Polling systems and multitype branching processes in random environment counted by random characteristics

TL;DR

This paper investigates the tail distribution of busy periods and other characteristics in multitype branching process-based polling systems with randomly changing service disciplines, input parameters, and service time distributions.

Contribution

It introduces a novel approach using multitype branching processes in random environments to analyze polling systems with random variations.

Findings

Derived tail distribution formulas for busy periods

Analyzed impact of random environment on system performance

Provided insights into system behavior under stochastic changes

Abstract

By the methods of multitype branching processes in random environment counted by random characteristics we study the tail distribution of busy periods and some other characteristics of the branching type polling systems in which the service disciplines, input parameters and service time distributions are changing in a random manner.

Equations

$${\mathbf{s}}^{\mathbf{k}}:=s_1^{k_1}\cdots s_m^{k_m},k_i\in {\mathbb{N}}_0=\left\{0,1,2,\dots \right\},$$

$${\varphi }^{(i)}(\mathbf{s};\lambda ):=\mathbf{E}\left[s_1^{{\theta }_{i1}}{s}_2^{{\theta }_{i2}}\cdots s_m^{{\theta }_{im}}{e}^{-\lambda {\varphi }_i}\right],i=1,...,m$$

$$\varphi (\mathbf{s};\lambda ):=\left({\varphi }^{(1)}(\mathbf{s};\lambda ),\dots ,{\varphi }^{(m)}(\mathbf{s};\lambda )\right)$$

$${\varphi }_0(\mathbf{s};\lambda ),{\varphi }_1(\mathbf{s};\lambda ),{\varphi }_2(\mathbf{s};\lambda ),\dots ,$$

$${\varphi }_n^{(i)}(\mathbf{s};\lambda ):=\mathbf{E}\left[s_1^{{\theta }_{i1}(n)}{s}_2^{{\theta }_{i2}(n)}\cdots s_m^{{\theta }_{im}(n)}{e}^{-\lambda {\varphi }_i(n)}\right].$$

$$\mathbf{E}\left[s_1^{{\theta }_{i1}(n,j)}{s}_2^{{\theta }_{i2}(n,j)}\cdots s_m^{{\theta }_{im}(n,j)}{e}^{-\lambda {\varphi }_i(n,j)}\right]={\varphi }_n^{(i)}(\mathbf{s};\lambda ).$$

$$\sum _{j=0}^m{\gamma }_{ij}=1,i=1,...,m.$$

$${\mathcal{E}}_0={\left({\epsilon }_{ij}(0)\right)}_{i,j=1}^m,{\Gamma }_0={\left({\gamma }_{ij}(0)\right)}_{i=1,j=0}^m,{\mathbf{T}}_0=\left(T_{10},...,T_{m0}\right)$$

$${\mathcal{E}}_n={\left({\epsilon }_{ij}(n)\right)}_{i,j=1}^m,{\Gamma }_n={\left({\gamma }_{ij}(n)\right)}_{i=1,j=0}^m,{\mathbf{T}}_n=\left(T_{1n},...,T_{mn}\right)$$

$$t_{in}(\lambda ):={\int }_0^{\infty }{e}^{-\lambda x}d{T}_{in}(x)$$

$${\varphi }_n^{(i)}(\mathbf{s};\lambda )$$

$$h_n^{(i)}(\mathbf{s}):={\varphi }_n^{(i)}(\mathbf{s};0)=\left({\gamma }_{i0}(n)+\sum _{j=1}^m{\gamma }_{ij}(n)s_j\right)t_{in}\left(\sum _{j=1}^m{\epsilon }_{ij}(n)(1-s_j)\right).$$

$$w_{in}(\lambda )=\frac{1-{\gamma }_{ii}(n)}{1-{\gamma }_{ii}(n)t_{in}(\lambda )}{t}_{in}(\lambda ).$$

$$y_{in}(\mathbf{s}):=\frac{{\gamma }_{i0}(n)+{\sum }_{j\ne i}{\gamma }_{ij}(n)s_j}{1-{\gamma }_{ii}(n)}.$$

