Generalizing the Pollaczek-Khintchine Formula to Account for Arbitrary Work Removal

Abstract

Recently, a Pollaczek-Khintchine-like formulation for M/G/l queues with disasters has been obtained. A disaster is said to occur if a negative arrival causes all the customers (and therefore work) to depart from the system immediately. This study generalizes this result further, as it is shown to hold even when negative arrivals cause only part of the work to be demolished. In other words, an arbitrary amount of work, following a known distribution, is allowed to be removed at a negative event. Under these circumstances, a general approach for obtaining the Pollaczek–Khintchine Formula is proposed, which is then illustrated via several examples. Typically, it is seen that the formulainvolves certain parameters that are not explicitly known. The formula itself is made possible due to the number in system being geometric under preemptive last in-first out discipline.

Fraud risk assessment within blockchain transactions

“…where ξ (S z−1 ) = S(σ S z−1 ) − S z−1 denotes the overshoot, immediately after ruin, in model (21). The following Cramér-Lundberg upper bound holds:…”

“…Assumption (A2) implies that the process {e γ S * (n) , n ≥ 1} is a martingale as a consequence of [ →' stands for convergence almost surely, follows from assumption (A1) and the law of large numbers. Let the initial reserves S z−1 = s ≥ 0 be fixed in (21). The application of [2, Proposition 3.1] allows us to rewrite the ultimate ruin probability as…”

“…where ξ (u) = S * (σ s ) − s denotes the overshoot given that ruin occurred in model (21). Thanks to the connection (23), by conditioning on the values of S z−1 , it holds that…”

“…[9], [15], [18], [20], and [24]. To the best of my knowledge, the closest link to queueing theory is the single-server queue with either work or customer removal introduced by Gelenbe et al [13] and further considered in [5], [19], and [21] for different queues. The related risk process includes lump addition; see [6].…”

A Note on Negative Customers, GI/G/1 Workload, and Risk Processes

“…Consider the M/G/1 queue with additional removal of work as studied by Boucherie and Boxma [2] and Jain and Sigman [8]. Customers arrive according to a Poisson process with rate f..+.…”