Markovian bulk-arrival and bulk-service queues with general state-dependent control

Markovian queues occupy a significant niche in applied probability. Indeed, Markovian queues play a very important role both in the development of general queueing models and in the theory and applications of continuous-time Markov chains. Good references, among many others, for the former are Asmussen [4], Gross and Harris [18], Kleinrock [23] and Medhi [27], and, for the latter, Anderson [1], Chung [12], Freedman [15] and Syski [36].

Within the framework of queueing theory, there are two particularly interesting topics which have attracted much attention. The first one is bulk queues, which have wide and very important applications. The theory of bulk queues, (including bulk arrivals and/or bulk service) has attracted extensive attention and is well developed. Note that bulk arrivals (sometimes to be called batch arrivals) and bulk service queues are commonplace in scenarios such as industrial assembly lines, road traffic flow, the movement of aircraft passengers, etc., and thus the related models have extensive and important applications. One important reference in this topic is the good monograph by Chaudhry and Templeton [7]. For advances in this topic, see Armero and Conesa [2], Arumuganathan and Ramaswami [3], Chang, Choi and Kim [6], Fakinos [14], Srinivasan, Renganathan and Kalyanaraman [33], Sumita and Masuda [35] and Ushakumari and Krishnamoorthy [37], Stadje [34], V. Ramaswami [31], Lucantoni [25] and many others. The second topic is state-dependent input and output mechanisms, usually called state-dependent controls, which have also attracted considerable attention. For example, Chen and Renshaw [10, 11] considered models which have allowed the possibility of clearing the entire workload.

It seems that it is Chen et al. [8] who first combined the two modifications together by interweaving the bulk-arrival and bulk-service queues with state-dependent control either at idle time or at time with empty waiting line, which thus generalises the Chen-Renshaw models [10, 11] to make them more relevant and applicable. See also Chen et al. [9].

The model discussed in Chen et al. [8] has close links with so-called negative arrivals. It seems to us that Gelenbe [16] and Gelenbe, Glynn and Sigman [17] first introduced the particularly useful concept of negative arrivals, and this was followed up by many other authors, including Bayer and Boxma [5], Harrison and Pitel [19], Henderson [20] and Jain and Sigman [22]. The model also has close theoretical links with the versatile Markovian arrival processes introduced by Neuts [28], which includes several kinds of batch-arrival processes. Additionally, Neuts [29] described a number of interesting batch-arrival models together with useful methods for analysing them. For further developments, see, for example, Lucantoni and Neuts [26], Nishimura and Sato [30] and Dudin and Nishimura [13].

However, the model discussed in Chen et al. [8] has the serious limitation that the control effect only happens when the queue is empty or has only one customer, which prevents the model from having extensive applications. The main aim of this paper is therefore to overcome this limitation. That is, the current paper combines the bulk-arrival and bulk-service mechanism with general state-dependent control. More specifically, based on the bulk-arrival and bulk-service structure, the control effect can happen for arbitrary (but finitely) many states. This makes the model have much wider applications.

We now give a formal definition of our model. Our model is a Markovian one and thus the model is usually specified by an infinitesimal q-matrix; see Asmussen [4] or Anderson [1]. Now our model discussed in this paper has an infinitesimal q-matrix \(Q=\{q_{ij};\, i,j \in Z_+\}\), where \(Z_+=\{0,1,2\ldots \}\), which takes the following form: there exist a positive integer \(N\ge 1\) such thatwhereBy the above definition, we see that the underlying structure of our model is something like an \(M^X/M^Y/1\) queue. However, mainly due to the applications, the state-dependent input and output mechanisms have been incorporated into this underlying structure. Intuitively speaking, this extra structure indicates that when the queueing length is less than some specific level, N say, then the manager may wish, randomly, to move some “customers” or “workload”, stored somewhere else, to the system in order to speed up working and thus make the work more effectively. However, because of applications, we shall not only consider increasing working loads but also consider some other aspects, and thus make the extra structure arbitrary. This is exactly the reason why we name this extra structure “state-dependent control” even though it may not be an appropriate terminology. It should be noticed that due to this arbitrary effect, the underlying structure is seriously affected and, in particular, the original “arrival process” and “service process” are closely interwoven and correlated with each other. This makes some traditional methods and techniques, including the powerful matrix-analytic method, less effective in analysing our current model. It should be also noticed that our model is a Markovian one and thus our model has more deep properties than, for example, the M/G/1-type Markov chains. Being a Markovian model, we also have more powerful methods and techniques such as Kolmogorov backward and forward equations and Ito’s excursion law which enable us to get many more fruitful results than for the M/G/1-type Markov chains, say.

