The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates

Abstract

We study a queueing system with Erlang arrivals with k phases and Erlang service with m phases. Transition rates among phases vary periodically with time. For these systems, we derive an analytic solution for the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as infinite series. We provide bounds for the error of the estimate obtained by truncating the series. Examples are provided comparing the solution of the system of ordinary differential equation with a truncated state space to these asymptotic solutions involving remarkably few terms of the infinite series. The method can be generalized to other level independent quasi-birth-death processes if the singularities of the generating function are known.

Appendix: Evolution operators

A Banach space is a complete normed vector space.

Let X be a Banach space. For every t. $0\le t\le T$ let $A(t):D(A(t))\subset X\rightarrow X$ be a linear operator in X. Consider the initial value problem

$$\begin{aligned} \dot{u}(t)=u(t)A(t)\\ u(s)=x.\nonumber \end{aligned}$$

(58)

A X valued function $u:[s,T]\rightarrow X$ is a classical solution of (58) if u is a continuous function on [s, T], $u(t)\in D(A(t))$, for $s<t\le T$, u is continuously differentiable on $s<t\le T$ and satisfies (58).

Theorem 4

(Pazy) [25] Let X be a Banach space for every t, $0\le t\le T$. Let A(t) be a bounded linear operator on X. If the function $t\rightarrow A(t)$ is continuous in the uniform operator topology then for every $x\in X$ the initial value problem (58) has a unique classical solution u.

Definition 3

We define the evolution operator of the initial value Problem (58) by $xU(t,s)=u(t)$ for $0\le s \le t\le T$ where u is the solution of (58).

The evolution operator is also called a solution operator or propagator in the literature. If we meet the conditions in theorem 4, then our evolution operator has the following properties:

Theorem 5

(Evolution operator properties) [25] For every $0\le s\le t\le T$, U(t, s) is a bounded linear operator and

• 1. $||U(t,s)||\le \exp \left\{ \int _s^t A(\tau )d\tau \right\}$.

• 2. $U(t,t)=I$, $U(t,s)=U(t,r)U(r,s)$ for $0\le s\le r\le t\le T$.

• 3. $(t,s)\rightarrow U(t,s)$ is continuous in the uniform operator topology for $0\le s\le t\le T$.

• 4. $\frac{\partial U(t,s)}{\partial t}=U(t,s)A(t)$ for $0\le s\le t\le T$.

• 5. $\frac{\partial U(t,s)}{\partial s}=-A(s)U(t,s)$ for $0\le s\le t\le T$.

Definition 4

A two parameter family of bounded linear operators U(t, s), $0\le s\le t\le T$, on X is called an evolution system if the following two conditions are met:

• 1. $U(s,s)=I$, $U(t,r)U(r,s)=U(t,s)$ for $0\le s\le r\le t\le T$.

• 2. $(t,s)\rightarrow U(t,s)$ is strongly continuous for $0\le s\le t\le T$.

In this paper, we observe evolution operators in one of two contexts: 1) the u(t) are probability vectors, x is the probability distribution at some initial time s, and $\mathbf{A}(t)$ is an infinite-dimensional matrix with bounded non-negative entries; or 2) u(z, t) are vectors of probability generating functions and A(z, t) are finite dimensional matrices with parameter z defined in Eq. (19).

The evolution operator is a generalization of the exponential function. If x and A are scalars, U(t, s) is a scalar exponential function. If x is a vector and $A(t)=A$ an appropriately dimensioned matrix, then U(t, s) is the matrix exponential $\mathrm{e}^{A(t-s)}$. The matrix exponential may be defined in terms of its Taylor series expansion:

$$\begin{aligned} \mathrm{e}^{A t}=\sum _{n=0}^\infty \frac{t^n}{n!}A^n. \end{aligned}$$

(59)

When A(t) depends on time, we have the Peano series representation for U(t, s). Define

$$\begin{aligned} \mathcal {I}_n(t,s)=\underbrace{\int _s^t A(\tau _1)\int _s^{\tau _1}A(\tau _2)\cdots \int _s^{\tau _{n-1}} A(\tau _n)d\tau _1\cdots d\tau _n}_{n\text { integrals}}, \end{aligned}$$

with $\mathcal {I}_0=I$, then:

$$\begin{aligned} U(t,s)=\sum _{n=0}^\infty \mathcal {I}_n(t,s). \end{aligned}$$

(60)

In the event that $A(t)=A$ is a constant matrix, the Peano series given in Eq. (60) yields the matrix exponential $\mathrm{e}^{A(t-s)}$.

U(t, s) solves the Volterra equation

$$\begin{aligned} U(t,s)=\mathbf{I}+\int _s^t U(\tau ,s)A(\tau )d\tau . \end{aligned}$$

If the matrix A(t) is finite, then U(t, s) is given by the product integral

$$\begin{aligned} U(t,s)=\lim _{\Delta \tau _i\rightarrow 0}\prod _i\mathrm{e}^{A(\tau _i)\Delta \tau _i}. \end{aligned}$$

This product integral is sometimes called the time-ordered exponential from s to t, or the multiplicative integral. See Dollard and Friedman [5] or Gill and Johansen [9].

There are several examples in the body of the paper.