service functions are arbitrary positive. On a truncated state space such as IN N, the service functions are much more complex. Furthermore, to determine completely the functions f(), we have to express all solutions i for 1 i N. But solving the trac equations (11) in Henderson's model has roughly the same complexity than solving the global balance equation for our model. Despite our model belongs to the class of G-networks with batches dened in [14], building the various functions concerned with their model is as dicult as substitution of the solution in Chapman-Kolmogorov equation. 5 Conclusion G-network were originally designed to model neural networks []. Signals and customers respectively represent inhibitory and excitatory signals in models of neural networks, while queue lengths represent the neurons input potentials. Recently, new applications of these networks have been proposed in the eld of reliability of performability (see [1], [8] for instance). In these models the deletion capabilities of signals is used to model breakdowns which cause the loss of some customers (or even all the customers) in a queue. It may be worthy to remark that in [1] a G-network with triggers in a loop is used to approximate a network of queues with ushing. Clearly, our result allows to model complex joint departures processes and G-networks with partial synchronized ushing may provide a new approximation for assembly lines, parallel systems or for some new polling mechanisms. We also expect that it will possible to generalize this result to more complex synchronized transitions. References [1] X. Chao. A queueing network model with catastrophes and product form solution. Operations Research Letters, 18:75{79,

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