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

Abstract

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 𝜀₀ and 𝑠₀ 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. And we give the parameters 𝜀₁ and 𝑠₁ of the rate of convergence and the decay of the stationary tail, respectively, for the subgeometric rate 𝑟(𝑛)=exp(𝑠𝑛^1/(1+𝛼)), 𝑠>0, 𝛼>0.

1. 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 𝜆. The service times 𝜈₁,𝜈₂,… for the customers are independently identically distributed random variables with a common distribution function 𝐵(𝑥). Let 1𝜇≡^∞₀(𝑥)𝜆𝑥𝑑𝐵,𝜌≡𝜇, (1) where 𝜇>0 is a constant, and 𝜌 is called the service intensity. Denote the workload process of the M/G/1 queueing system by 𝑊(𝑡); then, {𝑊(𝑡),𝑡≥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 [1–3]). 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-and-death 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 geometric-ergodicity for random walk-type Markov chains. We will discuss several types of ergodicity and the tail asymptotic behavior of the stationary distribution by Foster-Lyapounov- drift 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 𝑠 lies and prove that it is determined by the tail of the service distribution. The parameters 𝜀₀ and 𝑠₀ 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 𝑟(𝑛)=exp(𝑠𝑛^1/(1+𝛼)),𝛼>0,𝑠>0; Case 2: polynomial rate function 𝑟(𝑛)=𝑛^𝛼,𝛼>0; Case 3: logarithmic rate function 𝑟(𝑛)=log^𝛼𝑛,𝛼>0). And we give the parameters 𝜀₁ and 𝑠₁ 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.

2. Basic Definitions and the Main Results

Let {𝑋_𝑛,𝑛≥0} be a discrete time Markov chain on the state space (𝐸,ℰ) with transition kernel 𝑃. Assume that it is 𝜓-irreducible, aperiodic, and positive recurrent. Now, we discuss the convergence in 𝑓-norm of the iterates 𝑃^𝑛 of the kernel to the stationary distribution 𝜋 at rate 𝑟∶=(𝑟(𝑛),𝑛≥0); that is, for all 𝐴∈ℰ, lim_𝑛→∞𝑟(𝑛)𝑃^(𝑛)(𝑥,𝐴)−𝜋(𝐴)_𝑓=0,𝜋-𝑎.𝑒., (2) where 𝑓∶𝐸→[1,+∞) satisfies 𝜋(𝑓)<+∞, and for all signed measures 𝜎, the 𝑓-norm ||𝜎||_𝑓 is defined as sup_{|𝑔|≤𝑓}|𝜎(𝑔)|.

Geometric Rate Function. That is, the function 𝑟 satisfies (𝑛)0<liminflog𝑟𝑛(𝑛)≤limsuplog𝑟𝑛<+∞. (3)

Subgeometric Rate Function. That is, the function 𝑟 satisfies lim_𝑛→∞log𝑟(𝑛)𝑛=0. (4) The class of subgeometric rates function includes polynomial rates functions; that is, 𝑟(𝑛)=𝑛^𝛼,𝛼>0, and rate functions which increase faster than the polynomial ones; 𝑟(𝑛)=exp(𝑠𝑛^1/(1+𝛼)),𝛼>0,𝑠>0.

We shall discuss geometric rates of convergence 𝑟(𝑛)=exp(𝑠𝑛),𝑠>0, subgeometric rate of convergence 𝑟(𝑛)=exp(𝑠𝑛^1/(1+𝛼)),𝛼>0,𝑠>0, polynomial rate of convergence 𝑟(𝑛)=𝑛^𝛼,𝛼>0, and logarithmic rate of convergence 𝑟(𝑛)=log^𝛼𝑛,𝛼>0.

Condition 𝐷(𝜙,𝑉,𝐶). There exist a function 𝑉∶𝐸→[1,∞), a concave monotone nondecreasing differentiable function 𝜙∶[1,∞]→(0,∞], a measurable set 𝐶, and a finite constant 𝑏 such that Δ𝑉(𝑥)=𝑃𝑉(𝑥)−𝑉(𝑥)≤𝜙∘𝑉+𝑏𝐼_𝐶, (5) where 𝐼_𝐶 is the indicator function of the set 𝐶.

Now we shall give Theorems 1 and 2 which we will use in this paper.

Theorem 1 (Theorem  14.0.1 in [1]). If 𝐷(𝜙,𝑉,𝐶) holds for some petite set 𝐶 and there exists 𝑥₀∈𝐸 such that 𝑉(𝑥₀)<∞, then there exists a unique invariant distribution 𝜋, 𝜋(𝜙∘𝑉)<∞ and lim_𝑛→∞𝑃^(𝑛)(𝑥,𝐴)−𝜋(𝐴)_𝜙∘𝑉=0,𝜋-𝑎.𝑒., (6) where 𝜙∘𝑉(𝑥)≥1,𝑥∈𝐸.

Theorem 2 (Proposition 2.5 in Douc et al. [12]). Let 𝑃 be a 𝜓-irreducible and aperiodic kernel. Assume that 𝐷(𝜙,𝑉,𝐶) holds for function 𝜙 with lim_𝑡→+∞𝜙^′(𝑡)=0, a petite set 𝐶, and a function 𝑉 with {𝑉<+∞}≠∅. Then, there exists an invariant probability measure 𝜋, and for all 𝑥 in the full and absorbing set {𝑉<∞}, lim_𝑛→∞𝑟_𝜙(𝑛)𝑃^(𝑛)(𝑥,𝐴)−𝜋(𝐴)=0, (7) where (𝑟_𝜙(𝑛))=𝜙∘𝐻_𝜙⁻¹(𝑛), 𝐻_𝜙(𝑣)∶=∫^𝑣₁(1/𝜙(𝑥))𝑑𝑥.

Since 𝜙 is a concave monotone nondecreasing differentiable function, 𝜙^′ is nonincreasing. Then, there exists 𝑐∈[0,1), such that lim_𝑡→+∞𝜙^′(𝑡)=𝑐. In Theorem 2, for the case 𝑐∈(0,1), condition 𝐷(𝜙,𝑉,𝐶) implies that the chain is geometric ergodic, but the rate in the geometric convergence property cannot be achieved under the condition that lim_𝑡→+∞𝜙^′(𝑡)=𝑐>0.

