Analytical evaluation of the output variability in production systems with general Markovian structure

Abstract

Performance evaluation models are used by companies to design, adapt, manage and control their production systems. In the literature, most of the effort has been dedicated to the development of efficient methodologies to estimate the first moment performance measures of production systems, such as the expected production rate, the buffer levels and the mean completion time. However, there is industrial evidence that the variability of the production output may drastically impact on the capability of managing the system operations, causing the observed system performance to be highly different from what expected. This paper presents a general methodology to analyze the variability of the output of unreliable single machines and small-scale multi-stage production systems modeled as General Markovian structure. The generality of the approach allows modeling and studying performance measures such as the variance of the cumulated output and the variance of the inter-departure time under many system configurations within a unique framework. The proposed method is based on the characterization of the autocorrelation structure of the system output. The impact of different system parameters on the output variability is investigated and characterized. Moreover, managerial actions that allow reducing the output variability are identified. The computational complexity of the method is studied on an extensive set of computer experiments. Finally, the limits of this approach while studying long multi-stage production lines are highlighted.

Appendices

Appendix A: Proof of Theorem 1

According to Eqs. (2) and (3):

$$\begin{aligned} {\hbox {var}} \left[ Z_{t}\right] = t e(1-e) + 2 \sum _{k= 1}^{t-1} (t-k) {\hbox {cov}}_{k} \left[ Y \right] \end{aligned}$$

(36)

Rearranging the previous expression

$$\begin{aligned} {\hbox {var}} \left[ Z_{t}\right] = t e(1-e) + 2 t \sum _{k= 1}^{t} {\hbox {cov}}_{k} \left[ Y \right] - 2 \sum _{k= 1}^{t} k\,\, {\hbox {cov}}_{k} \left[ Y \right] \!, \end{aligned}$$

(37)

where

$$\begin{aligned} {\hbox {cov}}_k[Y]={\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P }^k {\varvec{\mu }}- \left( {\varvec{\pi }}{\varvec{\mu }}\right) ^2={\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P }^k {\varvec{\mu }}- e^2 \end{aligned}$$

(38)

The sums in the second and the third terms of Eq. (37) can be expressed as known geometric sums:

$$\begin{aligned} \sum _{k= 1}^{t} {\hbox {cov}}_{k}= \sum _{k= 1}^{t} \left( {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P }^k {\varvec{\mu }}- e^2 \right) ={\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} \mathbf{P }^k {\varvec{\mu }}- te^2 \end{aligned}$$

(39)

By adding and removing the term ${\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \left( \sum \nolimits _{k= 1}^{t} \mathbf{A }^k\right) {\varvec{\mu }}$, where $A$ is a square ($s$ $\times$ $s$) matrix with identical rows formed by the transpose of the steady-state probability vector ${\varvec{\pi }}$, the following can be obtained:

$$\begin{aligned} \sum _{k= 1}^{t} {\hbox {cov}}_{k}&= {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} ({ \mathbf P }-\mathbf{A })^k {\varvec{\mu }}+{\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} \mathbf A ^k {\varvec{\mu }}- te^2\nonumber \\&= {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} ( \mathbf{P }-\mathbf{A })^k {\varvec{\mu }}+t {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag}\mathbf A {\varvec{\mu }}- te^2=\nonumber \\&= {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} ( \mathbf{P }-\mathbf{A })^k {\varvec{\mu }}+t e^2 - te^2={\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} ( \mathbf{P }-\mathbf{A })^k {\varvec{\mu }}\end{aligned}$$

(40)

It is worth to recall that, due to the general solution of discrete time Markov chains, $\mathbf{P }\mathbf{A }=\mathbf{A }$ and $\mathbf{A }^{k}=\mathbf{A }$ for each value of $k>0$. Using the known sum results it is possible to write

$$\begin{aligned} \sum _{k= 1}^{t} {\hbox {cov}}_{k} = {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} {(\mathbf{P }-\mathbf{A })(\mathbf{I }-(\mathbf{P }-\mathbf{A })^{t})}{(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-1}}{\varvec{\mu }}\end{aligned}$$