Another reason for us to use the current approach is that the results obtained in this paper open the door and paves the way to study another advanced and extremely important topic of quasi-limiting distributions, including determining the decay parameter and invariant measure, and functions which reveal deep properties regarding transient behaviour of our current queuing models. It is remarkable that the quasi-limiting behaviour is quite different for the continuous time and discrete time processes. Therefore, as far as quasi-limiting distributions are concerned, the behaviour of our current model is also remarkably different to say, the M/G/1-type Markov chains. We shall discuss this topic in a couple of subsequent papers.

The paper is organised as follows: In Sect. 2, we present a fundamental construction theorem together with some key lemmas. All later developments depend heavily on these results. Section 3 concentrates on discussing the so-called stopped queue with both bulk arrivals and bulk service. A delicate structure is revealed. The results obtained in this section regarding stopped queues are not only of their own interest but also crucial in our later analysis. Sect. 4 concerns questions of recurrence and ergodicity, as well as equilibrium distributions. The generating function of the equilibrium distribution is derived. The queue length distribution is also derived in this section. In Sect. 5, the bulk-arrival and bulk-service queues stopped at the idle state is analysed in detail, which paves the way to study the busy period distributions. In Sect. 6, the important hitting time distributions and the busy period distributions are discussed. Many important properties regarding these hitting time distributions and busy period distributions are revealed and many deep results regarding these distributions are presented. In the final Sect. 7, an example is provided to illustrate the conclusions obtained in the previous sections.

We first pack our known data specified in (1.1)–(1.3) into a few generating functions. For any \(0 \le i\le N-1\), defineandHere we view B(s) and \(H_i(s)~(0\le i\le N-1)\) as complex functions. Note that B(s) and \(H_i(s)~(0\le i\le N-1)\) may have their (usually different) convergence radii. But due to conditions (1.2) and (1.3), they are all well-defined at least on the closed unit disc \(\{s;|s|\le 1\}\) and analytic on the open unit disc \(\{s;|s|<1\}\). Considering that the q-matrix Q given in (1.1)–(1.3) is conservative and bounded, the corresponding Q-process is unique and is just the Feller minimal Q-process. It follows that the Feller minimal Q-resolvent \(R(\lambda )\) satisfies both the Kolmogorov backward and the forward equations. Using the Kolmogorov forward equations, we immediately obtain the following construction theorem which will be the starting point for our further analysis.

By Theorem 2.1, particularly by (2.24), it is clear we need to define, for each \(\lambda >0\),which is \(C^{\infty }\) at least on \((-1,1)\). Similarly to B(s), we view \(B_{\lambda }(s)\) as complex functions of s and note that \(B_\lambda (s)\) is well defined, at least on the closed unit disc \(\{s; |s|\le 1\}\), and is analytic on the open unit disc \(\{s; |s|< 1\}\).

We now provide a couple of fundamental lemmas which will be our stepping stones for further analysis. To make these lemmas, as well as the conclusions obtained thereafter, enjoy probabilistic meanings, letdenote the mean “arrival” and “service” rates, respectively. Note that \(0< m_d< +\infty \) and \(0<m_b\le +\infty \). Clearly,which explains the probabilistic meaning of \(B'(1)\).

The following conclusions are just corollaries of results obtained in Li and Chen [24].