The workload process {𝑊(𝑡),𝑡≥0} of the M/G/1 queueing system is a Markov process on the state space {𝑅₊,ℬ(𝑅₊)}. {𝑊(𝑛ℎ)} is an ℎ-skeleton of {𝑊(𝑡),𝑡≥0}. We choose ℎ=1, and denote {𝑊(𝑛ℎ)}_ℎ=1 by {𝑋_𝑛}. Suppose that the workload can be decreased by min{1,𝑋_𝑛} during the time interval [𝑛,𝑛+1]. And suppose that the transition kernel of {𝑋_𝑛} is 𝑃(𝑥,⋅). For convenience, let 𝜎_𝑘=𝜈₁+𝜈₂+⋯+𝜈_𝑘. Then, 𝑋_𝑛+1=^+∞_𝑘=1𝐼_{𝜉=𝑘}𝜎_𝑘+min𝑋_𝑛−1,0,𝑛=1,2,…, (8) where 𝜉 is the number of arrivals in a time interval of unit length.

Lemma 3. {𝑋_𝑛} is irreducible and aperiodic.

Proof. Let 𝜑 be a measure on 𝑅₊ with 𝜑({0})=1,𝜑({0}^𝑐)=0. For all 𝑥∈𝑅₊, there exists a 𝑘 satisfying 𝑘−1<𝑡≤𝑘, such that 𝑃^𝑘(𝑥,{0})≥exp(−𝜆𝑘)>0. (9) Hence, {𝑋_𝑛} is irreducible. From 𝑃^𝑛(𝑥,{0})>0,𝑛≥1, (10) we know that {𝑋_𝑛} is also aperiodic.

Lemma 4. 𝐶=[0,𝑐] is petite set, where 𝑐(𝑐≥0) is a real number.

Proof. Let [𝑐] be the maximum integer no more than 𝑐. Since 𝑃^[𝑐]+1(𝑥,{0})≥exp{−𝜆([𝑐]+1)}>0,∀𝑥∈𝐶, (11) and 𝐶 is a closed set, we know that min_𝑥∈𝐶𝑃^[𝑐]+1(𝑥,{0})>0. Let 𝜈₂ be a measure on 𝑅₊ satisfying, for all 𝐵∈ℬ(𝑅₊), 𝜈₂(𝐵)=0,{0}∉𝐵,min_𝑥∈𝐶𝑃^[𝑐]+1(𝑥,{0}),{0}∈𝐵. (12) Obviously, for all 𝑥∈𝐶, 𝑃^[𝑐]+1(𝑥,𝐵)≥𝜈₂(𝐵),∀ℬ𝑅₊. (13) Thus, we get that 𝐶 is a petite set.

Lemma 5. The Markov chain {𝑋_𝑛} is stochastically monotonic.

Proof. For every fixed 𝑦, from 𝑃𝑋_𝑛+1≤𝑦∣𝑋_𝑛=𝑥=𝑃^+∞_𝑘=1𝐼_{𝜉=𝑘}𝜎_𝑘[],𝑃𝑋≤𝑦,∀𝑥∈0,1_𝑛+1≤𝑦∣𝑋_𝑛=𝑥=𝑃^+∞_𝑘=1𝐼_{𝜉=𝑘}𝜎_𝑘+𝑥−1≤𝑦,=𝑃^+∞_𝑘=1𝐼_{𝜉=𝑘}𝜎_𝑘−𝑦−1≤−𝑥,∀𝑥∈[1,+∞), (14) we obtain that 𝑃{𝑋_𝑛+1≤𝑦∣𝑋_𝑛=𝑥} is nonincreasing in 𝑥. That is, {𝑋_𝑛} is stochastically monotonic.

For two sequences 𝑢_𝑛 and 𝑣_𝑛, we write 𝑢_𝑛≍𝑣_𝑛, if there exist positive constants 𝑐₁ and 𝑐₂ such that, for large 𝑛, 𝑐₁𝑢_𝑛≤𝑣_𝑛≤𝑐₂𝑢_𝑛.

Let us say that the distribution function 𝐺 of a random variable 𝜉 is in 𝒢⁺(𝑟) if 𝐸𝑒^𝑠𝜉=₀^+∞𝑒^𝑠𝑥𝐺(𝑑𝑥)<+∞,0<𝑠≤𝑟; (15) the distribution function 𝐺 of a random variable 𝜉 is in 𝒮⁺(𝑟,𝛼) if 𝐸𝑒^{𝑠𝜉1/(1+𝛼)}=₀^+∞𝑒^{𝑠𝑥1/(1+𝛼)}𝐺(𝑑𝑥)<+∞,0<𝑠≤𝑟, (16) where 𝑟>0, and 𝛼>0.

Now, we give the main result.

