47 research outputs found
Large fork-join queues with nearly deterministic arrival and service times
In this paper, we study an server fork-join queue with nearly
deterministic arrival and service times. Specifically, we present a fluid limit
for the maximum queue length as . This fluid limit depends on the
initial number of tasks. In order to prove these results, we develop extreme
value theory and diffusion approximations for the queue lengths.Comment: 36 pages, 15 figure
Optimization of Inventory and Capacity in Large-Scale Assembly Systems Using Extreme-Value Theory
High-tech systems are typically produced in two stages: (1) production of components using specialized equipment and staff and (2) system assembly/integration. Component production capacity is subject to fluctuations, causing a high risk of shortages of at least one component, which results in costly delays. Companies hedge this risk by strategic investments in excess production capacity and in buffer inventories of components. To optimize these, it is crucial to characterize the relation between component shortage risk and capacity and inventory investments. We suppose that component production capacity and produce demand are normally distributed over finite time intervals, and we accordingly model the production system as a symmetric fork-join queueing network with N statistically identical queues with a common arrival process and independent service processes. Assuming a symmetric cost structure, we subsequently apply extreme value theory to gain analytic insights into this optimization problem. We derive several new results for this queueing network, notably that the scaled maximum of N steady-state queue lengths converges in distribution to a Gaussian random variable. These results translate into asymptotically optimal methods to dimension the system. Tests on a range of problems reveal that these methods typically work well for systems of moderate size
Large fork-join queues with nearly deterministic arrival and service times
In this paper, we study an N server fork-join queue with nearly deterministic arrival and service times. Specifically, we present a fluid limit for the maximum queue length as N → ∞. This fluid limit depends on the initial number of tasks. In order to prove these results, we develop extreme value theory and diffusion approximations for the queue lengths
Maximum waiting time in heavy-tailed fork-join queues
In this paper, we study the maximum waiting time
in an -server fork-join queue with heavy-tailed services as .
The service times are the product of two random variables. One random variable
has a regularly varying tail probability and is the same among all servers,
and one random variable is Weibull distributed and is independent and
identically distributed among all servers. This setup has the physical
interpretation that if a job has a large size, then all the subtasks have large
sizes, with some variability described by the Weibull-distributed part. We
prove that after a temporal and spatial scaling, the maximum waiting time
process converges in to the supremum of an extremal process with
negative drift. The temporal and spatial scaling are of order
, where is the shape
parameter in the regularly varying distribution, is a slowly
varying function, and is a sequence for which holds that
, as ,
where are i.i.d.\ Weibull-distributed random variables. Finally, we prove
steady-state convergence
Tail asymptotics for the delay in a Brownian fork-join queue
We study the tail behavior of maxi≤Nsups>0Wi(s)+WA(s)−βs as N→∞, with (Wi,i≤N) i.i.d. Brownian motions and WA an independent Brownian motion. This random variable can be seen as the maximum of N mutually dependent Brownian queues, which in turn can be interpreted as the backlog in a Brownian fork-join queue. In previous work, we have shown that this random variable centers around [Formula presented]logN. Here, we analyze the rare event that this random variable reaches the value ([Formula presented]+a)logN, with a>0. It turns out that its probability behaves roughly as a power law with N, where the exponent depends on a. However, there are three regimes, around a critical point a⋆; namely, 0a⋆. The latter regime exhibits a form of asymptotic independence, while the first regime reveals highly irregular behavior with a clear dependence structure among the N suprema, with a nontrivial transition at a=a⋆