For the full proof of these important conclusions, readers could consult Li and Chen [24]. Clearly the key point in Lemma 2.2 is the fact that there exist exactly N roots on the unit open disc \(\{s;|s|<1\}\). This important fact can be easily proved by using Rouché’s Lemma. Indeed, by Rouché’s lemma we can show that \(B_{\lambda }(s)\) and \(f(s)=s^N\) have the same number of zeros within the open disc \(\{s;|s|<1\}\). For details, see Li and Chen [24].

The most important root is the positive root \(u_0(\lambda )\) of the equation \(B_{\lambda }(z)=0\) on [0, 1). Hence, in the following lemma, we concentrate on discussing important properties of \(u_0(\lambda )\). First note that \(u_0(\lambda )\) is defined only for \(\lambda >0\) at the current stage.

In this section, we assume that all the first N states are absorbing. That is, we assume \(h_{ij}\equiv 0\) \((0\le i\le N-1,~j\ge 0)\) and thus \(H_i(s)\equiv 0\) \((0\le i\le N-1)\). The corresponding q-matrix is denoted by \(Q^*\). There are two main reasons for us to study the \(Q^*\)-process first. On the one hand, the properties of the \(Q^*\)-process will serve as a tool in investigating the main models which will be discussed in detail in the following sections and, on the other hand, to study the corresponding \(Q^*\)-process has its own interests since it can be viewed as a model of generalised Markov branching processes rather than a queueing model. Just because of the latter reason, we shall use some notation and terminology, such as extinction probabilities, within this section. For both reasons, we are now interested in getting closed forms of the Feller minimal \(Q^*\)-resolvent \(\{\phi _{ik}^{*}(\lambda )\}\).

By Lemma 2.2 the equation \(B_\lambda (s)=0\) has exactly N roots on the open unit disc \(\{s; |s|< 1\}\), denoted by \(\{u_0(\lambda ), u_1(\lambda ),\ldots ,u_{N-1}(\lambda )\}\), with \(u_0(\lambda )\) as the unique positive root on (0, 1). We now use the N-dimensional column vector \(\mathbf{U} (\lambda )\) to denote these N roots, i.e.,where T denote the transpose. Also, for any non-negative integers \(k\ge 0\), denoteOf course \(\mathbf{U} ^{1}(\lambda )=\mathbf{U} (\lambda )\) and \(\mathbf{U} ^0(\lambda )=\mathbf{1 }\); here \(\mathbf{1 }\) denotes the column vector whose components are all one. Similarly, we letdenote the N roots of the equation \(B(s)=0\) on the unit closed disc \(\{s; |s|\le 1\}\) as Lemma 2.1 shows. In fact, we have \(\lim _{\lambda \rightarrow 0}{} \mathbf{U} (\lambda )=\mathbf{U} (0)\). In many cases, we shall just denote \(\mathbf{U }=\mathbf{U} (0)\), or in component form,Note that, however, if \(m_d<m_b\le +\infty \), \(u_0(0)=u<1\), and the trivial root 1 is not included in (3.3). As a result, all the component of \(\mathbf{U} (0)\equiv \mathbf{U }\) are totally distinct. It is also convenient to denoteThat is, \({\hat{\mathbf{U }}}(\lambda )\) is nothing but cutting off the first component of \(\mathbf{U} (\lambda )\). Similarly, \(\hat{\mathbf{U }}(0)\hat{=}\hat{\mathbf{U }}\) is just the column vector obtained by cutting off the first component from U. Finally, the determinant of the \(N\times N\) matrix \((\mathbf{1 },\mathbf{U} (\lambda )\), \(\mathbf{U} \,^2(\lambda ),\ldots ,\mathbf{U} \,^{N-1}(\lambda )\)) will be denoted byBy applying properties regarding determinants, it is easy to see that \(\Delta (\lambda )\) defined in the above (3.6) can be rewritten in the following forms:Also, for any \(0\le i\le N-1\), denoteBy comparing the forms of \(\Delta (\lambda )\) and \(\Delta ^{(i)}(\lambda )\) given in (3.6) and (3.8), we see that they are the same except that the kth column of the former is replaced by the ith column of the latter.

Keeping the above notation in mind, we may claim the following simple yet interesting result.

