Consider a multiclass M/G/1 queue where queued customers are served in their order of arrival at a rate which depends on the customer class. We model this system using a chain with states represented by a tree. Since the service time distribution depends on the customer class, the stationary distribution is not of product form so there is no simple expression for the stationary distribution. Nevertheless, we can find a harmonic function on this chain which provides information about the asymptotics of this stationary distribution. The associated h-transformation produces a change of measure that increases the arrival rate of customers and decreases the departure rate thus making large deviations common. The Canadian Journal of Statistics 37: 327–346; 2009 © 2009 Statistical Society of Canada
Considerons une file d'attente M/G/1 multicategorie ou les consommateurs dans la file d'attente sont servis selon leur ordre d'arrivee a un taux dependant de leur categorie de consommateurs. Nous modelisons ce systeme en utilisant une chaine ou les etats sont representes a l'aide d'un arbre. Puisque la distribution du temps de service depend du type de consommateurs, la distribution stationnaire ne peut pas s'ecrire sous la forme d'un produit. Par consequent, il n'existe pas d'expression simple pour representer la distribution stationnaire. Cependant, nous pouvons obte-nir une transformation harmonique de cette chaine contenant de l'information sur les proprietes asymptotiques de cette distribution stationnaire. La transformation-h associee conduit a un chan-gement de mesure qui augmente le taux d'arrivee et decroit le taux de service ce qui augmente la probabilite de grandes deviations. La revue canadienne de statistique 37: 327–346; 2009 © 2009 Societe statistique du Canada
[1]
David McDonald.
Elements of applied probability for engineering, mathematics and systems science
,
2004
.
[2]
D. McDonald,et al.
Cell loss probability for M/G/1 and time-slotted queues
,
2000,
Journal of Applied Probability.
[3]
K. Mani Chandy,et al.
Open, Closed, and Mixed Networks of Queues with Different Classes of Customers
,
1975,
JACM.
[4]
Sung Ho Choi,et al.
On the M/G/1 Bernoulli feedback queue with multi-class customers
,
2000,
Comput. Oper. Res..
[5]
Harry Kesten,et al.
Renewal Theory for Functionals of a Markov Chain with General State Space
,
1974
.
[6]
S. Asmussen.
Equilibrium properties of the M/G/1 queue
,
1981
.
[7]
Tetsuya Takine,et al.
The M/G/1 FIFO Queue with Several Customer Classes
,
2003,
Queueing Syst. Theory Appl..
[8]
Tetsuya Takine,et al.
Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions
,
2001,
Queueing Syst. Theory Appl..
[9]
Rare events for stationary processes
,
2000
.
[10]
D. McDonald,et al.
Asymptotics of first passage times for random walk in an orthant
,
1999
.