Theorem 6. Suppose that 𝜌<1 and 𝜋 is the stationary distribution of {𝑋_𝑛}.
(1) If 𝐵∈𝒢⁺(𝑟), then one has exp(₁^+∞𝑠𝑥)𝜋(𝑑𝑥)<+∞,∀0<𝑠<𝑠₀, (17) where 𝑠₀ is the minimum positive root of the equation 𝐸𝑒^𝑠𝜈1=1+(𝑠/𝜆). Moreover, {𝑋_𝑛} is geometrically ergodic, lim_𝑛→∞𝑒^𝜀0𝑛𝑃^(𝑛)(𝑥,𝐴)−𝜋(𝐴)=0, (18) where 𝜀₀=𝜆+𝑠−𝜆𝐸𝑒^𝑠𝜈1, and 𝑠∈(0,𝑠₀) is a root of the equation 𝐸𝜈₁𝑒^𝑠𝜈1=1/𝜆.
(2) If 𝐵∈𝒮⁺(𝑟,𝛼), then one has ₁^+∞𝑥^{−𝛼/(1+𝛼)}exp𝑠𝑥^1/(1+𝛼)𝜋(𝑑𝑥)<∞,0<𝑠<𝑠₁, (19) where 𝑠₁ is the minimal positive solution of 𝛽(𝑠)=0(𝛽(𝑠)=𝑥^{𝛼/(1+𝛼)}{1−∑^∞_𝑘=0(𝜆^𝑘𝑒^−𝜆/𝑘!)𝐸exp{𝑠(𝜎_𝑘−1+𝑥)^1/(1+𝛼)−𝑠𝑥^1/(1+𝛼)}}). And lim_𝑛→∞𝑛^{(−𝛼/(1+𝛼))}exp𝜀₁𝑛^1/(1+𝛼)𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0, (20) where 𝜀₁=max_{𝑠∈(0,𝑠1)}[(𝛼+1)𝛽(𝑠)]^1/(𝛼+1).
(3) If there exists a constant 𝛼>1, such that 𝐸𝜈^𝛼₁=∫₀^+∞𝑥^𝛼𝐵(𝑑𝑥)<+∞, then lim_𝑛→∞𝑛^𝛼−1𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0,_1+∞𝑥^𝛼−1𝜋(𝑑𝑥)<+∞. (21)
(4) If there exists an integer number 𝛼>0, such that 𝐸𝜈₁log^𝛼(𝜈₁+1)=∫₀^+∞𝑥log^𝛼(𝑥+1)𝐵(𝑑𝑥)<+∞, then lim_𝑛→∞log^𝛼𝑛𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0,_1+∞log^𝛼𝑥𝜋(𝑑𝑥)<+∞. (22)

We shall prove Theorem 6 in Sections 3 and 4.

3. Geometric Rate of Convergence

The Markov chain {𝑋_𝑛} is geometrically ergodic if (2) holds with 𝑟(𝑛)=𝑒^𝑠𝑛 for some 𝑠>0. By Theorem  15.0.1 in [1], an equivalent condition of geometric ergodicity is that there exist a petite set 𝐶, constants 𝛽>0 and 𝑏<∞, and a function 𝑉≥1 finite for at least one 𝑥₀∈𝐸 satisfying Δ𝑉(𝑥)<−𝛽𝑉(𝑥)+𝑏𝐼_𝐶,𝑥∈𝐸. (23) By using the drift previous condition, we usually obtain the geometric ergodicity, but we could not get the parameters for the geometric rate of convergence. Now, we will study the geometric decay of the stationary tail and geometric rate of convergence to the stationary distribution.

Let 𝑉_𝑠(𝑥)=exp(𝑠𝑥),0<𝑠≤𝑟,𝑥∈𝑅₊. Taking the petite set 𝐶=[0,1], for all 𝑥∈[0,1], Δ𝑉_𝑠(𝑥)=𝑃𝑉_𝑠(𝑥)−𝑉_𝑠(𝑥)<^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑠𝜈𝑘!𝐸exp₁+⋯+𝜈_𝑘≤+𝑥_∞𝑘=0𝜆^𝑘𝑘!𝐸𝑒^{𝑠𝜈1𝑘}𝑒^𝑥=exp𝜆𝐸𝑒^𝑠𝜈1𝑒^𝑥. (24) Since 𝐵∈𝒢⁺(𝑟)(i.e., 𝐸𝑒^𝑠𝜈1<∞,0<𝑠≤𝑟), we know that Δ𝑉_𝑠(𝑥)<exp𝜆𝐸𝑒^𝑠𝜈1𝑒^𝑥<∞,∀𝑥∈[0,1]. (25) For all 𝑥>1 (i.e., 𝑥∈𝐶^𝐶), Δ𝑉_𝑠(𝑥)=𝑃𝑉_𝑠(𝑥)−𝑉_𝑠(𝑥)=−exp(𝑠𝑥)+exp(𝑠𝑥)^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑠𝜎𝑘!𝐸exp_𝑘(()−1=−exp𝑠𝑥)1−exp−𝜆−𝑠^∞_𝑘=0𝜆^𝑘𝑘!𝐸𝑒^{𝑠𝜈1𝑘}=−exp(𝑠𝑥)1−exp−𝜆−𝑠+𝜆𝐸𝑒^𝑠𝜈1. (26) Let 𝛽(𝑠)=1−exp(−𝜆−𝑠+𝜆𝐸𝑒^𝑠𝜈1). Now, we prove that there exists an 𝑠^′>0 such that 𝛽(𝑠^′)>0. By the stated condition 𝐵∈𝒢⁺(𝑟), we know that 𝛽(𝑠) is a finite differentiable function for 𝑠∈[0,𝑟). Furthermore, 𝛽(0)=1−exp−𝜆−𝑠+𝜆𝐸𝑒^𝑠𝜈1_𝑠=0𝛽=0,_′(𝑠)∣_𝑠=0=−−1+𝜆𝐸𝜈₁𝑒^𝑠𝜈1exp−𝜆−𝑠+𝜆𝐸𝑒^𝑠𝜈1_𝑠=0=1−𝜌>0. (27)

Proposition 7. Suppose that 𝜌<1 and 𝜋 is the stationary distribution of {𝑋_𝑛}. If 𝐵∈𝒢⁺(𝑟); then, exp(₁^+∞𝑠𝑥)𝜋(𝑑𝑥)<+∞,∀0<𝑠<𝑠₀, (28) where 𝑠₀ is the minimum positive root of the equation 𝐸𝑒^𝑠𝜈1=1+(𝑠/𝜆).