It is interesting to note that (3.10) and (3.11) provide a perfect solution to our process, since the whole resolvent can be simply expressed by the N roots of the known equation \(B(s)-\lambda s^N=0\). Therefore the expressions (3.10) and (3.11) will be extremely helpful in analysing the properties of the \(Q^{*}\)-process. In particular, let \(\{Z^{*}_{t},\, t\ge 0\}\) be the \(Q^{*}\)-process. Define the hitting time to state i \((i=0,1,\ldots , N-1)\) as \(\tau ^{*}_{i} = \inf \{ t>0: Z^{*}_{t}=i \}\) if \(Z^{*}_{t}=i\) for some \(t>0\), and \(\tau ^{*}_{i}=\infty \) if \(Z^{*}_{t}\ne i\) for all \(t>0\). Also, let \(\tau ^{*}:=\wedge _{k=0}^{N-1} \tau ^{*}_k\) denote the “overall” hitting time (the hitting time to the absorbing set \(\{0,1,\ldots , N-1\}\) ). We are now ready and interested in getting these important quantities. We first claim the following simple lemma.

As a consequence of Lemma 3.1, we have the following corollary which can be more easily applied to our later analysis.

We are now ready to reveal properties of the \(Q^{*}\)-process. Let \(P^{*}(t)=\{p_{ij}^{*}(t);\,i,j\ge 0\}\) be the \(Q^{*}\)-function. Thus, for any \(n\ge N\), \(0\le k\le N-1\), \(p_{nk}^{*}(t)\) is just the cumulative distribution function of \(\tau _k\) given that the \(Q^{*}\text {-}\)process starts at state \(n\ge N\). That is,Here we have denoted \({\mathbb {P}}\{\tau _k^{*}\le t|Z_0^{*}=n\}\) as \({\mathbb {P}}_n\{\tau _k^{*}\le t\}\).

Let \(a_{nk}^{*}={\mathbb {P}}_n\{\tau _k^{*}< \infty \}\) \((n\ge N;~0\le k\le N-1)\), and thus \(a_{nk}^{*}\) is the hitting probability to state \(k~(0\le k\le N-1)\), given that the process \(\{Z_t^{*}(\omega );~t\ge 0\}\) starts at the state \(n\ge N\). We shall use the terms “extinction probabilities”, etc., as though the \(Q^{*}\text {-}\)process is some kind of branching processes. Similarly, let \(a_{n}^{*}={\mathbb {P}}_n\{\tau ^{*}< \infty \}\) be the overall “extinction probability” to the “extinction set” \(\{0,1,2,\ldots ,N-1\}\) given that the \(Q^{*}\text {-}\) process starts at state \(n \ge N\). Now we may claim the following important conclusion by noting that \(\sum _{j=0}^{\infty }\phi ^* _{ij}(\lambda )s^j\) is the Laplace transform of the generating function of transition probabilities \(p_{ij}^{*}(t)\); i.e.,

By Theorem 3.2, we see that if \(m_d \ge m_b\), then the \(Q^*\)-process will definitely go to extinction. We are therefore interested in obtaining the mean extinction time, i.e., \({\mathbb {E}}(\tau ^{*}|Z_0^{*}(\omega )=n)\), where \(n\ge N\). We shall denote this important quantity as \({\mathbb {E}}_n(\tau ^{*})~(n\ge N).\)

By Theorem 3.3 we see that, if \(m_d < m_{b}\le \infty \), the mean time to overall extinction \({\mathbb {E}}_n(\tau ^{*})\) is infinite which is quite informative. The main reason is that if \(m_d < m_{b}\le \infty \) then the overall extinction probability \(a^{*}_{n}\) is strictly less than 1. This prompts us to consider the expectation of \(\tau ^{*}\) conditional on \(\tau ^{*}<\infty \) which will be much more informative.

In order to state our results more simply, we give the following non-standard definition: Suppose \(A=\{a_1, a_2, \ldots , a_n\}^\mathrm{T}\) and \(B=\{b_1, b_2, \ldots , b_n\}^\mathrm{T}\) are two n-dimensional column vectors. We defineas a new n-dimensional column vector.

