92 research outputs found
On deciding stability of multiclass queueing networks under buffer priority scheduling policies
One of the basic properties of a queueing network is stability. Roughly
speaking, it is the property that the total number of jobs in the network
remains bounded as a function of time. One of the key questions related to the
stability issue is how to determine the exact conditions under which a given
queueing network operating under a given scheduling policy remains stable.
While there was much initial progress in addressing this question, most of the
results obtained were partial at best and so the complete characterization of
stable queueing networks is still lacking. In this paper, we resolve this open
problem, albeit in a somewhat unexpected way. We show that characterizing
stable queueing networks is an algorithmically undecidable problem for the case
of nonpreemptive static buffer priority scheduling policies and deterministic
interarrival and service times. Thus, no constructive characterization of
stable queueing networks operating under this class of policies is possible.
The result is established for queueing networks with finite and infinite buffer
sizes and possibly zero service times, although we conjecture that it also
holds in the case of models with only infinite buffers and nonzero service
times. Our approach extends an earlier related work [Math. Oper. Res. 27 (2002)
272--293] and uses the so-called counter machine device as a reduction tool.Comment: Published in at http://dx.doi.org/10.1214/09-AAP597 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Queueing systems with many servers: Null controllability in heavy traffic
A queueing model has heterogeneous service stations, each consisting
of many independent servers with identical capabilities. Customers of
classes can be served at these stations at different rates, that depend on both
the class and the station. A system administrator dynamically controls
scheduling and routing. We study this model in the central limit theorem (or
heavy traffic) regime proposed by Halfin and Whitt. We derive a diffusion model
on with a singular control term that describes the scaling
limit of the queueing model. The singular term may be used to constrain the
diffusion to lie in certain subsets of at all times . We
say that the diffusion is null-controllable if it can be constrained to
, the minimal closed subset of containing all
states of the prelimit queueing model for which all queues are empty. We give
sufficient conditions for null controllability of the diffusion. Under these
conditions we also show that an analogous, asymptotic result holds for the
queueing model, by constructing control policies under which, for any given
, all queues in the system are kept empty on the time
interval , with probability approaching one. This introduces a
new, unusual heavy traffic ``behavior'': On one hand, the system is critically
loaded, in the sense that an increase in any of the external arrival rates at
the ``fluid level'' results with an overloaded system. On the other hand, as
far as queue lengths are concerned, the system behaves as if it is underloaded.Comment: Published at http://dx.doi.org/10.1214/105051606000000358 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Recommended from our members
Staffing and Scheduling to Differentiate Service in Many-Server Service Systems
This dissertation contributes to the study of a queueing system with a single pool of multiple homogeneous servers to which multiple classes of customers arrive in independent streams. The objective is to devise appropriate staffing and scheduling policies to achieve specified class-dependent service levels expressed in terms of tail probability of delays. Here staffing and scheduling are concerned with specifying a time-varying number of servers and assigning newly idle servers to a waiting customer from one of K classes, respectively. For this purpose, we propose new staffing-and-scheduling solutions under the critically-loaded and overloaded regimes. In both cases, the proposed solutions are both time dependent (coping with the time variability in the arrival pattern) and state dependent (capturing the stochastic variability in service and arrival times). We prove heavy-traffic limit theorems to substantiate the effectiveness of our proposed staffing and scheduling policies. We also conduct computer simulation experiments to provide engineering confirmation and practical insight
A skill based parallel service system under FCFS-ALIS : steady state, overloads and abandonments
We consider a queueing system with servers S={m1,...,mJ}, and with customer types C={a,b,...}. A bipartite graph G describes which pairs of server-customer types are compatible. We consider FCFS-ALIS policy: A server always picks the first, longest waiting compatible customer, and a customer is always assigned to the longest idle compatible server. We assume Poisson arrivals and server dependent exponential service times. We derive an explicit product-form expression for the stationary distribution of this system when service capacity is sufficient. We also calculate fluid limits of the system under overload, to show that local steady state exists. We distinguish the case of complete resource pooling when all the customers are served at the same rate by the pooled servers, and the case when the system has a unique decomposition into subsets of customer types, each of which is served at its own rate by a pooled subset of the servers. Finally, we discuss possible behavior of the system with generally distributed abandonments, under many server scaling. This paper complements and extends previous results of Kaplan, Caldentey and Weiss [18], and of Whitt and Talreja [34], as well as previous results of the authors [4, 35] on this topic. Keywords: Service systems, multi type customers, multi type skill based servers, matching of infinite sequences, product form solution, first come first served policy, assign longest idle server policy, complete resource pooling, local steady state, overloaded queues, abandonment
A Switching Fluid Limit of a Stochastic Network Under a State-Space-Collapse Inducing Control with Chattering
Routing mechanisms for stochastic networks are often designed to produce
state space collapse (SSC) in a heavy-traffic limit, i.e., to confine the
limiting process to a lower-dimensional subset of its full state space. In a
fluid limit, a control producing asymptotic SSC corresponds to an ideal sliding
mode control that forces the fluid trajectories to a lower-dimensional sliding
manifold. Within deterministic dynamical systems theory, it is well known that
sliding-mode controls can cause the system to chatter back and forth along the
sliding manifold due to delays in activation of the control. For the prelimit
stochastic system, chattering implies fluid-scaled fluctuations that are larger
than typical stochastic fluctuations. In this paper we show that chattering can
occur in the fluid limit of a controlled stochastic network when inappropriate
control parameters are used. The model has two large service pools operating
under the fixed-queue-ratio with activation and release thresholds (FQR-ART)
overload control which we proposed in a recent paper. We now show that, if the
control parameters are not chosen properly, then delays in activating and
releasing the control can cause chattering with large oscillations in the fluid
limit. In turn, these fluid-scaled fluctuations lead to severe congestion, even
when the arrival rates are smaller than the potential total service rate in the
system, a phenomenon referred to as congestion collapse. We show that the fluid
limit can be a bi-stable switching system possessing a unique nontrivial
periodic equilibrium, in addition to a unique stationary point
EUROPEAN CONFERENCE ON QUEUEING THEORY 2016
International audienceThis booklet contains the proceedings of the second European Conference in Queueing Theory (ECQT) that was held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT, Toulouse, France. ECQT is a biannual event where scientists and technicians in queueing theory and related areas get together to promote research, encourage interaction and exchange ideas. The spirit of the conference is to be a queueing event organized from within Europe, but open to participants from all over the world. The technical program of the 2016 edition consisted of 112 presentations organized in 29 sessions covering all trends in queueing theory, including the development of the theory, methodology advances, computational aspects and applications. Another exciting feature of ECQT2016 was the institution of the TakĂĄcs Award for outstanding PhD thesis on "Queueing Theory and its Applications"
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