Results 1  10
of
10
On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach Via Fluid Limit Models
 Annals of Applied Probability
, 1995
"... It is now known that the usual traffic condition (the nominal load being less than one at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified ..."
Abstract

Cited by 223 (17 self)
 Add to MetaCart
It is now known that the usual traffic condition (the nominal load being less than one at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified approach to this problem. In this paper, we prove that a queueing network is positive Harris recurrent if the corresponding fluid limit model eventually reaches zero and stays there regardless of the initial system configuration. As an application of the result, we prove that single class networks, multiclass feedforward networks and firstbufferfirstserved preemptive resume discipline in a reentrant line are positive Harris recurrent under the usual traffic condition. AMS 1991 subject classification: Primary 60K25, 90B22; Secondary 60K20, 90B35. Key words and phrases: multiclass queueing networks, Harris positive recurrent, stability, fluid approximation Running title: Stability of mu...
Piecewise Linear Test Functions for Stability and Instability of Queueing Networks
 Queueing Systems
"... We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is ..."
Abstract

Cited by 36 (3 self)
 Add to MetaCart
We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is stable and hence that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a particular LP admits a solution, then the network model is transient. Running head : Stability and Instability of Queueing Networks Keywords : Multiclass queueing networks, ergodicity, stability, performance analysis. 1 Introduction It has generally been taken for granted in queueing theory that stability of a network is guaranteed so long as the overall traffic intensity is less than unity and in recent years there has been much analysis which supports this belief for special classes of systems, such as single class queueing networks (see Borovkov [2], Sig...
The Policy Iteration Algorithm for Average Reward Markov Decision Processes with General State Space
 IEEE Trans. Automat. Control
, 1997
"... The average cost optimal control problem is addressed for Markov decision processes with unbounded cost. It is found that the policy iteration algorithm generates a sequence of policies which are cregular (a strong stability condition) , where c is the cost function under consideration. This result ..."
Abstract

Cited by 22 (9 self)
 Add to MetaCart
The average cost optimal control problem is addressed for Markov decision processes with unbounded cost. It is found that the policy iteration algorithm generates a sequence of policies which are cregular (a strong stability condition) , where c is the cost function under consideration. This result only requires the existence of an initial cregular policy, and an irreducibility condition on the state space. Furthermore, under these conditions the sequence of relative value functions generated by the algorithm is bounded from below, and "nearly" decreasing, from which it follows that the algorithm is always convergent. Under further conditions, it is shown that the algorithm does compute a solution to the optimality equations, and hence an optimal average cost policy. These results provide elementary criteria for the existence of optimal policies for Markov decision processes with unbounded cost, and recover known results for the standard LQG problem. When these results are specialize...
On the stability of open networks: an unified approach by stochastic dominance
 QUEUEING SYSTEMS
, 1994
"... Using stochastic dominance, in this paper we provide a new characterization of point processes. This characterization leads to a unified proof for various stability results of open Jackson networks where service times are i.i.d. with a general distribution, external interarrival times are i.i.d. wit ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
Using stochastic dominance, in this paper we provide a new characterization of point processes. This characterization leads to a unified proof for various stability results of open Jackson networks where service times are i.i.d. with a general distribution, external interarrival times are i.i.d. with a general distribution and the routing is Bernoulli. We show that if the traffic condition is satisfied, i.e., the input rate is smaller than the service rate at each queue, then the queue length process (the number of customers at each queue) is tight. Under the traffic condition, the p th moment of the queue length process is bounded for all t if the p+1 th moment of the service times at all queues are nite. If, furthermore, the moment generating functions of the service times at all queues exist, then all the moments of the queue length process are bounded for all t. When the interarrival times are unbounded and nonlattice (resp. spreadout), the queue lengths and the remaining service times converge in distribution (resp. in total variation) to a steady state. Also, the moments converge if the corresponding moment conditions are satisfied.
On the Stability of Greedy Polling Systems With General Service Policies
, 1998
"... We consider a polling system with a finite number of stations fed by compound Poisson arrival streams of customers asking for service. A server travels through the system. Upon arrival at a nonempty station i, say, with x > 0 waiting customers, the server tries to serve there a random number B of c ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
We consider a polling system with a finite number of stations fed by compound Poisson arrival streams of customers asking for service. A server travels through the system. Upon arrival at a nonempty station i, say, with x > 0 waiting customers, the server tries to serve there a random number B of customers if the queue length has not reached a random level C < x before the server has completed the B services. The random variable B may also take the value # so that the server has to provide service as long as the queue length has reached size C . The distribution H i,x of the pair (B, C) may depend on i and x while the service time distribution is allowed to depend on i. The station to be visited next is chosen among some neighbors according to a greedy policy. That is to say that the server always tries to walk to the fullest station in his welldefined neighborhood. Under appropriate independence assumptions two conditions are established which are sufficient for stability and su...
Stationary Ergodic Jackson Networks: Results and CounterExamples
, 1996
"... This paper gives a survey of recent results on generalized Jackson networks, where classical exponential or i.i.d. assumptions on services and routings are replaced by stationary and ergodic assumptions. We first show that the most basic features of the network may exhibit unexpected behavior. Sever ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
This paper gives a survey of recent results on generalized Jackson networks, where classical exponential or i.i.d. assumptions on services and routings are replaced by stationary and ergodic assumptions. We first show that the most basic features of the network may exhibit unexpected behavior. Several probabilistic properties are then discussed, including a strong law of large numbers for the number of events in the stations, the existence, uniqueness and representation of stationary regimes for queue size and workload.
Cyclic Queueing Networks with Subexponential Service Times
 J. Appl. Probab
, 2004
"... For a Kstage cyclic queueing network with N customers and general service times we provide bounds on the n th departure time from each stage. Furthermore, we analyze the asymptotic tail behavior of cycle times and waiting times given that at least one service time distribution is subexponential. ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
For a Kstage cyclic queueing network with N customers and general service times we provide bounds on the n th departure time from each stage. Furthermore, we analyze the asymptotic tail behavior of cycle times and waiting times given that at least one service time distribution is subexponential.
STABILITY OF JACKSONTYPE QUEUEING NETWORKS, I
, 1999
"... This paper gives a pathwise construction of Jacksontype queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanis ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This paper gives a pathwise construction of Jacksontype queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanism are jointly stationary and ergodic in a sense that is made precise in the paper. The main tools for these results are the subadditive ergodic theorem, which is used to derive a strong law of large numbers, and basic theorems on monotone stochastic recursive sequences. The techniques which are proposed here apply to other and more general classes of discrete event systems, like Petri nets or GSMP’s. The paper also provides new results on the Jacksontype networks with i.i.d. driving sequences which were studied in the past.
Convergence Rates in Monotone Separable Stochastic Networks
"... We study bounds on the rate of convergence to the stationary distribution in monotone separable networks which are represented in terms of stochastic recursive sequences. Monotonicity properties of this subclass of Markov chains allow us to formulate conditions in terms of marginal network character ..."
Abstract
 Add to MetaCart
We study bounds on the rate of convergence to the stationary distribution in monotone separable networks which are represented in terms of stochastic recursive sequences. Monotonicity properties of this subclass of Markov chains allow us to formulate conditions in terms of marginal network characteristics. Two particular examples, generalized Jackson networks and multiserver queues, are considered.