(41)

Similarly,

$$\begin{aligned} \sum _{k= 1}^{t} k \cdot {\hbox {cov}}_{k}= \sum _{k= 1}^{t} k\left( {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P }^k {\varvec{\mu }}- e^2 \right) \!=\!{\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} k\mathbf{P }^k {\varvec{\mu }}\!-\! \left( \frac{t^2}{2}+\frac{t}{2}\right) e^2\qquad \end{aligned}$$

(42)

Moreover, by adding and removing the term ${\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \left( \sum _{k= 1}^{t} \mathbf A ^k\right) {\varvec{\mu }}$, the first term can be expressed as

$$\begin{aligned} {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} k\mathbf{P }^k {\varvec{\mu }}&= {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} k(\mathbf{P }-\mathbf{A })^k {\varvec{\mu }}- {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} k\mathbf{A }^k {\varvec{\mu }}; \end{aligned}$$

(43)

therefore, using the known geometric sum

$$\begin{aligned} \sum _{k= 1}^{t} k(\mathbf{P }-\mathbf{A })^k&= {(\mathbf{P }-\mathbf{A })(\mathbf{I }-(\mathbf{P }-\mathbf{A })^{t})}{(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-2}}\nonumber \\&-\, t {(\mathbf{P }-\mathbf{A })^{t+1}}{(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-1}} \end{aligned}$$

(44)

and, for the second term

$$\begin{aligned} {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \sum _{k= 1}^{t} k\mathbf{A }^k {\varvec{\mu }}= \left( \frac{t^2}{2}+\frac{t}{2} \right) {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{A } {\varvec{\mu }}= \left( \frac{t^2}{2}+\frac{t}{2}\right) e^2 \end{aligned}$$

(45)

by substituting Eqs. (44), (45) and (43) into Eq. (42) the following can be obtained:

$$\begin{aligned} \sum _{k= 1}^{t} k {\hbox {cov}}_{k}&= {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \left[ (\mathbf{P }-\mathbf{A })(\mathbf{I }-(\mathbf{P }-\mathbf{A })^{t}){(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-2}}\right. \nonumber \\&-\,\left. t {(\mathbf{P }-\mathbf{A })^{t+1}}{(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-1}} \right] {\varvec{\mu }}\end{aligned}$$

(46)

Finally, by substituting Eqs. (41) and (46) into Eq. (37) we obtain

$$\begin{aligned} {\hbox {var}} \left[ Z_{t}\right]&= t e(1-e) + 2 t {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} { (\mathbf{P }-\mathbf{A })}{(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-1}}{\varvec{\mu }}\nonumber \\&-\, 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} {(\mathbf{P }-\mathbf{A })(\mathbf{I }-(\mathbf{P }-\mathbf{A })^{t})}{(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-2}} {\varvec{\mu }}\end{aligned}$$

(47)

This is a closed-form expression of the variance of the cumulated number of parts produced at time $t$ by a general Markovian system with transition probability matrix $\mathbf{P }$ and reward vector ${\varvec{\mu }}$. It can be rewritten in the following form:

$$\begin{aligned} {\hbox {var}} \left[ Z_{t}\right] = t \alpha + \beta (t) \end{aligned}$$

where:

$$\begin{aligned} \alpha = e(1-e) + 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} { (\mathbf{P }-\mathbf{A })}{(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-1}}{\varvec{\mu }}\end{aligned}$$

(48)

and:

$$\begin{aligned} \beta (t)= - 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} {(\mathbf{P }-\mathbf{A })(\mathbf{I }-(\mathbf{P }-\mathbf{A })^{t})}{(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-2}} {\varvec{\mu }}\end{aligned}$$

