Performance Analysis of a Block-Structured Discrete-Time Retrial Queue with State-Dependent Arrivals

Abstract

In this paper, we introduce a new discrete block state-dependent arrival (D-BSDA) distribution which provides fresh insights leading to a successful generalization of the discrete-time Markovian arrival process (D-MAP). The D-BSDA distribution is related to structured Markov chains and the method of stages. The consideration of this new discrete-time state-dependent block description gives one the ability of construct new stochastic models. The retrial queue analyzed in this paper gives an example of application of the D-BSDA distribution to construct more general and sophisticated models. We assume that the primary arrivals and the retrials follow the D-BSDA description and the service times are of discrete phase-type (PH). We study the underlying level dependent Markov chain of M/G/1-type at the epochs immediately after the slot boundaries. To this end, we employ the UL-type RG-factorization which provides an expression for the stationary probabilities. We also perform an analysis of waiting times. Numerical experiments are presented to study the system performance.

Appendices

Appendix A

Here are two examples of non-homogeneous block arrival patterns subsumed under the D-BSDA distribution as special cases.

• (1) A state-dependent PH arrival process

First of all, we assume that the fundamental system state (e.g., queue length) takes values on a state space $S_{\overrightarrow{X}}$ which can be partitioned as $S_{\overrightarrow{X}}=\cup _{q=1}^{Q}S(q)$, where S(q), for 1 ≤ q ≤ Q, are disjoint classes. Consider a finite family of PH distributions with representation $(\boldsymbol{\alpha }(q),\mathbf{T}(q))$ of order s(q), for 1 ≤ q ≤ Q. We also define $\mathbf{t}^{0}(q)=( \mathbf{I}_{s(q)}-\mathbf{T}(q))\mathbf{e}_{s(q)}$, for 1 ≤ q ≤ Q. Assume that at time t + the PH distribution of index q(t + ) is in progress and let it evolve to absorption. Then, as with the discrete PH renewal process (Latouche and Ramaswami 1999), the arrival process is automatically restarted by choosing a PH distribution whose index q ^′ depends on the system state at the absorption time. The arrival pattern defined in this way is called a state-dependent PH arrival process.

We now introduce the notation of the D-BSDA distribution governing the state-dependent PH arrival process.

Without loss of generality we may assume that the description of the fundamental system state is one-dimensional. It means that k = 1 and $\overrightarrow{x}=i\in S_{\overrightarrow{X}}$. For example, we consider $S_{\overrightarrow{X}}=\{0,1,...\}$ to represent an infinite capacity queue.

The phase vector $\overrightarrow{y}=(q,j)$, for 1 ≤ q ≤ Q and 0 ≤ j ≤ s(q), has dimension l = 2 and denotes the index of the PH distribution in progress and its current phase.

During a time slot we may have 0 or 1 arrivals, so p = 1 and $\overrightarrow{n}=a$, where a denotes the number of arrivals. We notice that $S_{\overrightarrow{N}|_{(i,q,j)}}=\{0,1\}$, for any initial state $( \overrightarrow{x},\overrightarrow{y})=(i,q,j)$.

The kernel of the D-BSDA distribution is given by the matrices $\{\mathbf{D}_{i}^{a}; i\geq 0$, 0 ≤ a ≤ 1}, with elements

$$ \mathbf{D}_{i}^{0}(q,j;q^{\prime },j^{\,\prime})=\delta _{qq^{\prime }}T_{jj^{\,\prime}}(q),\text{ }i\geq 0, $$

(7.3)

$$ \mathbf{D}_{i}^{1}(q,j;q^{\prime },j^{\,\prime})=\delta _{\overline{i} q^{\prime }}t_{j}^{0}(q)\alpha _{j^{\,\prime}}(q^{\prime }),\text{ }i\geq 0, $$

(7.4)

where $\overline{i}$ denotes the index q such that $i\in S( \overline{i})$.

