Busy period, time of the first loss of a customer and the number of customers in M^\varkappa|G^δ|1|B

2011/03/21 by Tetyana Kadankova, Kadankova, Tetyana, V. F. Kadankov +5
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math.PR

paper · pdf · doi:10.48550/arxiv.1103.4064

arxiv created 2011/03/21 · openalex publication_date 2011/03/21 · arxiv updated 2011/03/22 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28

Abstract

A two-sided exit problem is solved for a difference of a compound Poisson process and a compound renewal process. More precisely, the Laplace transforms of the joint distribution of the first exit time, the value of the overshoot and the value of a linear component at this instant are found. Further, we study the process reflected in its supremum. We determine the main two-boundary characteristics of the process reflected in its supremum. These results are then applied for studying the M^\varkappa|G^δ|1|B system. We derive the distribution of a busy period and the numbers of customers in the system in transient and stationary regimes. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process.

Related

• Exit problems for oscillating compound Poisson process
• On Busy Periods of the Critical GI/G/1 Queue and BRAVO
• Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes
• Renewal Processes Represented as Doubly Stochastic Poisson Processes
• Law of two-sided exit by a spectrally positive strictly stable process