Service with a queue and a random capacity cart: random processing batches and E-limited policies

Abstract

In this paper we examine a queueing model with Poisson arrivals, service phases of random length, and vacations, and its applications to the analysis of production systems in which material handling plays an important role. The length of a service phase can be interpreted as a “processing batch”, leading to a varying E-limited M/G/1 queue and the analysis is carried out separately for processing batch distributions with bounded and unbounded support. In the first case, standard techniques from the analysis of limited service systems are used, involving Rouché’s theorem, while in the second the analysis proceeds via Wiener–Hopf factorization techniques. Processing batches with size that is either geometrically distributed or distributed according to a combination of geometric factors lead to particularly simple solutions related to Bernoulli vacation models. In all cases, care is taken in the analysis in order to obtain the steady state distribution of the system under minimal assumptions, namely the finiteness of the first moment of the service and vacation distributions together with the stability condition. This is in contrast to most of the literature where usually the assumption that the service and vacation distribution is light-tailed is either explicitly stated or tacitly adopted.

Appendix

1.1 Stability

In this section of the Appendix we complete the discussion on the stability of the system by furnishing the proofs of the assertions made in the last paragraphs of Sect. 2. We begin by recalling Foster’s criterion for positive recurrence (see Asmussen 2003, p.19).

Theorem 7

Suppose that $(P_{ij})$ is the transition probability matrix of a discrete time Markov chain with countable state space E which is irreducible and let $E_0$ be a finite subset of E. Then:

• (i) the chain is recurrent if there exists a function $h:E\rightarrow \mathbf{R}$ which is not bounded on E and satisfies

  $$\begin{aligned} \sum _{k\in E}P_{jk}h(k)\le h(j),\quad j\notin E_0. \end{aligned}$$

• (ii) the chain is positive recurrent if for some $h:E\rightarrow \mathbf{R}$ and some $\epsilon >0$ we have $\inf _{j\in E}h(j)>-\infty$ and

  $$\begin{aligned} \sum _{k\in E}P_{jk}h(k)< & {} \infty ,\quad j\in E_0 \\ \sum _{k\in E}P_{jk}h(k)< & {} h(j)-\epsilon ,\quad j\notin E_0. \end{aligned}$$

We will apply the above theorem to the embedded Markov chain $\{{\varPhi }_m \}$ of the number of customers at the beginning of each active period which has state space $\mathbf{N}_0$ and is clearly irreducible. Taking $h(j)=j$ to be the identity function on E we shall establish that $\{{\varPhi }_m \}$ is positive recurrent by showing that $E[{\varPhi }_1 -{\varPhi }_0 \left| {\varPhi }_0 =k\right. ]<0$ for all k greater than some $k_0$.

1.1.1 Stability under the complete batch policy

We first show that when the stability condition (1) is satisfied then the system under the complete batch policy is stable. To this end, as we have already seen, it suffices to show that the embedded Markov chain $\{{\varPhi }_m \}$ is positive recurrent.

With the notation of Sect. 3 let $X_{d_n^m }$ be the number of customers in the system immediately after the nth service completion of the mth cycle. Let us set $X_{d_n^m }:=\chi _n^m$ and denote by $\xi _n^m$ the number of arrivals during the nth service time of the mth cycle (but excluding the arrival that initiates the service time if the server happens to be idle and waiting for a new arrival). Also, let $\zeta _m$ denote the number of Poisson arrivals during the vacation phase of the mth cycle. Then we clearly have

$$\begin{aligned} \chi _{n+1}^m= & {} \left( \chi _n^m -1\right) ^{+}+\xi _n^m \quad \text{ for }\quad n=0,1,2,\ldots ,{\varTheta }_m -1, \nonumber \\ \chi _0^m= & {} {\varPhi }_m , \end{aligned}$$

(80)

and hence

$$\begin{aligned} {\varPhi }_{m+1}=\sum _{n=1}^\infty \mathbf {1}({\varTheta }_m =n)\chi _n^m +\zeta _m. \end{aligned}$$

(81)

