1,088 research outputs found
Stochastic decomposition in discrete-time queues with generalized vacations and applications
For several specific queueing models with a vacation policy, the stationary system occupancy at the beginning of a rantdom slot is distributed as the sum of two independent random variables. One of these variables is the stationary number of customers in an equivalent queueing system with no vacations. For models in continuous time with Poissonian arrivals, this result is well-known, and referred to as stochastic decomposition, with proof provided by Fuhrmann and Cooper. For models in discrete time, this result received less attention, with no proof available to date. In this paper, we first establish a proof of the decomposition result in discrete time. When compared to the proof in continuous time, conditions for the proof in discrete time are somewhat more general. Second, we explore four different examples: non-preemptive proirity systems, slot-bound priority systems, polling systems, and fiber delay line (FDL) buffer systems. The first two examples are known results from literature that are given here as an illustration. The third is a new example, and the last one (FDL buffer systems) shows new results. It is shown that in some cases the queueing analysis can be considerably simplified using this property
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
Content Based Status Updates
Consider a stream of status updates generated by a source, where each update
is of one of two types: high priority or ordinary (low priority). These updates
are to be transmitted through a network to a monitor. However, the transmission
policy of each packet depends on the type of stream it belongs to. For the low
priority stream, we analyze and compare the performances of two transmission
schemes: (i) Ordinary updates are served in a First-Come-First-Served (FCFS)
fashion, whereas, in (ii), the ordinary updates are transmitted according to an
M/G/1/1 with preemption policy. In both schemes, high priority updates are
transmitted according to an M/G/1/1 with preemption policy and receive
preferential treatment. An arriving priority update discards and replaces any
currently-in-service high priority update, and preempts (with eventual resume
for scheme (i)) any ordinary update. We model the arrival processes of the two
kinds of updates, in both schemes, as independent Poisson processes. For scheme
(i), we find the arrival and service rates under which the system is stable and
give closed-form expressions for average peak age and a lower bound on the
average age of the ordinary stream. For scheme (ii), we derive closed-form
expressions for the average age and average peak age of the high priority and
low priority streams. We finally show that, if the service time is
exponentially distributed, the M/M/1/1 with preemption policy leads to an
average age of the low priority stream higher than the one achieved using the
FCFS scheme. Therefore, the M/M//1/1 with preemption policy, when applied on
the low priority stream of updates and in the presence of a higher priority
scheme, is not anymore the optimal transmission policy from an age point of
view
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