Proof. By (27), we know that 𝛽(0)=0 and 𝛽^′(0)>0. So, there exists an 𝑠^′∈(0,𝑟) such that 𝛽(𝑠^′)>0. The function 𝛽(𝑠) is continuous in the interval [𝑠^′,𝑟], and it is easy to see that 𝛽(𝑟)<0,𝛽(𝑠^′)>0. By the zero theorem, we know that there exists at least one root of the equation 𝛽(𝑠)=0 (i.e., 𝐸𝑒^𝑠𝜈1=1+(𝑠/𝜆)). Let 𝑠₀ be the minimum positive root; then, 𝛽(𝑠)>0, for all 0<𝑠<𝑠₀.
Let 𝑏=sup_{𝑠∈(0,𝑠0)}exp(𝜆𝐸𝑒^𝑠𝜈1)<∞; then, we have Δ𝑉_𝑠(𝑥)≤−𝛽(𝑠)𝑉_𝑠(𝑥)+𝑏𝐼_[0,1]=−𝜙∘𝑉_𝑠(𝑥)+𝑏𝐼_[0,1],0<𝑠<𝑠₀,𝑥∈𝑅₊, (29) where 𝜙(𝑥)=𝛽(𝑠)𝑥 (i.e., condition 𝐷(𝜙,𝑉_𝑠,𝐶) holds). By Theorem 1, we know that 𝜋(𝜙∘𝑉_𝑠)<∞; that is, exp(₁^+∞𝑠𝑥)𝜋(𝑑𝑥)<∞,0<𝑠<𝑠₀. (30)

Proposition 8. Suppose that 𝜌<1 and 𝜋 is the stationary distribution of {𝑋_𝑛}. If 𝐵∈𝒢⁺(𝑟); then, lim_𝑛→∞𝑒^𝜀0𝑛𝑃^(𝑛)(𝑥,𝐴)−𝜋(𝐴)=0, (31) where 𝜀₀=𝛼(𝑠), 𝛼(𝑠)=𝜆+𝑠−𝜆𝐸𝑒^𝑠𝜈1, and 𝑠∈(0,𝑠₀) is the root of the equation 1−𝜆𝐸𝜈₁𝑒^𝑠𝜈1=0.

Proof. From (29), Δ𝑉_𝑠(𝑥)≤−𝛽(𝑠)𝑉_𝑠(𝑥)+𝑏𝐼_[0,1],0<𝑠<𝑠₀,𝑥∈𝑅, (32) where 𝛽(𝑠)=1−exp(−𝜆−𝑠+𝜆𝐸𝑒^𝑠𝜈1), and 𝑠₀ is the minimum positive root of the equation 𝐸𝑒^𝑠𝜈1=1+(𝑠/𝜆). We have 𝑃𝑉_𝑠(𝑥)≤exp−𝜆−𝑠+𝜆𝐸𝑒^𝑠𝜈1𝑉_𝑠(𝑥)+𝑏𝐼_[0,1],0<𝑠<𝑠₀,𝑥∈𝑅₊. (33) From Lemma 5, we know that {𝑋_𝑛} is a stochastically monotonic Markov chain. By using Theorem  1.1 in [13], we have lim_𝑛→∞exp𝜆+𝑠−𝜆𝐸𝑒^𝑠𝜈1𝑛𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0,0<𝑠<𝑠₀. (34) Let 𝛼(𝑠)=𝜆+𝑠−𝜆𝐸𝑒^𝑠𝜈1. From 𝛼^′′(𝑠)=−𝜆𝐸𝜈²₁𝑒^𝑠𝜈1<0, we know that 𝛼(𝑥) is a concave function. Together with 𝛼(0)=0,𝛼(𝑠₀)=0, there exists a unique point 𝑠∈(0,𝑠₀), such that 𝛼^′(𝑠)=1−𝜆𝐸𝜈₁𝑒^𝑠𝜈1=0, and 𝛼(𝑠) has a maximum 𝛼(𝑠) at the point 𝑠 in the interval (0,𝑠₀). So, lim_𝑛→∞𝑒^𝜀0𝑛𝑃^(𝑛)(𝑥,𝐴)−𝜋(𝐴)=0, (35) where 𝜀₀=𝛼(𝑠). The proof is completed.

4. Subgeometric Rates of Convergence for Cases 1–3

Case 1 (The Rate Function 𝑟(𝑛)=exp(𝑠𝑛^1/(1+𝛼))). The rate function 𝑟(𝑛)=exp(𝑠𝑛^1/(1+𝛼)), which increases to infinity faster than the polynomial one, and slower than the geometrical one, has been discussed only recently in the literature.

Proposition 9. Suppose that 𝜌<1 and 𝜋 is the stationary distribution of {𝑋_𝑛}. If 𝐵∈𝒮⁺(𝑟,𝛼), then one has ₁^+∞𝑥^{−𝛼/(1+𝛼)}exp𝑠𝑥^1/(1+𝛼)𝜋(𝑑𝑥)<∞,0<𝑠<𝑠₁, (36) where 𝑠₁ is the minimal positive solution of 𝛽(𝑠)=0  (𝛽(𝑠)=𝑥^{𝛼/(1+𝛼)}{1−∑^∞_𝑘=0(𝜆^𝑘𝑒^−𝜆/𝑘!)𝐸exp{𝑠(𝜎_𝑘−1+𝑥)^1/(1+𝛼)−𝑠𝑥^1/(1+𝛼)}}). And lim_𝑛→∞𝑛^{−𝛼/(1+𝛼)}exp𝜀₁𝑛^1/(1+𝛼)𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0, (37) where 𝜀₁=max_{𝑠∈(0,𝑠1)}[(𝛼+1)𝛽(𝑠)]^(1/(1+𝛼)).