In the light of expression (7.3), we observe that matrices $\mathbf{D} _{i}^{0}$ (no arrival case) do not depend on i ≥ 0. In fact, $\mathbf{D} _{i}^{0}=diag\left( \mathbf{T}(1),...,\mathbf{T}(Q)\right)$, for all i; that is, the matrix describing the no arrival case is diagonal with sub-blocks T(q), for 1 ≤ q ≤ Q. According to our construction, $\mathbf{D}_{i}^{1}(q,j;q^{\prime },j^{\,\prime})$ in Eq. 7.4 is the probability that the PH(q) distribution in progress reaches its absorption and a phase is automatically assigned to the next PH(q ^′) interval. The index q ^′ of the restarted PH distribution depends on the fundamental state i.

For 1 ≤ q ^′ ≤ Q, let D ¹(q ^′) be the matrix having non-zero block entries $\mathbf{t}^{0}(q)\boldsymbol{\alpha} (q^{\prime })$, for 1 ≤ q ≤ Q, at the q ^′th column-block. Since $S_{\overrightarrow{X}}$ has been partitioned into Q classes, for each i the matrix $\mathbf{D}_{i}^{1}$ amounts one of the matrices D ¹(q ^′). Thus, the kernel reduces only to Q + 1 different matrices.

• (2) A state-dependent IPB arrival process

An ATM multiplexer transmits incoming cells (i.e., arrivals) generated by M input sources. All incoming cells are queued in a shared buffer of capacity K. In the literature (Hong et al. 2002; Lenin 2006), it is assumed that cells arrive from each source according to a homogeneous interrupted Bernoulli process (IBP). In what follows, we propose an extension of the IBP where arrivals depend on the queue length, which gives a chance to control the system congestion. Alternatively, we may consider that an incoming cell is admitted and consequently queued with a probability that depends on the system state. For simplicity, we assume that there is no need of introducing phases to model the service time distribution.

In this context, the underlying D-BSDA distribution has dimensions k = l = p = 1. The system state is described by the pair $(\overrightarrow{x}, \overrightarrow{y})=(i,m)$, where i and m respectively denote the queue size and the number of ON sources at time t +. Then, the system state is $S_{\overrightarrow{X},\overrightarrow{Y}}=S=\{(i,m); 0\leq i\leq K, 0\leq m\leq M\}$. The number of arrivals, a, takes values on $S_{ \overrightarrow{N}|_{(i,m)}}=\{0,...,m\}$; that is, in a time slot the number of incoming cells is bounded by the number of ON sources.

Each input source in a slot takes either ON state or OFF state. When an input source is in ON state and the queue size is i, one cell is generated with probability g _i . If the source is in OFF state, then no cell is generated. Suppose also that any OFF (or ON) source in a time slot change to the ON (or OFF) state with probability p (or q) in the next slot.

Now the kernel of the D-BSDA distribution is given by the matrices $\{\mathbf{D}_{i}^{a}; 0\leq i\leq K, 0\leq a\leq M\}$, with elements

$$ \mathbf{D}_{i}^{a}(m;m^{\prime })=\binom{m}{a} g_{i}^{a}(1-g_{i})^{m-a}f_{mm^{\prime }},\text{ }0\leq a\leq m, $$

(7.5)

where $f_{mm^{\prime}}$, for 0 ≤ m,m ^′ ≤ M, is given by

$$ f_{mm^{\prime }}=\sum\limits_{k=0}^{m}\binom{m}{k}q^{k}(1-q)^{m-k}\binom{M-m }{m^{\prime }+k-m}p^{m^{\prime }+k-m}(1-p)^{M-m^{\prime }-k}. $$

The binomial term in Eq. 7.5 is the probability of a arrivals during the current slot given that the system state is (i,m). On the other hand, $f_{mm^{\prime}}$ is the probability that in the next time slot there will m ^′ ON sources given that in the current slot there are m.

Appendix B

We next show the structure of the one-step transition probability matrix P (see formula (4.1)) for the simple case K = 2, s = 2 and M = N = 1.