From (80) we have $\chi _{n+1}^m -\chi _n^m =-\mathbf {1}\left( \chi _n^m >0\right) +\xi _n^m$ and hence (81) can be written as

$$\begin{aligned} {\varPhi }_{m+1}= & {} {\varPhi }_m +\sum _{n=1}^\infty \mathbf {1}({\varTheta }_m \ge n)\left( \chi _n^m -\chi _{n-1}^m \right) +\zeta _m \nonumber \\= & {} {\varPhi }_m +\zeta _m +\sum _{n=1}^\infty \mathbf {1}({\varTheta }_m \ge n)\left( \xi _{n-1}^m -\mathbf {1}\left( \chi _{n-1}^m >0\right) \right) . \end{aligned}$$

(82)

Thus

$$\begin{aligned} E\left[ {\Delta } {\varPhi }_m \mid {\varPhi }_m =k\right]= & {} \lambda EG + \sum _{n=1}^\infty P({\varTheta }_m \ge n)\left( \rho -E\left[ \mathbf {1}\left( \chi _{n-1}^m >0\right) \mid {\varPhi }_m =k\right] \right) \\= & {} \lambda EG+\rho E{\varTheta } -\sum _{n=0}^\infty P({\varTheta }_m >n)E\left[ \mathbf {1} \left( \chi _n^m >0\right) \mid {\varPhi }_m =k\right] . \end{aligned}$$

Now, using the Dominated Convergence Theorem, we have

$$\begin{aligned}&\lim _{k\rightarrow \infty }\sum _{n=1}^\infty P({\varTheta }_m >n)E\left[ \mathbf {1} \left( \chi _n^m >0\right) \mid {\varPhi }_m =k\right] \nonumber \\&\quad =\sum _{n=0}^\infty P({\varTheta }_m >n)\lim _{k\rightarrow \infty } E\left[ \mathbf {1}\left( \chi _n^m >0\right) \mid {\varPhi }_m =k\right] = \sum _{n=0}^\infty P({\varTheta }_m >n)=E{\varTheta } \end{aligned}$$

(83)

where in the last equation we have used the fact that $\lim _{k\rightarrow \infty }E\left[ \mathbf {1}(\chi _n^m >0)\mid {\varPhi }_m =k\right] =1.$ Thus, (83) together with (1) implies that there exists $k_0 \in \mathbf{N}_0$ such that $E\left[ {\Delta } {\varPhi }_m \mid {\varPhi }_m =k\right] <0$ for all $k\ge k_0$. This in turn implies the positive recurrence of $\{{\varPhi }_m \}$. Again, the fact that the expected cycle time is finite implies the stability of the system itself.

It remains to show that, when the inequality in (1) is reversed, then the system is unstable. To establish this it is enough to show that the Markov chain $\{{\varPhi }_m \}$ is transient. We do this by means of a stochastic dominance argument as follows. Consider an auxiliary Markov chain $\{\widetilde{{\varPhi } }_m \}$ defined by means of the recursion

$$\begin{aligned} \widetilde{{\varPhi } }_{m+1}=\left( \widetilde{{\varPhi } }_m +\sum _{n=1}^{{\varTheta }_m }\xi _n^m -{\varTheta }_m \right) ^{+}+\zeta _m. \end{aligned}$$

(84)

We will now argue inductively that, if $\widetilde{{\varPhi } }_0 ={\varPhi }_0$ with probability 1, then

$$\begin{aligned} \widetilde{{\varPhi } }_m \le {\varPhi }_m \text{ w.p. } \text{1 } \text{ for } \text{ each } m\in \mathbf{N}_0. \end{aligned}$$

(85)

Indeed, suppose that (85) holds for a given value of m. Note that (82) can be written also as

$$\begin{aligned} {\varPhi }_{m+1}={\varPhi }_m +\zeta _m +\sum _{n=1}^{{\varTheta }_m }\xi _{n-1}^m -\sum _{n=1}^{{\varTheta }_m }\mathbf {1}\left( \chi _{n-1}^m >0\right) . \end{aligned}$$