Expressions (3.9)–(3.11) in Theorem 3.1 can also be used to obtain information about the mean function of the \(Q^*\)-process. Let \(m^*_i(t)={\mathbb {E}}_i(Z_t^{*}(\omega ))\) and denote its Laplace transform by \(\xi ^*_i(\lambda )\). Then, since \(m^*_i(t)=\sum _{k=1}^{\infty }kp^*_{ik}(t)\), we havefrom which we can obtain the following result.

After studying the properties of the \(Q^*\)-processes in the previous section, we are now ready to study the full queueing model with bulk-arrival-bulk-service and general state-dependent control; this is the key section of this paper. Our q-matrix Q now takes the general form (1.1)−(1.3) with the feature that for all \(0\le k\le N-1\) we have \(h_{kk}\ne 0 \). It follows that \(H_k(s)\ne 0~(0\le k\le N-1)\). For convenience, we shall further assume that the q-matrix Q is irreducible. It follows that the (unique Feller minimal) Q-function and Q-resolvent are also irreducible. Denote the corresponding queueing process as \(\{X_t;~t\ge 0\}\).

This section consists of three sub-sections. In the first sub-section, we consider the structure of the Q-resolvent of the queueing process \(\{X_t;~t\ge 0\}\) which is the main tool to investigate properties of our full queuing model. Based on the conclusions obtained in the first sub-section, the ergodic property, particularly the equilibrium distribution, which is always one of the most important topics for all kinds of queuing models, is fully discussed in the second sub-section. In the final sub-section, the queue length distribution, which is also one of the important topics in queuing models, is investigated in detail. Another extremely important topic, the busy period distribution, will be investigated in the following Sects. 5 and 6.

In this, as well as the following, section, we concentrate on discussing the very important topic of the busy period distribution. As a preparation, the probabilistic laws regarding hitting time to the idle state zero are firstly revealed in this section. To achieve this aim, we need to study a special structure of our queueing model. More specifically, we need to examine the structure of the regular process \(\{Y_t;\,t\ge 0\}\) whose (regular) q-matrix \({\hat{Q}}\) is defined in (1.1)−(1.3) but with the special feature of \(\{h_{0j}\}=0~(\forall j\ge 0)\), but \(h_{ii}\ne 0~(1\le i\le N-1)\). We shall immediately see that there exists a close relationship between this process and our full queuing model. In particular, the hitting properties of the process \(\{Y_t;\,t\ge 0\}\) are directly related to the busy period properties of our full queueing process \(\{X_t\}\). For convenience, as in Sect. 3, let’s view \(\{Y_t;\,t\ge 0\}\) as a branching process and thus use terms such as extinction probability etc. In fact, the process \(\{Y_t;\,t\ge 0\}\) can indeed be viewed as a generalised Markov branching process with state-dependent immigration, and thus also has its own interest.

Similar to in Sect. 3, we now again provide a special structure for the \({\hat{Q}}\)-resolvent of the process \(\{Y_t;\,t\ge 0\}\). But this is easy. In fact, it is simply a direct consequence of Theorem 2.1. Indeed, letting \(H_0(s)=0\) in Theorem 2.1 immediately yields the following conclusion.

Similar to \({\hat{{\textit{\textbf{U}}}}}(\lambda )\) defined in (3.5), we define \({\hat{{\textit{\textbf{V}}}}_k(\lambda )}\) as follows:Note that the component form of \(\hat{{{\textit{\textbf{V}}}}}_k(\lambda )\) is, for \(0\le k\le N-1\),We now use Theorem 5.1 to investigate properties of the branching process (named as above) \(\{Y_t;\,t\ge 0\}\). Denote by \(\tau \) the extinction time to state 0, i.e. \(\tau :=\inf \{t>0;\, Y_t=0\}\) if \(Y_t=0\) for some \(t>0\) and \(\tau =\infty \) otherwise. Letanddenote the conditional distribution of the extinction time and the extinction probability, respectively, conditional on the process starting at state \(i\ge 1\). By noting that the Laplace transforms of \(q_{i0}(t)\) are actually given in (5.3), we can immediately obtain the following important result.