Proof. Let 𝑉_𝑠(𝑥)=exp(𝑠𝑥^1/(1+𝛼)),0<𝑠≤𝑟,𝑥∈𝑅₊. For all 𝑥∈𝐶, Δ𝑉_𝑠(𝑥)=𝑃𝑉_𝑠(𝑥)−𝑉_𝑠(𝑥)≤−exp𝑠𝑥^1/(1+𝛼)+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑠𝜎𝑘!𝐸exp_𝑘+𝑥^1/(1+𝛼)≤−exp𝑠𝑥^1/(1+𝛼)+exp𝑠𝑥^1/(1+𝛼)⋅^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑠𝜈𝑘!𝐸exp₁^1/(1+𝛼)+𝜈₂^1/(1+𝛼)+⋯+𝜈_𝑘^1/(1+𝛼)=−exp𝑠𝑥^1/(1+𝛼)+exp𝑠𝑥^1/(1+𝛼)⋅^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑘!𝐸exp𝑠𝜈₁^{1/(1+𝛼)𝑘}=−exp𝑠𝑥^1/(1+𝛼)+exp𝑠𝑥^1/(1+𝛼)exp𝜆𝐸exp𝑠𝜈₁^1/(1+𝛼), (38) where the second inequality holds by using the condition that 𝑓(𝑥)=𝑥^1/(1+𝛼) is concave. Let 𝛼(𝑠)=−exp(𝑠𝑥^1/(1+𝛼))+exp(𝑠𝑥^1/(1+𝛼))exp{𝜆𝐸exp(𝑠𝜈₁^1/(1+𝛼))}. Since 𝐸exp(𝑠𝜈₁^1/(1+𝛼))<∞, we know that Δ𝑉_𝑠(𝑥)≤𝛼(𝑠)<∞,𝑥∈𝐶. (39) For all 𝑥∈𝐶^𝐶, Δ𝑉_𝑠(𝑥)=𝑃𝑉_𝑠(𝑥)−𝑉_𝑠(𝑥)=−exp𝑠𝑥^1/(1+𝛼)+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑠𝜎𝑘!𝐸exp_𝑘−1+𝑥^1/(1+𝛼)=−exp𝑠𝑥^1/(1+𝛼)⋅1−^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑠𝜎𝑘!𝐸exp_𝑘−1+𝑥^1/(1+𝛼)−𝑠𝑥^1/(1+𝛼)=−exp𝑠𝑥^1/(1+𝛼)𝑥^1/(1+𝛼)𝑥^{𝛼/(1+𝛼)}⋅1−^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑠𝜎𝑘!𝐸exp_𝑘−1+𝑥^1/(1+𝛼)−𝑠𝑥^1/(1+𝛼). (40) Let 𝛽(𝑠)=𝑥^{𝛼/(1+𝛼)}⋅1−^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑠𝜎𝑘!𝐸exp_𝑘−1+𝑥^1/(1+𝛼)−𝑠𝑥^1/(1+𝛼). (41) Now, we prove that there exists an 𝑠₁>0 such that 𝛽(𝑠)>0 for all 𝑠∈(0,𝑠₁). Similar to the proof of the case 𝑥∈𝐶, we know that 𝛽(𝑠) is a finite function for 𝑠∈[0,𝑟). Furthermore, 𝛽(0)=𝑥^{𝛼/(1+𝛼)}1−^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝛽𝑘!=0,_′(𝑠)∣_𝑠=0=−𝑥^{∞𝛼/(1+𝛼)}_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝜎𝑘!^𝑘−1+𝑥^1/(1+𝛼)−𝑥^1/(1+𝛼)=𝑥1−^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝜎𝑘!1+^𝑘−1𝑥^1/(1+𝛼)≥𝑥1−^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑘!1+𝑘/𝜇−1𝑥^1/(1+𝛼)≥𝑥1−1+𝜌−1𝑥^1/(1+𝛼)>0. (42) Let 𝑠₁ be the minimum positive root of the equation 𝛽(𝑠)=0; then, we have 𝛽(𝑠)>0 for all 0<𝑠<𝑠₁.
Let 𝑉_𝑠(𝑥)=exp(𝑠𝑥^1/(1+𝛼)), for all 𝑠∈(0,𝑠₁), and let 𝑏=max_{𝑠∈(0,𝑠1)}{𝛼(𝑠)}<∞; then, we have Δ𝑉_𝑠(𝑥)≤−𝛽(𝑠)exp𝑠𝑥^1/(1+𝛼)𝑥^{𝛼/(1+𝛼)}+𝑏𝐼_𝐶,=−𝜙∘𝑉_𝑠(𝑥)+𝑏𝐼_𝐶,𝑥∈𝑅₊, (43) where 𝜙(𝑥)=𝛽(𝑠)(𝑥/log^𝛼(𝑥)), (i.e., Condition 𝐷(𝜙,𝑉_𝑠,𝐶) holds). By Theorem 1, we know that there exists a unique invariant distribution 𝜋, 𝜋(𝜙∘𝑉)<∞, that is ₁^+∞𝑥^{−𝛼/(1+𝛼)}exp𝑠𝑥^1/(1+𝛼)𝜋(𝑑𝑥)<∞. (44) From 𝐻_𝜙(𝑥)=^𝑥₁𝑑𝑥𝜙(𝑥)=^𝑥₁log^𝛼(𝑥)𝑑𝑥𝛽(𝑠)𝑥=log^1+𝛼(𝑥)()𝛽(𝑠),𝛼+1 (45) we have 𝐻_𝜙⁻¹(𝑥)=exp[((𝛼+1)𝛽(𝑠)𝑥)^1/(1+𝛼)]. So, 𝑟_𝜙(𝑛)=𝜙∘𝐻_𝜙⁻¹(𝑛)(𝑠))𝛽(𝑠)𝑛=𝛽exp(𝛼+1^1/(1+𝛼)(𝛼+1)𝛽(𝑠)𝑛^{𝛼/(1+𝛼)}=𝛽(𝑠)^1/(1+𝛼)[(𝛼+1)𝑛]^{−𝛼/(1+𝛼)}⋅exp(𝛼+1)𝛽(𝑠)𝑛^1/(1+𝛼)≍𝑛^{−𝛼/(1+𝛼)}exp(𝛼+1)𝛽(𝑠)𝑛^1/(1+𝛼). (46) Let 𝜀₁=max_{𝑠∈(0,𝑠1)}{[(𝛼+1)𝛽(𝑠)]^1/(1+𝛼)}; then we have, lim_𝑛→∞𝑛^{−𝛼/(1+𝛼)}exp𝜀₁𝑛^1/(1+𝛼)𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0. (47) The proof is completed.

Case 2 (Polynomial Rate of Convergence). Consider the following.