$$\mathbf{E}\left[s_i^{{\sigma }_{i1}(n)}\cdots s_m^{{\sigma }_{im}(n)}{e}^{-\lambda {\tau }_i(n)}\right]=y_{in}(\mathbf{s})w_{in}\left(\lambda +\sum _{j=1}^m{\epsilon }_{ij}(n)(1-s_j)\right).$$

$$l_{in}(\lambda )=w_{in}\left(\lambda +{\epsilon }_{ii}(n)\left(1-l_{in}(\lambda )\right)\right)$$

$${\varphi }_n^{(i)}(\mathbf{s};\lambda ):=\mathbf{E}\left[s_1^{{\theta }_{i1}(n)}{s}_2^{{\theta }_{i2}(n)}\cdots s_m^{{\theta }_{im}(n)}{e}^{-\lambda {\eta }_{i,tot}(n)}\right],i=1,2,\dots ,m$$

$${\varphi }_n^{(i)}(\mathbf{s};\lambda )=y_{in}(\mathbf{s})w_{in}\left(\lambda +\sum _{j\ne i}{\epsilon }_{ij}(n)(1-s_j)+{\epsilon }_{ii}(n)(1-{\varphi }_n^{(i)}(\mathbf{s};\lambda ))\right).$$

$$h_n^{(i)}(\mathbf{s})=y_{in}(\mathbf{s})w_{in}\left(\sum _{j\ne i}{\epsilon }_{ij}(n)(1-s_j)+{\epsilon }_{ii}(n)(1-h_n^{(i)}(\mathbf{s}))\right).$$

$$F(\mathbf{s};\lambda )=\mathbf{E}\left[s_1^{{\xi }_1}{s}_2^{{\xi }_2}\cdots s_m^{{\xi }_m}{e}^{-\lambda \phi }\right],\mathbf{s}=(s_1,\dots ,s_m)\in {\left[0,1\right]}^m,\lambda \ge 0,$$

$${\mathcal{F}}_{\lambda }^m:={\mathcal{F}}_{\lambda }\times {\mathcal{F}}_{\lambda }\times \cdots \times {\mathcal{F}}_{\lambda }=\left\{\mathbf{F}(\mathbf{s};\lambda )=\left(F^{(1)}(\mathbf{s};\lambda ),\dots ,F^{(m)}(\mathbf{s};\lambda )\right)\right\}$$

$${\mathcal{F}}_0:=\left\{f(\mathbf{s})=F(\mathbf{s};0),\mathbf{s}\in {\left[0,1\right]}^m\right\}$$

$$f(\mathbf{s})=\mathbf{E}\left[s_1^{{\xi }_1}{s}_2^{{\xi }_2}\cdots s_m^{{\xi }_m}\right]$$

$${\mathcal{F}}_0^m:={\mathcal{F}}_0\times \cdots \times {\mathcal{F}}_0=\left\{\mathbf{f}(\mathbf{s})=\left(f^{(1)}(\mathbf{s}),\dots ,f^{(m)}(\mathbf{s})\right)\right\}$$

$${\mathbf{F}}_0(\mathbf{s};\lambda ),{\mathbf{F}}_1(\mathbf{s};\lambda ),\dots ,{\mathbf{F}}_k(\mathbf{s};\lambda ),\dots \text{ with }{\mathbf{F}}_n(\mathbf{s};\lambda ):=\left(F_n^{(1)}(\mathbf{s};\lambda ),\dots ,F_n^{(m)}(\mathbf{s};\lambda )\right)$$

$$F_n^{(i)}(\mathbf{s};\lambda )=\mathbf{E}\left[s_1^{{\xi }_{i1}(n)}{s}_2^{{\xi }_{i2}(n)}\cdots s_m^{{\xi }_{im}(n)}{e}^{-\lambda {\phi }_i(n)}\right].$$

$$\mathbf{R}(n)=\left(\mathbf{Z}(n);\Phi (n)\right),n=0,1,\dots ,$$

$${\mathbf{R}}_{\phi }(n)=\mathbf{R}(n):=\left(\mathbf{Z}(n);\Phi (n)\right)=\left(Z_1(n),\dots ,Z_m(n);\Phi (n)\right),n\in {\mathbb{N}}_0,$$