By the above theorem, we see that if \(m_d\ge m_b\), then, starting from any state \(i\ge 1\), the process will go to extinction with probability one. Therefore we are interested in obtaining the mean extinction times \({\mathbb {E}}_i(\tau )\ (i\ge 1)\) as well as the conditions under which these mean extinction times are finite.

Theorem 5.3 provides useful information regarding the mean extinction time \({\mathbb {E}}_i(\tau )\) under the conditions that \(m_d > m_b\) and \(H_k'(1)<+\infty \)\((1\le k\le N-1)\). If \(m_d<m_b\le +\infty \), then \({\mathbb {E}}_i(\tau )\) (for all \(i\ge 1\)) will be infinite since, by the above Theorem 5.2, the extinction probabilities \(q_{i0}<1\). To get rid of such uninteresting situations, we turn our attention to the conditional mean extinction time \({\mathbb {E}}_i(\tau \mid \tau <\infty )\) which will be much more informative.

We are now ready to consider the busy period distributions of our full queueing processes. As in Sect. 4, the full irreducible queueing process will be denoted by \(\{X_t;\,t\ge 0\}\) which denotes the number of customers in the queue (including the customers being served) at time \(t\ge 0\). Without loss of any generality, we assume that \(X_0=0\). We now define a sequence of random variables \(\{\sigma _n; \,n\ge 0\}\) as follows:and, for all \(n\ge 1\),It is clear that \(\{\sigma _n; \,n\ge 0\}\) is a sequence of increasing stopping times. Note that the random variables \(\{\sigma _{2n}-\sigma _{2n-1};\,n\ge 1\}\) are just the busy periods. By \(\hbox {It}{\hat{o}}\)’s excursion law, see [21], we know that \(\{\sigma _{2n}-\sigma _{2n-1};\,n\ge 1\}\) are independent identically distributed random variables. For more details regarding this important excursion law, one can also see Rogers and Williams [32]. Our main aim now is to find this common distribution. Now, by the strong Markov property, we have, for all \(n\ge 1\),However, under the condition that \( X_{\sigma _1}=k~~(k\ge 1)\), the distribution of \(\sigma _{2}-\sigma _{1}\) is just the extinction distribution of the \(\{Y_t(\omega );\,t\ge 0\}\) discussed in the previous Sect. 5. In other words, the Laplace transform of the conditional distribution \({\mathbb {P}}(\sigma _{2}-\sigma _{1}\le t\mid X_{\sigma _1}=k)\) is just \(\phi _{k0}(\lambda )~(k\ge 1)\) given in (5.3). More specifically, let T denote the length of the first busy period (recall that all busy periods are independent identically distributed random variables, and thus, it is enough to consider T only), and denote the cumulative distribution function of T byWe then can find the Laplace transform of \(F_k(t)\), i.e. \(\int _0^{\infty }e^{-\lambda t}\hbox {d}F_k(t)\), using the results obtained in the previous section. To this end, we claim the following conclusion.

We now can claim the following important conclusion.

By (6.4), we may get many properties regarding busy periods. For example, we may get an elegant form of the mean time of the busy periods as follows: The importance of such conclusions in queueing theory is well known.

If \(m_d<m_b\le +\infty \), then, although the mean busy period is infinite, we may still get much information by providing an expression regarding the conditional mean busy period. By the i.i.d property, it is enough to consider \({\mathbb {E}}(\sigma _2-\sigma _1\mid \sigma _2<\infty )\) only.

It is interesting to note that, as Theorem 6.4 shows, so long as \(m_d<m_b\le +\infty \), the conditional expectation of the busy period is always finite even if \(B'(1)\) and all \(H_k'(1)\ (0\le k\le N-1)\) are infinite. Note that, however, neither Theorem 6.3 nor Theorem 6.4 covers the case \(m_d=m_b\). In fact, this critical case is more subtle. Indeed, although the busy period is almost surely finite, the expected value is infinite. Nevertheless, we are still able to provide interesting information regarding the asymptotic behaviour of the busy period for this subtle case. For this purpose, we first provide the following lemma. Recall that \(\{\phi _{k0}(\lambda ); \,k\ge 1\}\) given in (5.3) is the Laplace transforms of the \(p_{k0}(t)\) for the absorbing process \(\{Y_t;\,t\ge 0\}\) which was constructed in Theorem 5.1.