Proposition 10. If 𝜌<1 and there exists a constant 𝛼>1 such that 𝐸𝜈^𝛼₁=₀^+∞𝑥^𝛼𝐵(𝑑𝑥)<+∞, (48) then lim_𝑛→∞𝑛^𝛼−1𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0,_1+∞𝑥^𝛼−1𝜋(𝑑𝑥)<+∞. (49)

Proof. Let 𝑉(𝑥)=(𝑥+1)^𝛼≥1,𝑥∈𝑅₊, and let 𝑚_𝛼 be the 𝛼th moment of the poisson distribution with parameter 𝜆. From (𝑎+𝑏)^𝛼≤2^𝛼(𝑎^𝛼+𝑏^𝛼), where 𝑎>0,𝑏>0, we have, for all 𝑥∈𝐶 (where 𝐶 is the petite [0,𝑐]), Δ𝑉(𝑥)=𝑃𝑉(𝑥)−𝑉(𝑥)≤−(𝑥+1)^𝛼+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝜈𝑘!¹+𝜈₂+⋯+𝜈_𝑘+𝑥+1^𝛼≤−(𝑥+1)^𝛼+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆2𝑘!^𝛼𝐸𝜎_𝑘^𝛼+(𝑥+1)^𝛼≤−(𝑥+1)^𝛼+2^𝛼(𝑥+1)^𝛼+2^𝛼∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑘𝑘!^𝛼𝐸𝜈^𝛼₁=−(𝑥+1)^𝛼+2^𝛼(𝑥+1)^𝛼+2^𝛼𝑚_𝛼𝐸𝜈^𝛼₁. (50) Let 𝑏=−(𝑐+1)^𝛼+2^𝛼(𝑐+1)^𝛼+2^𝛼𝑚_𝛼𝐸𝜈^𝛼₁. Since 𝐸𝜈^𝛼₁<∞, we know that Δ𝑉(𝑥)≤𝑏<∞,𝑥∈𝐶. (51) Let 𝐶^𝑖_𝑛 denote the binomial coefficient, and let 𝑛=⌊𝛼⌋,𝜃=𝛼−𝑛; then; 𝛼=𝑛+𝜃. For all 𝑥∈𝐶^𝐶, Δ𝑉(𝑥)=𝑃𝑉(𝑥)−𝑉(𝑥)=−(𝑥+1)^𝛼+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝜎𝑘!^𝑘−1+𝑥+1^𝛼=−(𝑥+1)^𝛼+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝜎𝑘!^𝑘−1+𝑥+1^𝑛+𝜃=−(𝑥+1)^𝛼+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆)𝑘!⋅𝐸(𝑥+1^𝑛+𝐶¹_𝑛(𝑥+1)^𝑛−1𝜎_𝑘+−1_𝑛𝑖=2𝐶^𝑖_𝑛(𝑥+1)^𝑛−𝑖𝜎_𝑘−1^𝑖𝜎_𝑘+𝑥^𝜃=^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸)𝑘!(𝑥+1^𝑛𝜎_𝑘+𝑥^𝜃−(𝑥+1)^𝛼+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝐶𝑘!^1𝑛(𝑥+1)^𝑛−1𝜎_𝑘−1𝜎_𝑘+𝑥^𝜃+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑘!⋅𝐸^𝑛_𝑖=2𝐶^𝑖_𝑛(𝑥+1)^𝑛−𝑖𝜎_𝑘−1^𝑖𝜎_𝑘+𝑥^𝜃. (52) Since 𝑓₁(𝜉)=𝜉^𝜃 is a concave function, we know that 𝐸(𝜉+𝑥)^𝜃≤(𝐸𝜉+𝑥)^𝜃. (53) Thus, the first part of (52) is ^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸)𝑘!(𝑥+1^𝑛𝜎_𝑘+𝑥^𝜃−(𝑥+1)^𝛼≤(𝑥+1)^𝑛𝜌+𝑥^𝜃−(𝑥+1)^𝛼<0. (54) If 𝛼 is integer (i.e., 𝜃=0), then the second part of (52) is ^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝐶𝑘!^1𝑛(𝑥+1)^𝑛−1𝜎_𝑘−1=𝐶¹_𝑛𝜌−1(𝑥+1)^𝛼−1. (55) If 𝛼 is not integer (i.e., 𝜃≠0), then the second part of (52) is ^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝐶𝑘!^1𝑛(𝑥+1)^𝑛−1𝜎_𝑘−1𝜎_𝑘+𝑥^𝜃=𝐶¹_𝑛(𝑥+1)^∞𝑛−1_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝜎𝑘!^𝑘−1(𝑥+1)^𝜃+𝐶¹_𝑛(𝑥+1)^∞𝑛−1_𝑘=0𝜆^𝑘𝑒^−𝜆𝜎𝑘!⋅𝐸_𝑘−1𝜎_𝑘+𝑥^𝜃−(𝑥+1)^𝜃≤𝐶¹_𝑛𝜌−1(𝑥+1)^𝛼−1+𝐶¹_𝑛(𝑥+1)^∞𝑛−1_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{𝜎𝑘>1}𝜎_𝑘^𝜃+1+𝐸1_{𝜎𝑘<1}(𝑥+1)^𝜃−𝑥^𝜃≤𝐶¹_𝑛𝜌−1(𝑥+1)^𝛼−1+𝐶¹_𝑛(𝑥+1)^𝑛−1𝑚_𝜃+1𝐸𝜈₁^𝜃+1+1=𝐶¹_𝑛𝜌−1(𝑥+1)^𝛼−1+𝑎₁(𝑥+1)^𝑛−1, (56) where 𝑎₁=𝐶¹_𝑛𝑚_𝜃+1(𝐸𝜈₁^𝜃+1+1). From (55) and (56), we obtain that the second part of (52) is ^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝐶𝑘!^1𝑛(𝑥+1)^𝑛−1𝜎_𝑘−1𝜎_𝑘+𝑥^𝜃=𝐶¹_𝑛𝜌−1(𝑥+1)^𝛼−1+𝑎₁(𝑥+1)^𝑛−1, (57) where 𝑎₁=0 if 𝛼 is integer. The third part of (52) is ^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑘!^𝑛_𝑖=2𝐶^𝑖_𝑛(𝑥+1)^𝑛−𝑖𝐸𝜎_𝑘−1^𝑖𝜎_𝑘+𝑥^𝜃≤^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑘!^𝑛_𝑖=2𝐶^𝑖_𝑛(𝑥+1)^𝑛−𝑖⋅𝐸𝜎_𝑘^𝑖+1𝜎_𝑘^𝜃+(𝑥+1)^𝜃≤^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝑘!^𝑛_𝑖=2𝐶^𝑖_𝑛(𝑥+1)^𝑛−𝑖⋅𝐸𝜎_𝑘^𝑖+𝜃+𝜎_𝑘^𝑖(𝑥+1)^𝜃+𝜎_𝑘^𝜃+(𝑥+1)^𝜃=^𝑛_𝑖=2𝐶^𝑖_𝑛(𝑥+1)^𝑛−𝑖⋅𝑚_𝑖+𝜃𝐸𝜈₁^𝑖+𝜃+𝑚_𝑖𝐸𝜈^𝑖₁(𝑥+1)^𝜃+𝑚_𝜃𝐸𝜈^𝜃₁+(𝑥+1)^𝜃≤^𝑛_𝑖=2𝐶^𝑖_𝑛𝑚_𝑖+𝜃𝐸𝜈₁^𝑖+𝜃+𝑚_𝑖𝐸𝜈^𝑖₁+𝑚_𝜃𝐸𝜈^𝜃₁+1(𝑥+1)^{𝑛−𝑖+𝜃}=^𝑛_𝑖=2𝑎_𝑖(𝑥+1)^𝛼−𝑖, (58) where 𝑎_𝑖=𝐶^𝑖_𝑛(𝑚_𝑖+𝜃𝐸𝜈₁^𝑖+𝜃+𝑚_𝑖𝐸𝜈^𝑖₁+𝑚_𝜃𝐸𝜈^𝜃₁+1)<∞,1≤𝑖≤𝑛 (by 𝐸𝜈^𝛼₁<∞). Combining (52), (54), (57), and (58), we have Δ𝑉(𝑥)≤𝐶¹_𝑛𝜌−1(𝑥+1)^𝛼−1+𝑎₁(𝑥+1)^𝑛−1+^𝑛_𝑖=2𝑎_𝑖(𝑥+1)^𝛼−𝑖, (59) where 𝑎₁=0 if 𝛼 is integer. Choose 𝑐 large enough such that, for all 𝑥>𝑐 (i.e., 𝑥∈𝐶^𝑐,𝐶=[0,𝑐]), 𝑎₁(𝑥+1)^𝑛−1+^𝑛_𝑖=2𝑎_𝑖(𝑥+1)^𝛼−𝑖1<−2𝐶¹_𝑛𝜌−1(𝑥+1)^𝛼−1. (60) Thus (𝑥)≤1Δ𝑉2𝐶¹_𝑛𝜌−1(𝑥+1)^𝛼−1,∀𝑥∈𝐶^𝑐. (61) Together with (51), we have Δ𝑉(𝑥)≤−𝛽(𝑥+1)^𝛼−1+𝑏𝐼_𝐶,=−𝜙∘𝑉(𝑥+1)+𝑏𝐼_𝐶,∀𝑥∈𝑅₊, (62) where 𝛽=(1/2)𝐶¹_𝑛(1−𝜌)>0, 𝜙(𝑧)=𝛽𝑧^{(𝛼−1)/𝛼} (i.e., condition 𝐷(𝜙,𝑉,𝐶) holds). By Theorem 1, we know that there exists a unique invariant distribution 𝜋, 𝜋(𝜙∘𝑉)<∞ (i.e., ∫₁^+∞𝑥^𝛼−1𝜋(𝑑𝑥)<∞) and lim_𝑛→∞𝑃^(𝑛)(𝑥,𝐴)−𝜋(𝐴)_𝜙∘𝑉=0,𝜋-𝑎.𝑒.. (63) From 𝐻_𝜙(𝑥)=^𝑥₁𝑑𝑧𝜙(𝑧)=1𝛽^𝑥₁𝑧^{(1−𝛼)/𝛼}𝛼𝑑𝑧=𝛽𝑥^1/𝛼−1, (64) we have 𝐻_𝜙⁻¹(𝑥)=((𝛽/𝛼)𝑥+1)^𝛼. So, 𝑟_𝜙(𝑛)=𝜙∘𝐻_𝜙⁻¹(𝑛)𝛽=𝛽𝛼𝑛+1^{𝛼(𝛼−1)/𝛼}𝛽=𝛽𝛼𝑛+1^𝛼−1≍𝑛^𝛼−1. (65) That is, lim_𝑛→∞𝑛^𝛼−1𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0. (66)