$${\mathbb{N}}_0^m\times {\mathbb{R}}_0:=\left\{z=(z_1,\dots ,z_m;w),z_i\in {\mathbb{N}}_0;w\in [0,\infty )\right\}$$

$$\mathbf{R}(0)=\left(\mathbf{Z}(0);\Phi (0)\right)=(\mathbf{z};{\phi }_0),$$

Full text

Key words and phrases: polling systems, multitype branching processes in random environment, final product, busy period, random matrices

1

Polling systems with service policies of branching type

We consider a polling system consisting of a single server and $m$ stations with infinite-buffer queues indexed by $i\in\left\{1,\ldots,m\right\}.$ Initially there are no customers in the system. When customers arrive to the system the server starts immediately the service by visiting the stations in cyclic order $(1\rightarrow 2\rightarrow\cdots\rightarrow m\rightarrow 1\rightarrow\cdots)$ starting at station $1$ according to a selected service policy (to be described later on) and with zero switchover times between queues. Later on the initial stage of services $(1\rightarrow 2\rightarrow\cdots\rightarrow m)$ will be called the zero cycle. The subsequent routes of the server will be called the first cycle, the second cycle and so on. When the system is empty the server waits their arrival at a parking place $R$. Customers arrive to the queues in accordance with a point process whose parameters are changing in a random manner each time when the server switches from station to station .

To give a rigorous description of the arrival and service processes for the system in question we need some notions. Let $\mathbf{s:}=(s_{1},\ldots,s_{m})\in\left[0,1\right]^{m}$ be a $m$-dimensional variable,

[TABLE]

$\mathbf{e}_{i}:=\left(0,\ldots,1,\ldots,0\right)$ be a $m-$dimensional vector with zero components except the $i$-th equal to $1,$ $\mathbf{0}$ and $\mathbf{1}$ are $m-$dimensional vectors whose all components are equal to [math] and $1$, respectively. Let

[TABLE]

be the mixed probability generating function (m.p.g.f.’s) of $(m+1)-$ dimensional vectors $\left(\theta_{i1},\ldots,\theta_{im};\phi_{i}\right)$, where $\theta_{ij}$ are nonnegative integer-valued random variables and $\phi_{i}$ is a nonnegative random variable. Denote

[TABLE]

the respective vector-valued m.p.g.f. and let $\mathcal{F}=\left\{\mathbf{\phi}(\mathbf{s;}\lambda)\right\}$ be the set of all such m.p.g.f. Assume that a probability measure $\mathbb{P}$ is specified on the natural $\sigma$-algebra of $\mathcal{F}$ and let

[TABLE]

be a sequence selected in iid manner from $\mathcal{F}$ in accordance with $\mathbb{P},$ where $\mathbf{\phi}_{n}(\mathbf{s;}\lambda):=\left(\phi_{n}^{(1)}(\mathbf{s;}\lambda),\ldots,\phi_{n}^{(m)}(\mathbf{s;}\lambda)\right)$ with

[TABLE]

In the present paper we investigate such polling systems whose arrival and service procedures of customers meet the following property.

Branching property. When the server arrives to station $i$ for the $n$-th time and find there, say, $k_{i}$ customers labelled $1,2,...,k_{i}$ , then, during the course of the server’s visit, the arrival of customers is arranged in such a way that after the end of each stage of service of customer $j$ (the number of such stages may be more than one if the service discipline admits feedback) the queues at the system will be increased by a random population of customers $\left(\theta_{i1}(n,j),\ldots,\theta_{im}(n,j)\right),$ where $\theta_{il}(n,j)$ is the number of customers added to the $l$-th station, and, in addition, a final product of size $\phi_{i}(n,j)\geq 0$ will be added to the system. It is assumed that the vectors $\left(\theta_{i1}(n,j),\ldots,\theta_{im}(n,j);\phi_{i}(n,j)\right),$