In this case, we have

$$ \mathbf{P=}\left( \begin{array}{cc} \mathbf{P}_{00}\,\,\,\, & \mathbf{P}_{01} \\ \mathbf{P}_{10}\,\,\,\, & \mathbf{P}_{11} \end{array} \right) , $$

where the blocks $\mathbf{P}_{ii^{\prime}}$ of dimension g are given by

$$ \mathbf{P}_{00}=\left( \begin{array}{ccc} \mathbf{D}_{00}^{0,0}\,\,\,\, & \mathbf{D}_{00}^{1,0}\alpha _{1}\,\,\,\, & \mathbf{D} _{00}^{1,0}\alpha _{2} \\[3pt] t_{1}^{0}\mathbf{D}_{01}^{0,0}\,\,\,\, & t_{1}^{0}\mathbf{D}_{01}^{1,0}\alpha _{1}+T_{11}\mathbf{D}_{01}^{0,0}\,\,\,\, & t_{1}^{0}\mathbf{D}_{01}^{1,0}\alpha _{2}+T_{12}\mathbf{D}_{01}^{0,0} \\[3pt] t_{2}^{0}\mathbf{D}_{02}^{0,0}\,\,\,\, & t_{2}^{0}\mathbf{D}_{02}^{1,0}\alpha _{1}+T_{21}\mathbf{D}_{02}^{0,0}\,\,\,\, &t_{2}^{0}\mathbf{D}_{02}^{1,0}\alpha _{2}+T_{22}\mathbf{D}_{02}^{0,0} \end{array} \right) , $$

$$ \mathbf{P}_{01}=\left( \begin{array}{ccc} \mathbf{0}_{h\times h}\,\,\,\, & \mathbf{D}_{00}^{2,0}\alpha _{1}\,\,\,\, & \mathbf{D} _{00}^{2,0}\alpha _{2} \\ \mathbf{0}_{h\times h}\,\,\,\, & T_{11}\mathbf{D}_{01}^{1,0}\,\,\,\, & T_{12}\mathbf{D} _{01}^{1,0} \\ \mathbf{0}_{h\times h}\,\,\,\, & T_{21}\mathbf{D}_{02}^{1,0}\,\,\,\, & T_{22}\mathbf{D} _{02}^{1,0} \end{array} \right) , $$

$$ \mathbf{P}_{10}=\left( \begin{array}{ccc} \mathbf{0}_{h\times h}\,\,\,\, & \mathbf{D}_{10}^{0,1}\alpha _{1}\,\,\,\, & \mathbf{D} _{10}^{0,1}\alpha _{2} \\ \mathbf{0}_{h\times h}\,\,\,\, & t_{1}^{0}\mathbf{D}_{11}^{0,1}\alpha _{1}\,\,\,\, & t_{1}^{0}\mathbf{D}_{11}^{0,1}\alpha _{2} \\ \mathbf{0}_{h\times h}\,\,\,\, & t_{2}^{0}\mathbf{D}_{12}^{0,1}\alpha _{1}\,\,\,\, & t_{2}^{0}\mathbf{D}_{12}^{0,1}\alpha _{2} \end{array} \right) , $$

$$ \mathbf{P}_{11}=\left( \begin{array}{ccc} \mathbf{D}_{10}^{0,0}\,\,\,\, & \left( \mathop{\textstyle \sum }_{r=0}^{1}\mathbf{D}_{10}^{1,r}\right) \alpha _{1}\,\,\,\, & \left( \mathop{\textstyle \sum }_{r=0}^{1}\mathbf{D}_{10}^{1,r}\right) \alpha _{2} \\ t_{1}^{0}\mathbf{D}_{11}^{0,0}\,\,\,\, & T_{11}\left( \mathop{\textstyle \sum }_{r=0}^{1}\mathbf{D} _{11}^{0,r}\right)\,\,\,\, & T_{12}\left( \mathop{\textstyle \sum }_{r=0}^{1}\mathbf{D}_{11}^{0,r}\right) \\ t_{2}^{0}\mathbf{D}_{12}^{0,0}\,\,\,\, & T_{21}\left( \mathop{\textstyle \sum }_{r=0}^{1}\mathbf{D} _{12}^{0,r}\right)\,\,\,\, & T_{22}\left( \mathop{\textstyle \sum }_{r=0}^{1}\mathbf{D}_{12}^{0,r}\right) \end{array} \right) . $$

To explicitly compute the sub-blocks $\mathbf{P}_{(i,j)(i^{\prime },j^{\,\prime})}$, it remains to specify the kernel of the D-BSDA distribution and the PH distribution for the service times. In Section 7 (see formulas (7.1) and (7.2)), we considered a kernel consisting of geometric arrivals and retrials which phases vary randomly. Moreover, a discrete law consisting at most of three slots was assumed as PH service distribution.