Exact Tail Asymptotics for a Queueing System with a Retrial Orbit and Batch Service

This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts serving, it serves all customers in the queue in a single batch, which is the so-called batch service. If a new customer or a retrial customer finds all the customers’ rooms are occupied, he will decide whether or not to join the retrial orbit. By using the censoring technique and the matrix analysis method, we first obtain the decay function of the stationary distribution for the quantity of customers in the retrial orbit and the quantity of customers in the queue. Then based on the form of decay rate function and the Karamata Tauberian theorem, we finally get the exact tail asymptotics of the stationary distribution.

Exact Tail Asymptotics for a Discrete-time Preemptive Priority Queue

In this paper, we consider a discrete-time preemptive priority queue with different service com- pletion probabilities for two classes of customers, one with high-priority and the other with low-priority. This model corresponds to the classical preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server. Due to the possibility of customers＇ arriving and departing at the same time in a discrete-time queue, the model considered in this paper is more complicated than the continuous- time model. In this model, we focus on the characterization of the exact tail asymptotics for the joint stationary distribution of the queue length of the two types of customers, for the two boundary distributions and for the two marginal distributions, respectively. By using generating functions and the kernel method, we get the exact tail asymptotic properties along the direction of the low-priority queue, as well as along the direction of the high-priority queue.

Stationary distributions for two-dimensional sticky Brownian motions:Exact tail asymptotics and extreme value distributions

Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions,which find applications in many areas including queueing theory and mathematical finance.In this paper,we focus on stationary distributions for sticky Brownian motions.Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions.The kernel method,copula concept and extreme value theory are the main tools used in our analysis.

Tail Asymptotics of Two Parallel Queues with Transfers of Customers

This paper studies a system consisting of two parallel queues with transfers of customers.In the system,one queue is called main queue and the other one is called auxiliary queue.The main queue is monitored at exponential time instances.At a monitoring instant,if the number of customers in main queue reaches L(>K),a batch of L−K customers is transferred from the main queue to the auxiliary queue,and if the number of customers in main queue is less than or equal to K,the transfers will not happen.For this system,by using a Foster-Lyapunov type condition,we establish a sufficient stability condition.Then,we provide a sufficient condition under which,for any fixed number of customers in the auxiliary queue,the stationary probability of the number of customers in the main queue has an exact geometric tail asymptotic as the number of customers in main queue increases to infinity.Finally,we give some numerical results to illustrate the impact of some critical model parameters on the decay rate.

Extremes of Shepp statistics for fractional Brownian motion

Define the incremental fractional Brownian field with parameter H ∈ （0, 1） by ZH（τ, s） = BH（s-＋τ） - BH（S）, where BH（s） is a fractional Brownian motion with Hurst parameter H ∈ （0, 1）. We firstly derive the exact tail asymptoties for the maximum MH＊（T） = max（τ,s）∈[a,b]×[0,T] ZH（τ, s）/τH of the standardised fractional Brownian motion field, with any fixed 0 〈 a 〈 b 〈 ∞ and T 〉 0; and we, furthermore, extend the obtained result to the ease that T is a positive random variable independent of {BH（s）, s ≥ 0}. As a by-product, we obtain the Gumbel limit law for MH＊r（T） as T →∞.