Case 3 (Logarithmic Rate of Convergence). Now, we consider the logarithmic case which is slower than that for any polynomial.

Proposition 11. If 𝜌<1 and there exists a positive integer 𝛼 such that 𝐸𝜈₁log^𝛼𝜈₁+1=₀^+∞𝑥log^𝛼(𝑥+1)𝐵(𝑑𝑥)<+∞, (67) then lim_𝑛→∞log^𝛼𝑛𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0,_1+∞log^𝛼𝑥𝜋(𝑑𝑥)<∞. (68)(69)

Proof. For all 𝑥∈𝑅₊, let 𝑉(𝑥)=(𝑥+𝑒)log^𝛼(𝑥+𝑒); then, we have 𝑉(𝑥)>𝑒,𝑥∈𝑅₊. Let 𝑐>𝑒^𝛼, and choose 𝐶=[0,𝑐]. For all 𝑥∈𝐸, (𝑥)(𝑥)(𝑥)=Δ𝑉=𝑃𝑉−𝑉_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝜎𝑘!^𝑘−1+𝑥+𝑒log^𝛼𝜎_𝑘−()−1+𝑥+𝑒𝑥+𝑒log^𝛼(𝑥+𝑒)=^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝜎𝑘!^𝑘−1log^𝛼()+𝑥+𝑒_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝐸𝜎𝑘!^𝑘⋅−1+𝑥+𝑒log^𝛼𝜎_𝑘−1+𝑥+𝑒−log^𝛼(𝑥+𝑒)=𝜌−1log^𝛼()+𝑥+𝑒_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝜎𝑘!⋅𝐸_𝑘⋅−1+𝑥+𝑒log^𝛼()𝜎𝑥+𝑒1+^𝑘−1𝑥+𝑒−log^𝛼(≤𝑥+𝑒)𝜌−1log^𝛼()+𝑥+𝑒_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{𝜎𝑘>1}𝜎_𝑘⋅−1+𝑥+𝑒_𝛼𝑖=1𝐶^𝑖_𝛼log^𝛼−𝑖(𝑥+𝑒)log^𝑖𝜎1+^𝑘−1≤𝑥+𝑒𝜌−1log^𝛼()+𝑥+𝑒_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{{1<𝜎𝑘≤𝑥+𝑒}}2()⋅𝑥+𝑒_𝛼𝑖=1𝐶^𝑖_𝛼log^𝛼−𝑖(𝑥+𝑒)log^𝑖𝜎1+^𝑘−1+𝑥+𝑒_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{{𝜎𝑘>𝑥+𝑒}}2𝜎_𝑘^𝛼_𝑖=1𝐶^𝑖_𝛼log^𝛼−𝑖⋅(𝑥+𝑒)log^𝑖𝜎1+^𝑘−1≤𝑥+𝑒𝜌−1log^𝛼()+𝑥+𝑒_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{{1≤𝜎𝑘≤𝑥+𝑒}}2𝜎_𝑘−1^𝛼_𝑖=1𝐶^𝑖_𝛼log^𝛼−𝑖(+𝑥+𝑒)_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{{𝜎𝑘>𝑥+𝑒}}2𝜎_𝑘⋅^𝛼_𝑖=1𝐶^𝑖_𝛼log^𝛼−𝑖(𝑥+𝑒)log^𝑖𝜎_𝑘≤𝜌−1log^𝛼()+𝑥+𝑒_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{{1≤𝜎𝑘≤𝑥+𝑒}}2^𝛼+1𝜎_𝑘log^𝛼−1(+𝑥+𝑒)_∞𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{{𝜎𝑘>𝑥+𝑒}}2^𝛼+1𝜎_𝑘⋅log^𝛼−1(𝑥+𝑒)log^𝛼𝜎_𝑘≤𝜌−1log^𝛼(𝑥+𝑒)+2^𝛼+1log^𝛼−1()⋅𝑥+𝑒𝜌+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{{𝜎𝑘>𝑥+𝑒}}𝜎_𝑘log^𝛼𝜎_𝑘≤𝜌−1log^𝛼(𝑥+𝑒)+2^𝛼+1log^𝛼−1()⋅𝑥+𝑒𝜌+^∞_𝑘=0𝜆^𝑘𝑒^−𝜆𝐸1𝑘!^{{𝜎𝑘>𝑥+𝑒}}𝜎_𝑘2^𝛼⋅log^𝛼𝑘+log^𝛼𝜈₁≤+1𝜌−1log^𝛼(𝑥+𝑒)+2^𝛼+1log^𝛼−1()⋅𝑥+𝑒𝜌+2^𝛼𝑚₂𝐸𝜈₁+𝑚₁𝐸𝜈₁log^𝛼𝜈₁=+1𝜌−1log^𝛼(𝑥+𝑒)+𝑎₀log^𝛼−1(𝑥+𝑒), (70) where 𝑎₀=2^𝛼+1[𝜌+2^𝛼(𝑚₂𝐸𝜈₁+𝑚₁𝐸(𝜈₁log^𝛼(𝜈₁+1)))]. Since 𝐸(𝜈₁log^𝛼(𝜈₁+1))<∞, we get that 𝑎₀<+∞. Let 𝑏=𝑎₀log^𝛼−1(𝑐+𝑒); then, we have Δ𝑉(𝑥)≤𝑏<∞,𝑥∈𝐶. (71) Choose 𝑐 large enough such that, if 𝑥>𝑐 (i.e., 𝑥∈[0,𝑐]^𝑐), 𝑎₀log^𝛼−1()<1𝑥+𝑒21−𝜌log^𝛼(𝑥+𝑒). (72) Thus, (𝑥)1Δ𝑉≤−21−𝜌log^𝛼(𝑥+𝑒),∀𝑥∈𝐶^𝑐. (73) Together with (71), we have Δ𝑉(𝑥)≤−𝛽log^𝛼(𝑥+𝑒)+𝑏𝐼_𝐶,=−𝜙∘𝑉(𝑥)+𝑏𝐼_𝐶,∀𝑥∈𝑅₊, (74) where 𝛽=(1/2)(1−𝜌)>0, 𝜙(𝑧)=𝛽log^𝛼𝑧 (i.e., condition 𝐷(𝜙,𝑉,𝐶) holds). A straightforward calculation shows that 𝑟_𝜙(𝑛)≍log^𝛼𝑛. (75) That is, lim_𝑛→∞log^𝛼𝑛𝑃^(𝑛)(𝑥,⋅)−𝜋(⋅)=0. (76) By Theorem 1, we know that there exists a unique invariant distribution 𝜋, 𝜋(𝜙∘𝑉)<∞ (i.e., ∫₁^+∞log^𝛼𝑥𝜋(𝑑𝑥)<∞) and lim_𝑛→∞𝑃^(𝑛)(𝑥,𝐴)−𝜋(𝐴)_𝜙∘𝑉=0,𝜋-𝑎.𝑒.. (77)

4.1. 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 𝜀₀ and 𝑠₀ 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 𝑟(𝑛)=exp(𝑠𝑛^1/(1+𝛼)),𝛼>0,𝑠>0; Case 2: polynomial rate function 𝑟(𝑛)=𝑛^𝛼,𝛼>0; Case 3: logarithmic rate function 𝑟(𝑛)=log^𝛼𝑛,𝛼>0, we gave explicit criteria for the rate of convergence and decay of stationary tail. We gave the parameters 𝜀₁ and 𝑠₁ of the rate of convergence and the decay of the stationary tail, respectively, for the subgeometric rate 𝑟(𝑛)=exp(𝑠𝑛^1/(1+𝛼)),𝛼>0,𝑠>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.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (BUPT2011RC0703).