[Paper Review] Subexponential tail equivalence of the queue length distributions of BMAP/GI/1 queues with and without retrials

Hiroyuki Masuyama|arXiv (Cornell University)|Oct 17, 2013

Advanced Queuing Theory Analysis40 references7 citations

TL;DR

This paper establishes subexponential tail equivalence between the stationary queue length distributions of BMAP/GI/1 queues with and without retrials, using a novel stochastic-decomposition-like result and matrix analytic methods. The key contribution is proving that the tail behavior of the retrial queue matches that of the standard queue without retrials, even when batch sizes are heavy-tailed—extending prior results that required light-tailed batch arrivals.

ABSTRACT

The main contribution of this paper is to prove the subexponential tail equivalence of the stationary queue length distributions in the BMAP/GI/1 queues with and without retrials. We first present a stochastic-decomposition-like result of the stationary queue length in the BMAP/GI/1 retrial queue, which is an extension of the stochastic decomposition of the stationary queue length in the M${}^X$/GI/1 retrial queue. The stochastic-decomposition-like result shows that the stationary queue length distribution in the BMAP/GI/1 retrial queue is decomposed into two parts: the stationary conditional queue length distribution given that the server is idle; and a certain matrix sequence associated with the stationary queue length distribution in the corresponding standard BMAP/GI/1 queue (without retrials). Using the stochastic-decomposition-like result and matrix analytic methods, we prove the subexponential tail equivalence of the stationary queue length distributions in the BMAP/GI/1 queues with and without retrials. This tail equivalence result does not necessarily require that the size of an arriving batch is light-tailed, unlike Yamamuro's result for the M${}^X$/GI/1 retrial queue (Queueing Syst. 70:187--205, 2012). As a by-product, the key lemma to the roof of the main theorem presents a subexponential asymptotic formula for the stationary distribution of a level-dependent M/G/1-type Markov chain, which is the first reported result on the subexponential asymptotics of level-dependent block-structured Markov chains.

Motivation & Objective

• To establish subexponential tail equivalence between the stationary queue length distributions of BMAP/GI/1 queues with and without retrials.
• To extend the stochastic decomposition structure known for M^X/GI/1 retrial queues to the more general BMAP/GI/1 retrial model.
• To remove the light-tailed batch size assumption required in prior work, such as Yamamuro (2012), for subexponential tail equivalence.
• To derive a new asymptotic formula for the stationary distribution of level-dependent M/G/1-type Markov chains under subexponential tails.
• To provide a rigorous foundation for analyzing rare-event behavior in retrial queues with general batch arrivals and heavy-tailed service or interarrival times.

Proposed method

• Derives a stochastic-decomposition-like representation of the stationary queue length in the BMAP/GI/1 retrial queue, separating it into the idle-period queue length and a component linked to the standard BMAP/GI/1 queue.
• Applies matrix analytic methods to analyze the tail behavior of the queue length distribution in the retrial model.
• Uses a key lemma on subexponential asymptotics for level-dependent M/G/1-type Markov chains, which is the first such result for this class.
• Employs limit superior and inferior arguments on matrix sequences to bound tail probabilities, leveraging subexponential distribution properties.
• Establishes convergence of tail probabilities via inequalities involving matrix sequences and subexponential decay rates.
• Relies on convolution properties of matrix sequences with subexponential tails, formalized through Propositions B.1 and B.2 in the appendix.

Experimental results

Research questions

• RQ1 Does the stationary queue length distribution in the BMAP/GI/1 retrial queue exhibit the same subexponential tail behavior as in the corresponding standard BMAP/GI/1 queue?
• RQ2 Can the stochastic decomposition result for M^X/GI/1 retrial queues be generalized to BMAP/GI/1 retrial queues?
• RQ3 Is subexponential tail equivalence preserved when the batch size distribution is heavy-tailed, rather than light-tailed?
• RQ4 What is the asymptotic behavior of the stationary distribution for level-dependent M/G/1-type Markov chains with subexponential tails?
• RQ5 Can matrix analytic techniques be used to derive subexponential asymptotics for general retrial queueing systems with batch Markovian arrivals?

Key findings

• The stationary queue length distribution in the BMAP/GI/1 retrial queue is subexponentially equivalent to that in the standard BMAP/GI/1 queue, i.e., P(L^{(μ)}>x) ~ P(L^{(∞)}>x) as x→∞.
• The stochastic-decomposition-like result decomposes the retrial queue length into the idle-period queue length and a component tied to the standard queue, enabling tail analysis.
• The key lemma provides the first known subexponential asymptotic formula for the stationary distribution of level-dependent M/G/1-type Markov chains.
• The tail equivalence result holds without requiring the batch size distribution to be light-tailed, unlike Yamamuro’s (2012) result for M^X/GI/1 retrial queues.
• Propositions B.1 and B.2 establish asymptotic convolution rules for matrix sequences with subexponential tails, enabling precise tail probability bounds.
• The proof technique relies on matrix inequalities and limit superior/inferior analysis to control the tail decay rate, confirming equivalence under general conditions.