162 research outputs found
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
Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime. Asymptotics of the stationary distribution
We consider a heterogeneous queueing system consisting of one large pool of
identical servers, where is the scaling parameter. The
arriving customers belong to one of several classes which determines the
service times in the distributional sense. The system is heavily loaded in the
Halfin-Whitt sense, namely the nominal utilization is where
is the spare capacity parameter. Our goal is to obtain bounds on the
steady state performance metrics such as the number of customers waiting in the
queue . While there is a rich literature on deriving process level
(transient) scaling limits for such systems, the results for steady state are
primarily limited to the single class case.
This paper is the first one to address the case of heterogeneity in the
steady state regime. Moreover, our results hold for any service policy which
does not admit server idling when there are customers waiting in the queue. We
assume that the interarrival and service times have exponential distribution,
and that customers of each class may abandon while waiting in the queue at a
certain rate (which may be zero). We obtain upper bounds of the form
on both and the number of idle servers. The bounds
are uniform w.r.t. parameter and the service policy. In particular, we show
that . Therefore, the
sequence is tight and has a uniform exponential tail
bound. We further consider the system with strictly positive abandonment rates,
and show that in this case every weak limit of
has a sub-Gaussian tail. Namely .Comment: 21 page
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
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