(49)

It can be easily shown that, since $\mathbf{P }^{t}$ approaches $\mathbf{A }$ as $t$ tends to infinity, the term $\frac{\beta (t)}{t}$ tails off, and the asymptotic variance rate expression becomes

$$\begin{aligned} v = \alpha = e(1-e) + 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} (\mathbf{P }-\mathbf{A }){(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-1}}{\varvec{\mu }}\end{aligned}$$

(50)

The Fundamental Matrix $\mathbf{Z }$ of a discrete time Markov chain with transition probability matrix $P$ can be expressed as

$$\begin{aligned} \mathbf{Z }=(\mathbf{I }-\mathbf{P }+\mathbf{A })^{-1} \end{aligned}$$

(51)

The properties of the fundamental matrix are such that

$$\begin{aligned} \mathbf{A }\mathbf{Z }=\mathbf{A }\mathbf{Z }^2=\mathbf{A } \end{aligned}$$

(52)

Therefore, more compact expressions of $\alpha$ and $\beta$ can be obtained:

$$\begin{aligned} \alpha&= e(1-e) + 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} (\mathbf{P }-\mathbf{A })\mathbf{Z }{\varvec{\mu }}=e(1-e) + 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P }\mathbf{Z }{\varvec{\mu }}- 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag}{ \mathbf A }\mathbf{Z }{\varvec{\mu }}\nonumber \\&= e(1-e) + 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P }\mathbf{Z }{\varvec{\mu }}- 2 e^2=e(1-3e)+2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P }\mathbf{Z }{\varvec{\mu }}\end{aligned}$$

(53)

and

$$\begin{aligned} \beta (t)= 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \left( \mathbf{P }^{t+1}-\mathbf{P } \right) \mathbf{Z }^2 {\varvec{\mu }}\end{aligned}$$

(54)

Therefore, the final expression of the variance is

$$\begin{aligned} {\hbox {var}} \left[ Z_{t}\right] = t e(1-3e)+ 2t {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P } \mathbf{Z }{\varvec{\mu }}+ 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \left( \mathbf{P }^{t+1}-\mathbf{P } \right) \mathbf{Z }^2 {\varvec{\mu }}\end{aligned}$$

(55)

Appendix B: Proof of Theorem 2

By referring to the partitions of the transition probability matrix P defined in Sect. 3, the mean inter departure time can be obtained with the following equation:

$$\begin{aligned} \mathbb{E }[{\hbox {IDT}}]= \frac{1}{e} {\varvec{\pi }}_{U} \left( \mathbf{P }_{U,U} + \mathbf{P }_{U,D}\sum _{k= 2}^{\infty }k\mathbf{P }_{D,D}^{k-2}\mathbf{P }_{D,U} \right) {\varvec{\mu }}_{U} \end{aligned}$$

(56)

The first term in brackets reflects the situation in which the inter-departure time assumes value $1$ since the system makes a transition from one operational state to another operational state or it stays in the same operational state. The second term in brackets reflects the situation in which the system makes a transition to a non-operational state and the inter-departure time increases by one unit for any time step it remains in the down state, until the system goes back to an operational state. The multiplying factor is the conditional probability of the system being in any specific operational state.

Using the results for known sums of geometric series, the following can be written:

$$\begin{aligned} \sum _{k= 2}^{\infty }k\mathbf{P }_{D,D}^{k-2}=\sum _{q= 0}^{\infty }(q+2)\mathbf{P }_{D,D}^{q}=\left( 2\mathbf{I }-\mathbf{P }_{D,D}\right) {\left( \mathbf{I }-\mathbf{P }_{D,D}\right) ^{-2}} \end{aligned}$$

(57)