Then,

$$\begin{aligned} {\varPhi }_{m+1}= & {} \zeta _m +\left( {\varPhi }_m +\sum _{n=1}^{{\varTheta }_m }\xi _{n-1}^m -\sum _{n=1}^{{\varTheta }_m }\mathbf {1}\left( \chi _{n-1}^m >0\right) \right) ^{+}\\\ge & {} \zeta _m +\left( {\varPhi }_m +\sum _{n=1}^{{\varTheta }_m }\xi _{n-1}^m -{\varTheta }_m \right) ^{+} \\\ge & {} \zeta _m +\left( \widetilde{{\varPhi } }_m +\sum _{n=1}^{{\varTheta }_m }\xi _{n-1}^m -{\varTheta }_m \right) ^{+}=\widetilde{{\varPhi } }_{m+1} \end{aligned}$$

and thus we establish the inductive step.

$\{\widetilde{{\varPhi } }_m \}$ can be thought of as the Markov chain describing the queue length in a system with batch arrivals and batch services. The stability condition for this system is $-E{\varTheta } +Eu+Ev<0$ or equivalently $(1-\rho )E{\varTheta } +\lambda EG<0$ which is precisely (1). Thus, when the inequality in (1) is reversed, the auxiliary system is unstable (the Markov chain $\{\widetilde{{\varPhi } }_m \}$ is transient–see Meyn and Tweedie 1993). The stochastic ordering relation between the auxiliary chain and the original system implies thus that (1) is also necessary for the positive recurrence of the Markov chain $\{{\varPhi }_m \}$ for the complete batch policy.

1.1.2 Stability under the partial batch policy

When the partial batch policy is used,

$$\begin{aligned} {\varPhi }_{m+1}={\varPhi }_m +\zeta _m +\sum _{n=1}^{L_m }\left( \xi _{n-1}^m -1\right) \end{aligned}$$

where $L_m =\min \left( {\varTheta }_m ,\inf \{i\ge 0:{\varPhi }_m +\sum _{n=1}^i (\xi _{n-1}^m -1)=0\}\right)$. Thus, if we define again the process $\{\widetilde{{\varPhi } }_m \}$ by means of (84) and we assume that $\widetilde{{\varPhi }}_m \le {\varPhi }_m$ then

$$\begin{aligned} {\varPhi }_{m+1}\ge & {} \zeta _m +\left( {\varPhi }_m +\sum _{n=1}^{L_m }\left( \xi _{n-1}^m -1\right) \right) ^+ \ge \zeta _m +\left( \widetilde{{\varPhi }}_m +\sum _{n=1}^{{\varTheta }_m }\left( \xi _{n-1}^m -1\right) \right) ^+ \\= & {} \widetilde{{\varPhi }}_{m+1}. \end{aligned}$$

This establishes inductively the stochastic ordering relationship (85) in the case where the partial batch policy is used. Thus when (1) holds with the sense of the inequality reversed $\{\widetilde{{\varPhi } }_m \}$ is transient and the stochastic inequality just established implies that $\{{\varPhi }_m \}$ is transient as well.

Here we have only considered the issue of stability. Geometric ergodicity can also be addressed in this framework using the techniques in Spieksma and Tweedie (1994).

1.2 Roots within the unit disk

Here we show that equation (19) has N roots within the unit disk. Variations of this equation abound in the bulk service literature. (See for instance Chaudhry and Templeton 1983 and also Coffman and Gilbert 1992.) However in these treatments it is (either explicitly or tacitly) assumed that the service and vacation distributions are light-tailed, i.e. that the corresponding moment generating functions exist in an open interval containing the origin. We assume only the natural conditions for the existence of a stationary version of the process i.e. the finiteness of first moments plus the stability condition. We will use the following theorem established in Boudreau et al. (1962). (See also the paper by Adan et al. 2006).

Theorem 8

(Boudreau, Griffin, and Kac) Suppose that $\varphi (z):=\sum _{n=0}^\infty f_n z^n$ is the p.g.f. of $f_n ,$ $n=0,1,2,\ldots ,$ a non-degenerate probability distribution on the non-negative integers with finite mean $\mu :=\sum _{n=0}^\infty nf_n$ and N is a natural number. If the condition

$$\begin{aligned} N>\mu \end{aligned}$$

(86)

holds, then the equation

$$\begin{aligned} z^n -\varphi (z)=0 \end{aligned}$$

(87)

has N roots within the unit disk $\{z\in \mathbf{C}:\left| z\right| \le 1\}$. $z=1$ is a single root of (87) while the remaining $N-1$ roots have modulus strictly smaller than 1.

