Laws of Little in an open queueing network

The object of this research in the queueing theory is theorems about the functional strong laws of large numbers (FSLLN) under the conditions of heavy traffic in an open queueing network (OQN). The FSLLN is known as a fluid limit or fluid approximation. In this paper, FSLLN are proved for the values of important probabilistic characteristics of the OQN investigated as well as the virtual waiting time of a customer and the queue length of customers. As applications of the proved theorems laws of Little in OQN are presented

Product-form solutions for integrated services packet networks and cloud computing systems

We iteratively derive the product-form solutions of stationary distributions of priority multiclass queueing networks with multi-sever stations. The networks are Markovian with exponential interarrival and service time distributions. These solutions can be used to conduct performance analysis or as comparison criteria for approximation and simulation studies of large scale networks with multi-processor shared-memory switches and cloud computing systems with parallel-server stations. Numerical comparisons with existing Brownian approximating model are provided to indicate the effectiveness of our algorithm.Comment: 26 pages, 3 figures, short conference version is reported at MICAI 200

A Hierarchical Approach to Robust Stability of Multiclass Queueing Networks

We re-visit the global - relative to control policies - stability of multiclass queueing networks. In these, as is known, it is generally insufficient that the nominal utilization at each server is below 100%. Certain policies, although work conserving, may destabilize a network that satisfies the nominal load conditions; additional conditions on the primitives are needed for global stability. The global-stability region was fully characterized for two-station networks in [13], but a general framework for networks with more than two stations remains elusive. In this paper, we offer progress on this front by considering a subset of non-idling control policies, namely queue-ratio (QR) policies. These include as special cases also all static-priority policies. With this restriction, we are able to introduce a complete framework that applies to networks of any size. Our framework breaks the analysis of QR-global stability into (i) global state-space collapse and (ii) global stability of the Skorohod problem (SP) representing the fluid workload. Sufficient conditions for both are specified in terms of simple optimization problems. We use these optimization problems to prove that the family of QR policies satisfies a weak form of convexity relative to policies. A direct implication of this convexity is that: if the SP is stable for all static-priority policies (the "extreme" QR policies), then it is also stable under any QR policy. While QR-global stability is weaker than global stability, our framework recovers necessary and sufficient conditions for global stability in specific networks

Sample path large deviations for multiclass feedforward queueing networks in critical loading

We consider multiclass feedforward queueing networks with first in first out and priority service disciplines at the nodes, and class dependent deterministic routing between nodes. The random behavior of the network is constructed from cumulative arrival and service time processes which are assumed to satisfy an appropriate sample path large deviation principle. We establish logarithmic asymptotics of large deviations for waiting time, idle time, queue length, departure and sojourn-time processes in critical loading. This transfers similar results from Puhalskii about single class queueing networks with feedback to multiclass feedforward queueing networks, and complements diffusion approximation results from Peterson. An example with renewal inter arrival and service time processes yields the rate function of a reflected Brownian motion. The model directly captures stationary situations.Comment: Published at http://dx.doi.org/10.1214/105051606000000439 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The BAR approach for multiclass queueing networks with SBP service policies

The basic adjoint relationship (BAR) approach is an analysis technique based on the stationary equation of a Markov process. This approach was introduced to study heavy-traffic, steady-state convergence of generalized Jackson networks in which each service station has a single job class. We extend it to multiclass queueing networks operating under static-buffer-priority (SBP) service disciplines. Our extension makes a connection with Palm distributions that allows one to attack a difficulty arising from queue-length truncation, which appears to be unavoidable in the multiclass setting. For multiclass queueing networks operating under SBP service disciplines, our BAR approach provides an alternative to the "interchange of limits" approach that has dominated the literature in the last twenty years. The BAR approach can produce sharp results and allows one to establish steady-state convergence under three additional conditions: stability, state space collapse (SSC) and a certain matrix being "tight." These three conditions do not appear to depend on the interarrival and service-time distributions beyond their means, and their verification can be studied as three separate modules. In particular, they can be studied in a simpler, continuous-time Markov chain setting when all distributions are exponential. As an example, these three conditions are shown to hold in reentrant lines operating under last-buffer-first-serve discipline. In a two-station, five-class reentrant line, under the heavy-traffic condition, the tight-matrix condition implies both the stability condition and the SSC condition. Whether such a relationship holds generally is an open problem

On base-stock policies for make-to-order/make-to-stock program

"April 1995."Includes bibliographical references (p. [29]-[30]).Vien Nguyen