Several Types of Convergence Rates of the M/G/1 Queueing System

We study the workload process of the M/G/1 queueing system. Firstly, we give the explicit criteria for the geometric rate of convergence and the geometric decay of stationary tail. And the parameters ε 0 and s 0 for the geometric rate of convergence and the geometric decay of the stationary tail are obtained, respectively. Then, we give the explicit criteria for the rate of convergence and decay of stationary tail for three specific types of subgeometric cases. Andwe give the parameters ε 1 and s 1 of the rate of convergence and the decay of the stationary tail, respectively, for the subgeometric rate r(n) = exp(sn), s > 0, α > 0.


Introduction
We consider several types of convergence rates of the M/G/1 queueing system by using drift conditions. The M/G/1 queueing system discussed here is that the arrivals form a Poisson process with parameter . The service times 1 , 2 , . . . for the customers are independently identically distributed random variables with a common distribution function ( ).
where > 0 is a constant, and is called the service intensity. Denote the workload process of the M/G/1 queueing system by ( ); then, { ( ), ≥ 0} is a Markov process. Ergodicity, specially ordinary ergodicity, has been well studied for Markov processes. There are a large volume of references devoted to the geometric case (or exponential case) and the subgeometric case (e.g., see [1][2][3]). Hou and Liu [4,5] discussed ergodicity of embedded M/G/1 and GI/M/n queues, polynomial and geometric ergodicity for M/G/1-type Markov chain, and processes by generating function of the first return probability. Hou and Li [6,7] obtained the explicit necessary and sufficient conditions for polynomial ergodicity and geometric ergodicity for the class of quasi-birth-anddeath processes by using matrix geometric solutions.
There is much work on decay of the tail in the stationary distribution. Li and Zhao [8,9] studied heavy-tailed asymptotic and light-tailed asymptotic of stationary probability vectors of Markov chains of GI/G/1 type. Jarner and Roberts [10] discussed Foster-Lyapounov-type drift conditions for Markov chains which imply polynomial rate convergence to stationarity in appropriate V-norms. Jarner and Tweedie [11] proved that the geometric decay of the tail in the stationary distribution is a necessary condition for the geometricergodicity for random walk-type Markov chains. We will discuss several types of ergodicity and the tail asymptotic behavior of the stationary distribution by Foster-Lyapounovdrift conditions. We give the relationship of ergodicity and the decay of the tail in the stationary distribution for ℎ-skeleton chain in M/G/1 queueing system, which is different from the former; ergodicity and the decay of the tail are discussed, respectively. We shall give the bounded interval in which geometric and subexponential parameter lies and prove that it is determined by the tail of the service distribution. The parameters 0 and 0 for geometric rate of convergence and the geometric decay of the stationary tail are obtained, respectively. We shall also give explicit criteria for the rate of convergence and decay of stationary tail for three specific types of subgeometric cases (Case 1: the rate function ( ) = exp( 1/(1+ ) ), > 0, > 0; Case 2: polynomial rate function 2 Discrete Dynamics in Nature and Society ( ) = , > 0; Case 3: logarithmic rate function ( ) = log , > 0). And we give the parameters 1 and 1 of the rate of convergence and the decay of the stationary tail, respectively, for the subgeometric rate in Case 1.
We organize the paper as follows. In Section 2, we shall introduce basic definitions and theorems, including the main result, Theorem 6. In Section 3, we shall prove the geometric rates of convergence in Theorem 6. In Section 4, we shall prove the rates of convergence for the subgeometric Cases 1-3 in Theorem 6.

Basic Definitions and the Main Results
Let { , ≥ 0} be a discrete time Markov chain on the state space ( , E) with transition kernel . Assume that it is -irreducible, aperiodic, and positive recurrent. Now, we discuss the convergence in -norm of the iterates of the kernel to the stationary distribution at rate := ( ( ), ≥ 0); that is, for all ∈ E, where : → [1, +∞) satisfies ( ) < +∞, and for all signed measures , the -norm || || is defined as sup | |≤ | ( )|.
where is the indicator function of the set . Now we shall give Theorems 1 and 2 which we will use in this paper.
Theorem 2 (Proposition 2.5 in Douc et al. [12]). Let be airreducible and aperiodic kernel. Assume that ( , , ) holds for function with lim → +∞ ( ) = 0, a petite set , and a function with { < +∞} ̸ = 0. Then, there exists an invariant probability measure , and for all in the full and absorbing set { < ∞}, where ( ( )) = ∘ −1 ( ), ( ) : Since is a concave monotone nondecreasing differentiable function, is nonincreasing. Then, there exists ∈ [0, 1), such that lim → +∞ ( ) = . In Theorem 2, for the case ∈ (0, 1), condition ( , , ) implies that the chain is geometric ergodic, but the rate in the geometric convergence property cannot be achieved under the condition that lim → +∞ ( ) = > 0. The workload process { ( ), ≥ 0} of the M/G/1 queueing system is a Markov process on the state space Suppose that the workload can be decreased by min{1, } during the time interval [ , + 1]. And suppose that the transition kernel of { } is ( , ⋅). For convenience, let where is the number of arrivals in a time interval of unit length.

Lemma 3. { } is irreducible and aperiodic.
Proof. Let be a measure on For all ∈ + , there exists a satisfying − 1 < ≤ , such that we know that { } is also aperiodic.
Proof. Let [ ] be the maximum integer no more than . Since and is a closed set, we know that min ∈ [ ]+1 ( , {0}) > 0 . Let 2 be a measure on + satisfying, for all ∈ B( + ), Obviously, for all ∈ , Thus, we get that is a petite set.
For two sequences and , we write ≍ , if there exist positive constants 1 and 2 such that, for large , 1 ≤ ≤ 2 . Let us say that the distribution function of a random variable is in the distribution function of a random variable is in where > 0, and > 0. Now, we give the main result.
Theorem 6. Suppose that < 1 and is the stationary distribution of { }.

(4) If there exists an integer number
> 0, such that We shall prove Theorem 6 in Sections 3 and 4.
where the second inequality holds by using the condi- For all ∈ , Now, we prove that there exists an 1 > 0 such that ( ) > 0 for all ∈ (0, 1 ). Similar to the proof of the case ∈ , we know that ( ) is a finite function for ∈ [0, ). Furthermore, ] > 0.
Case 2 (Polynomial Rate of Convergence). Consider the following.

Conclusion and Future Research.
We studied the M/G/1 queueing system, and the waiting time process of the queueing system is a Markov process. For the workload process of the M/G/1 queueing system, we got an ℎ-skeleton process and discussed its properties of the irreducible and aperiodic and the property of stochastic monotone. Then, we got the parameters 0 and 0 for geometric rate of convergence and the geometric decay of the stationary tail, respectively. For three specific types of subgeometric cases: Case 1: the rate function ( ) = exp( 1/(1+ ) ), > 0, > 0; Case 2: polynomial rate function ( ) = , > 0; Case 3: logarithmic rate function ( ) = log , > 0, we gave explicit criteria for the rate of convergence and decay of stationary tail. We gave the parameters 1 and 1 of the rate of convergence and the decay of the stationary tail, respectively, for the subgeometric rate ( ) = exp( 1/(1+ ) ), > 0, > 0. These results are important in the study of the stability of M/G/1 queueing system.
For future research, much could be done. Our work could be used to the convergence analysis of Markov chain Monte Carlo (MCMC) theory. It could also be used to further discuss queue length, congestion, and so forth. Using similar techniques, these results may be extended to storage models, nonlinear autoregressive model, stochastic unit root models, multidimensional random walk, and other queueing systems.