We are now able to tackle the subtle case of \(m_d=m_b\).

In this final section, we provide an example to illustrate conclusions obtained in the previous sections. For notational convenience, we shall use k to replace N in this section. We assume that the control sequences \(\{h_{ij}\}\) takes a birth-death structure of the following forms: for any \(0\le i\le k-1\),Here we assume \(\mu _{-1}= \mu _0=0,~\mu _i>0~(1\le i\le k-1)\) and \(\lambda _i>0~(0\le i \le k-1)\). Hence, by (7.1) we have obviouslyfor \(1\le i \le k-1\),and thusWe further assume that the arrival-service sequence \(\{b_j;j\ge 0\}\) takes the following special forms:and hence, by (7.5), all the other components of the sequence \(\{b_j;j\ge 0\}\) are just zero. Hence this example is a queueing model with a Poisson arrival rate \(b > 0\) together with a full loading service rate \(a>0\). That is, suppose the service is taken by some vans, say, then only when the van is fully packed will the van begin taking service. For this example, we know thatwith \(m_d= ka\) and \(m_b= b\), and, for any \(\lambda >0\),By the results obtained before, we know that \(B_\lambda (s)\) has a unique positive zero in (0, 1), denoted by \(u(\lambda )\), together with the other \((k-1)\) zeros on the open unit disc \(\{z;|z|<1\}\). Note that, however, for this special case, \(B_\lambda (s)\) is simply a polynomial with degree \((k+1)\), and thus \(B_\lambda (s)\) has exactly \((k+1)\) zeros. By noting that \(B_\lambda (1)=-\lambda <0\) and \(\lim \limits _{s\rightarrow +\infty }B_\lambda (s)=+\infty \), we know that, in addition to the k zeros on the open unit disc \(\{z;|z|<1\}\), \(B_\lambda (s)\) has a unique positive zero on \(\{z;|z|>1\}\). We now let \(v(\lambda )\) denote this zero. Therefore, \(B_\lambda (s)\) has exactly two positive zeros \(u(\lambda )\) and \(v(\lambda )\) which satisfy, for all \(\lambda >0\),Similarly, B(s) is also a polynomial with degree \((k+1)\) and thus has exactly \((k+1)\) zeros on the complex plane. By noting that \(B'(1)=b-ka\), we can get the following simple conclusion.

Using the notation introduced in (3.4) and (3.5), we then havewhere \((\omega _1,\omega _2,\ldots \omega _{k-1})\) are the \((k-1)\) zeros of B(s) specified in (ii) of the above Lemma 7.1. We can now obtain many results regarding this example by citing conclusions obtained in the previous sections. However, we shall be mainly interested in the ergodic property and, in particular, the limiting distributions.

For this purpose, we introduce some new notations as follows:and, for \(1\le j\le k-1\),Hence \(\mathbf{D }^{(j)}~(0\le j\le k-1)\) are all \((k-1)\)-dimensional column vectors and the ith component of \(\mathbf{D }^{(j)}~(1\le j\le k-1)\) is justNow, the meaning of the following determinants should be self-explanatory:Applying the results obtained in the previous section, particularly Theorem 4.3, we can claim the following conclusion.

In order to get a more concrete and convincing example, we consider the further special case where \(\lambda _i\equiv \lambda >0\) \((0\le i\le k-1)\) and \(\mu _i\equiv \mu \) \((1\le i\le k-1)\). That is, we assume that the control sequence takes an M|M|1 structure. Then the forms (7.14)–(7.20) take particularly simple expressions. In particular, the Vandermonde determinant will play an interesting role. For simplicity, we further assume that \(k=3\). Then, as a direct consequence of Theorem 7.1, we can get the following simple yet interesting corollary.