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Jean Mairesse

Publications and source records attributed to Jean Mairesse.

32 records · Page 2Linked to original sources
arXiv2010

Deficiency Zero Petri Nets and Product Form

Consider a Markovian Petri net with race policy. The marking process has a "product form" stationary distribution if the probability of viewing a given marking can be decomposed as the product over places of terms depending only on the local marking. First we observe that the Deficiency Zero Theorem of Feinberg, developped for chemical reaction networks, provides a structural and simple sufficient condition for the existence of a product form. In view of this, we study the classical subclass of free-choice nets. Roughly, we show that the only such Petri nets having a product form are the state machines which can alternatively be viewed as Jackson networks.

cs.DM↗
arXiv2010

Stability of the bipartite matching model

We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan, and Weiss (Adv. Appl. Probab., 2009). Customers and servers play symmetrical roles. There is a finite set C resp. S, of customer, resp. server, classes. Time is discrete and at each time step, one customer and one server arrive in the system according to a joint probability measure on CxS, independently of the past. Also, at each time step, pairs of matched customer and server, if they exist, depart from the system. Authorized matchings are given by a fixed bipartite graph. A matching policy is chosen, which decides how to match when there are several possibilities. Customers/servers that cannot be matched are stored in a buffer. The evolution of the model can be described by a discrete time Markov chain. We study its stability under various admissible matching policies including: ML (Match the Longest), MS (Match the Shortest), FIFO (match the oldest), priorities. There exist natural necessary conditions for stability (independent of the matching policy) defining the maximal possible stability region. For some bipartite graphs, we prove that the stability region is indeed maximal for any admissible matching policy. For the ML policy, we prove that the stability region is maximal for any bipartite graph. For the MS and priority policies, we exhibit a bipartite graph with a non-maximal stability region.

cs.DM↗
arXiv2007

Series which are both max-plus and min-plus rational are unambiguous

Consider partial maps from the free monoid into the field of real numbers with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an unambiguous automaton from a max-plus automaton and a min-plus one that recognize the same series.

cs.DM↗
arXiv2007

Blocking a transition in a Free Choice net and what it tells about its throughput

In a live and bounded Free Choice Petri net, pick a non-conflicting transition. Then there exists a unique reachable marking in which no transition is enabled except the selected one. For a routed live and bounded Free Choice net, this property is true for any transition of the net. Consider now a live and bounded stochastic routed Free Choice net, and assume that the routings and the firing times are independent and identically distributed. Using the above results, we prove the existence of asymptotic firing throughputs for all transitions in the net. Furthermore the vector of the throughputs at the different transitions is explicitly computable up to a multiplicative constant.

cs.OH↗
arXiv2007

Queues, stores, and tableaux

Consider the single server queue with an infinite buffer and a FIFO discipline, either of type M/M/1 or Geom/Geom/1. Denote by A the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by D the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (D,r) has the same law as (A,s) which is an extension of the classical Burke Theorem. In fact, r can be viewed as the departures from a dual storage model. This duality between the two models also appears when studying the transient behavior of a tandem by means of the RSK algorithm: the first and last row of the resulting semi-standard Young tableau are respectively the last instant of departure in the queue and the total number of departures in the store.

cs.DM↗
arXiv2007

Services within a busy period of an M/M/1 queue and Dyck paths

We analyze the service times of customers in a stable M/M/1 queue in equilibrium depending on their position in a busy period. We give the law of the service of a customer at the beginning, at the end, or in the middle of the busy period. It enables as a by-product to prove that the process of instants of beginning of services is not Poisson. We then proceed to a more precise analysis. We consider a family of polynomial generating series associated with Dyck paths of length 2n and we show that they provide the correlation function of the successive services in a busy period with (n+1) customers.

cs.DM↗
arXiv2007

Products of irreducible random matrices in the (Max,+) Algebra

We consider the recursive equation ``x(n+1)=A(n)x(n)'' where x(n+1) and x(n) are column vectors of size k and where A(n) is an irreducible random matrix of size k x k. The matrix-vector multiplication in the (max,+) algebra is defined by (A(n)x(n))_i= max_j [ A(n)_{ij} +x(n)_j ]. This type of equation can be used to represent the evolution of Stochastic Event Graphs which include cyclic Jackson Networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence (A(n))_n is i.i.d or more generally stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices C such that P {A(0) in C} > 0, and the matrices in C have a unique periodic regime.

cs.OH↗
arXiv2007

Zero-automatic queues and product form

We introduce and study a new model: 0-automatic queues. Roughly, 0-automatic queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The salient result is that all stable 0-automatic queues have a product form stationary distribution and a Poisson output process. When considering the two simplest and extremal cases of 0-automatic queues, we recover the simple M/M/1 queue, and Gelenbe's G-queue with positive and negative customers.

cs.DM↗
arXiv2007

Randomly growing braid on three strands and the manta ray

Consider the braid group $B_3=< a,b| aba=bab>$ and the nearest neighbor random walk defined by a probability $ν$ with support $\{a,a^{-1},b,b^{-1}\}$. The rate of escape of the walk is explicitly expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitly describe the harmonic measure of the induced random walk on $B_3$ quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups.

math.PR↗
arXiv2007

Appendix to the paper "Randomly Growing Braid on Three Strands and the Manta Ray"

This paper is an appendix to the paper ``Randomly Growing Braid on Three Strands and the Manta Ray'' by J. Mairesse and F. Mathéus (to appear in the Annals of Applied Probability). It contains the details of some computations, and the proofs of some results concerning the examples treated there, as well as some extensions.

math.PR↗
arXiv2005

Random walks on free products of cyclic groups

Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitely solved, to get closed form formulas for the drift. The examples considered are the modular group Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy, and the volume of the group.

math.PR↗
arXiv2001

Computing the average parallelism in trace monoids

The {\em height} of a trace is the height of the corresponding heap of pieces in Viennot's representation, or equivalently the number of factors in its Cartier-Foata decomposition. Let $h(t)$ and $|t|$ stand respectively for the height and the length of a trace $t$. Roughly speaking, $|t|$ is the `sequential' execution time and $h(t)$ is the `parallel' execution time. We prove that the bivariate commutative series $\sum_t x^{h(t)}y^{|t|}$ is rational, and we give a finite representation of it. We use the rationality to obtain precise information on the asymptotics of the number of traces of a given height or length. Then, we study the average height of a trace for various probability distributions on traces. For the uniform probability distribution on traces of the same length (resp. of the same height), the asymptotic average height (resp. length) exists and is an algebraic number. To illustrate our results and methods, we consider a couple of examples: the free commutative monoid and the trace monoid whose independence graph is the ladder graph.

cs.DM↗