Therefore,

$$\begin{aligned} \mathbb{E }[{\hbox {IDT}}]= \frac{1}{e} {\varvec{\pi }}_{U} \left( \mathbf{P }_{U,U} + \mathbf{P }_{U,D}\left( 2\mathbf{I }-\mathbf{P }_{D,D}\right) \left( \mathbf{I }-\mathbf{P }_{D,D}\right) ^{-2}\mathbf{P }_{D,U} \right) {\varvec{\mu }}_{U} \end{aligned}$$

(58)

It can be easily proved that, for any system, the mean inter-departure time is the inverse of the throughput, i.e.,

$$\begin{aligned} \mathbb{E }[{\hbox {IDT}}]= \frac{1}{e} \end{aligned}$$

(59)

The variance of the inter-departure time can be expressed as a function of the second and the first moments:

$$\begin{aligned} {\hbox {var}}[{\hbox {IDT}}]= \mathbb{E }[{\hbox {IDT}}^{2}]-\mathbb{E }[{\hbox {IDT}}]^{2} \end{aligned}$$

(60)

The second moment of the inter-departure time can be expressed as follows:

$$\begin{aligned} \mathbb{E }[{\hbox {IDT}}^{2}]= \frac{1}{e} {\varvec{\pi }}_{U} \left( \mathbf{P }_{U,U} + \mathbf{P }_{U,D}\sum _{k= 2}^{\infty }k^{2}\mathbf{P }_{D,D}^{k-2}\mathbf{P }_{D,U} \right) {\varvec{\mu }}_{U} \end{aligned}$$

(61)

Using the results for known sums of geometric series, the following can be written:

$$\begin{aligned} \sum _{k= 2}^{\infty }k^{2}\mathbf{P }_{D,D}^{k-2}=\sum _{q= 0}^{\infty }(q^{2}+4q+4)\mathbf{P }_{D,D}^{q}={\left( \mathbf{P }_{D,D}^{2}-3\mathbf{P }_{D,D}+4\mathbf{I }\right) }{\left( \mathbf{I }-\mathbf{P }_{D,D}\right) ^{-3}}\nonumber \\ \end{aligned}$$

(62)

Therefore, the second moment of the inter-departure time can be expressed as

$$\begin{aligned} \mathbb{E }[{\hbox {IDT}}^{2}]= \frac{1}{e} {\varvec{\pi }}_{U} \left[ \mathbf{P }_{U,U} + \mathbf{P }_{U,D}\left( \mathbf{P }_{D,D}^{2}-3\mathbf{P }_{D,D}+4\mathbf{I }\right) \left( \mathbf{I }-\mathbf{P }_{D,D}\right) ^{-3}\mathbf{P }_{D,U} \right] {\varvec{\mu }}_{U}\nonumber \\ \end{aligned}$$

(63)

By rearranging the previous equation we obtain

$$\begin{aligned} \mathbb{E }[{\hbox {IDT}}^{2}]&= \frac{1}{e} {\varvec{\pi }}_{U} \left[ \mathbf{P }_{U,U} + \mathbf{P }_{U,D}\left( \left( 2\mathbf{I }-\mathbf{P }_{D,D}\right) \left( \mathbf{I }-\mathbf{P }_{D,D}\right) ^{-2}\right. \right. \nonumber \\&+\left. \left. 2\mathbf{I }\left( \mathbf{I }-\mathbf{P }_{D,D}\right) ^{-3}\right) \mathbf{P }_{D,U} \right] {\varvec{\mu }}_{U} \end{aligned}$$

(64)

Therefore, it is possible to express the second moment as a function of the first moment:

$$\begin{aligned} \mathbb{E }[{\hbox {IDT}}^{2}]= \mathbb{E }[{\hbox {IDT}}]+\frac{1}{e} {\varvec{\pi }}_{U}\mathbf{P }_{U,D}2\mathbf{I }\left( \mathbf{I }-\mathbf{P }_{D,D}\right) ^{-3}\mathbf{P }_{D,U}{\varvec{\mu }}_{U} \end{aligned}$$

(65)

