Discrete and Dynamic Optimization Problems in Operation ManagementView this Special Issue
Research Article | Open Access
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 and 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 and of the rate of convergence and the decay of the stationary tail, respectively, for the subgeometric rate , , .
We consider several types of convergence rates of the queueing system by using drift conditions. The queueing system discussed here is that the arrivals form a Poisson process with parameter . The service times for the customers are independently identically distributed random variables with a common distribution function . Let where is a constant, and is called the service intensity. Denote the workload process of the queueing system by ; then, 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  discussed Foster-Lyapounov-type drift conditions for Markov chains which imply polynomial rate convergence to stationarity in appropriate V-norms. Jarner and Tweedie  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 -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 and 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 ; Case 2: polynomial rate function ; Case 3: logarithmic rate function ). And we give the parameters and 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 be a discrete time Markov chain on the state space 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 ; that is, for all , where satisfies , and for all signed measures , the -norm is defined as sup.
Geometric Rate Function. That is, the function satisfies
Subgeometric Rate Function. That is, the function satisfies The class of subgeometric rates function includes polynomial rates functions; that is, , and rate functions which increase faster than the polynomial ones; .
We shall discuss geometric rates of convergence , subgeometric rate of convergence , polynomial rate of convergence , and logarithmic rate of convergence .
Condition . There exist a function , a concave monotone nondecreasing differentiable function a measurable set , and a finite constant such that where is the indicator function of the set .
Theorem 1 (Theorem 14.0.1 in ). If holds for some petite set and there exists such that , then there exists a unique invariant distribution , and where .
Theorem 2 (Proposition 2.5 in Douc et al. ). Let be a -irreducible and aperiodic kernel. Assume that holds for function with , a petite set , and a function with . Then, there exists an invariant probability measure , and for all in the full and absorbing set , where , .
Since is a concave monotone nondecreasing differentiable function, is nonincreasing. Then, there exists , such that . In Theorem 2, for the case , condition implies that the chain is geometric ergodic, but the rate in the geometric convergence property cannot be achieved under the condition that .
The workload process of the queueing system is a Markov process on the state space . is an -skeleton of . We choose , and denote by . Suppose that the workload can be decreased by during the time interval . And suppose that the transition kernel of is . For convenience, let . Then, 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 with . For all , there exists a satisfying , such that Hence, is irreducible. From we know that is also aperiodic.
Lemma 4. is petite set, where is a real number.
Proof. Let be the maximum integer no more than . Since and is a closed set, we know that . Let be a measure on satisfying, for all , Obviously, for all , Thus, we get that is a petite set.
Lemma 5. The Markov chain is stochastically monotonic.
Proof. For every fixed , from we obtain that is nonincreasing in . That is, is stochastically monotonic.
For two sequences and , we write , if there exist positive constants and such that, for large , .
Let us say that the distribution function of a random variable is in if the distribution function of a random variable is in if where , and .
Now, we give the main result.
Theorem 6. Suppose that and is the stationary distribution of .
If , then one has where is the minimum positive root of the equation . Moreover, is geometrically ergodic, where , and is a root of the equation .
If , then one has where is the minimal positive solution of . And where .
If there exists a constant , such that , then
If there exists an integer number , such that , then
3. Geometric Rate of Convergence
The Markov chain is geometrically ergodic if (2) holds with for some . By Theorem 15.0.1 in , an equivalent condition of geometric ergodicity is that there exist a petite set , constants and , and a function finite for at least one satisfying 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 . Taking the petite set , for all , Since (i.e., ), we know that For all (i.e., ), Let . Now, we prove that there exists an such that . By the stated condition , we know that is a finite differentiable function for . Furthermore,
Proposition 7. Suppose that and is the stationary distribution of . If ; then, where is the minimum positive root of the equation .
Proof. By (27), we know that and . So, there exists an such that . The function is continuous in the interval , and it is easy to see that . By the zero theorem, we know that there exists at least one root of the equation (i.e., ). Let be the minimum positive root; then, , for all .
Let ; then, we have where (i.e., condition holds). By Theorem 1, we know that ; that is,
Proposition 8. Suppose that and is the stationary distribution of . If ; then, where , , and is the root of the equation .
Proof. From (29), where , and is the minimum positive root of the equation . We have From Lemma 5, we know that is a stochastically monotonic Markov chain. By using Theorem 1.1 in , we have Let . From , we know that is a concave function. Together with , there exists a unique point , such that , and has a maximum at the point in the interval . So, where . The proof is completed.
Case 1 (The Rate Function ). The rate function , 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 and is the stationary distribution of . If , then one has where is the minimal positive solution of . And where .
Proof. Let . For all ,
where the second inequality holds by using the condition that is concave. Let . Since , we know that
For all ,
Now, we prove that there exists an such that for all . Similar to the proof of the case , we know that is a finite function for . Furthermore,
Let be the minimum positive root of the equation ; then, we have for all .
Let , for all , and let ; then, we have where , (i.e., Condition holds). By Theorem 1, we know that there exists a unique invariant distribution , , that is From we have . So, Let ; then we have, The proof is completed.
Case 2 (Polynomial Rate of Convergence). Consider the following.
Proposition 10. If and there exists a constant such that then
Proof. Let , and let be the th moment of the poisson distribution with parameter . From , where , we have, for all (where is the petite ), Let . Since , we know that Let denote the binomial coefficient, and let ; then; . For all , Since is a concave function, we know that Thus, the first part of (52) is If is integer (i.e., ), then the second part of (52) is If is not integer (i.e., ), then the second part of (52) is where . From (55) and (56), we obtain that the second part of (52) is where if is integer. The third part of (52) is where (by ). Combining (52), (54), (57), and (58), we have where if is integer. Choose large enough such that, for all (i.e., ), Thus Together with (51), we have where , (i.e., condition holds). By Theorem 1, we know that there exists a unique invariant distribution , (i.e., ) and From we have . So, That is,
Case 3 (Logarithmic Rate of Convergence). Now, we consider the logarithmic case which is slower than that for any polynomial.
Proposition 11. If and there exists a positive integer such that then
Proof. For all , let ; then, we have . Let , and choose . For all ,