The main idea of the proof is to show that the equation $z^n -w\varphi (z)=0$ has N roots within the unit disk when $0<w<1$ and then use a continuity argument to show that this remains true as $w\rightarrow 1.$ In our case, $\varphi (z)=D(z)\sum _{n=1}^n \theta _n z^{N-n}U(z)^n$ and $\mu =\varphi ^{\prime }(1)=EG-(1-\rho )E{\varTheta } +N,$ thus (86) is equivalent to the stability condition for the system (1).

1.3 Determination of the constants

In this section of the Appendix we give an explicit procedure for the computation of the N constants, $F_0 ,\ldots ,F_{N-1}$ in terms of the quantities $y_i ,$ $i=1,2,\ldots ,N-1,$ defined in (20), and C, defined in (26), in the case of the partial batch policy with finite cart capacity. These constants can be obtained from the identity (22) as follows. Let us denote by $S_k :=S_k (y_1 ,y_2 ,\ldots ,y_{N-1})$, the elementary symmetric functions in $N-1$ variables defined as

$$\begin{aligned} S_k = \sum _{1\le i_1 <i_2 <\cdots <i_k \le N-1}y_{i_1 }y_{i_2 }\cdots y_{i_k },\quad k=1,2,\ldots , N-1. \end{aligned}$$

Then $P(y):=C(y-1)\prod _{i=1}^{N-1}\left( y-y_i \right)$ can be expressed as

$$\begin{aligned} P(y)= & {} Cy^N -y^{N-1}C(1+S_1 )+y^{N-2}C(S_1 +S_2 )-y^{N-3}C(S_2 +S_3 )+\cdots \\&+\,(-1)^{N-1}yC(S_{N-2}+S_{N-1})+(-1)^N CS_{N-1}, \end{aligned}$$

where the constant C is given in (26). On the other hand, from (21),

$$\begin{aligned} P(y)= & {} y^N \left( \sum _{k=0}^{N-1}F_k \sum _{j=k+1}^N \theta _j \right) -y^{N-1}\left( \sum _{k=0}^{N-1}F_k \theta _{k+1}\right) -\cdots \\&-\,y^{N-i}\left( \sum _{k=0}^{N-i}F_k \theta _{k+i}\right) -\cdots -y^{2}\left( F_0 \theta _{N-2}+F_1 \theta _{N-1}+F_2 \theta _{N-2}\right) \\&-\,y\left( F_0 \theta _{N-1}+F_1 \theta _N \right) -F_0 \theta _N. \end{aligned}$$

Equating the coefficients of $y^i ,$ $i=0,1,\ldots ,N-1$, in the above equations we obtain the following triangular linear system which allows us to determine the constants $F_k$.

$$\begin{aligned} \theta _N F_0= & {} (-1)^{N-1}CS_{N-1} \\ \theta _N F_1 + \theta _{N-1}F_0= & {} (-1)^{N-2}C\left( S_{N-1}+S_{N-2}\right) \\&\vdots&\\ \theta _N F_i + \cdots + \theta _{N-i-1}F_1 + \theta _{N-i}F_0= & {} C(-1)^{N-i-1}\left( S_{N-i}+S_{N-i-1}\right) \\&\vdots&\\ \theta _N F_{N-2} + \cdots + \theta _3 F_1 + \theta _2 F_0= & {} -C\left( S_2 +S_1 \right) \\ \theta _N F_{N-1} + \theta _{N-1}F_{N-2} + \cdots + \theta _2 F_1 + \theta _1 F_0= & {} C\left( S_1 +1\right) \end{aligned}$$

(One additional equation, namely $F_0 \left( \theta _1 +\cdots +\theta _N \right) +\cdots +F_k \left( \theta _{k+1}+\cdots +\theta _N \right) +\cdots +F_{N-1}\theta _N =C$, which is obtained by equating the coefficients of $y^N$, is redundant since it can be obtained by adding all the N equations above and noting that the right hand side reduces to C.)