By substituting Eqs. (59) and (65) into Eq. (60) the closed-form expression for the variance of the inter-departure time for any general system can be obtained as follows:

$$\begin{aligned} {\hbox {var}}[{\hbox {IDT}}]=\frac{e-1}{e^{2}}+\frac{1}{e} {\varvec{\pi }}_{U}\mathbf{P }_{U,D}2\mathbf{I }{\left( \mathbf{I }-\mathbf{P }_{D,D}\right) ^{-3}}\mathbf{P }_{D,U}{\varvec{\mu }}_{U} \end{aligned}$$

(66)

Appendix C: Implementation issues

In this section, useful rearrangements of the proposed equations that positively contribute to reduce the computational time of the proposed approaches are proposed, both for the approximate formula and the exact formulas proposed in this paper.

1.1 C.1 Approximate formula for the variance of the cumulated production

Equation (5) can be rearranged in order to obtain a recursive function to be evaluated for $k=1,\ldots ,K^*$.

$$\begin{aligned} {\hbox {cov}}_k[Y] = \sum _{g=1}^{s}C_{k,g}\mu _{g} - e^2, \end{aligned}$$

(67)

where

$$\begin{aligned} C_{k,g}&= \sum _{j=1}^{s}C_{k-1,j} \mathbf{P }_{j,g} \quad k=2,\ldots ,k^*, g=1,\ldots ,s \nonumber \\ C_{1,g}&= \sum _{j=1}^{s}\pi _{j} \mu _j \mathbf P _{j,g} \end{aligned}$$

(68)

In this way, $C_{k}$ is a row vector that is computed by recursion at each significant time step $k=1,\ldots ,K^*$, thus avoiding several computations of the power matrix. For a similar approach see Tan (2002). $\rho _\mathrm{total}(\epsilon )$ then becomes

$$\begin{aligned} \rho _\mathrm{total}(\epsilon )=\frac{\sum \nolimits _{k=1}^{k^*}\sum \nolimits _{g=1}^{s}C_{k,g}\mu _{g}-k^*e^2}{e(1-e)} \end{aligned}$$

(69)

1.2 C.2 Exact formula for the variance of the cumulated production

From a computational point of view, the most complex aspect of Eqs. (13) and (14) is the calculation of the elements of the fundamental matrix $Z$. A computationally efficient method to address this problem is described in the following. Equation (13) is rewritten in the following terms:

$$\begin{aligned} \alpha = e(1-3e)+2 Q {\varvec{\mu }}, \end{aligned}$$

(70)

where $Q$ is a row vector of the form:

$$\begin{aligned} Q={\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P }\mathbf{Z }=q(I-P+A)^{-1}\!, \end{aligned}$$

(71)

where $q$ is a row vector of $s$ elements, easily obtained as

$$\begin{aligned} q={\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \mathbf{P } \end{aligned}$$

(72)

The vector $Q$ can be thus obtained by solving the following system of equations:

$$\begin{aligned} Q(I-P+A)=q\nonumber \\ Q u=0, \end{aligned}$$

(73)

where $u$ is a column vector with $s$ elements equal to $1$. Similarly, for $\beta (t)$

$$\begin{aligned} \beta (t)= 2 {\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \left( \mathbf{P }^{t+1}-\mathbf{P } \right) \mathbf{Z }^2 {\varvec{\mu }}=2W {\varvec{\mu }}, \end{aligned}$$

(74)

where $W$ is a row vector of the form

$$\begin{aligned} W=W_1\mathbf{Z } \end{aligned}$$

(75)

and

$$\begin{aligned} W_1={\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \left( \mathbf{P }^{t+1}-\mathbf{P } \right) \mathbf{Z }=w(I-P+A)^{-1}, \end{aligned}$$

(76)

where $w$ is a row vector of $s$ elements, easily obtained as

$$\begin{aligned} w={\varvec{\pi }}{\varvec{\mu }}_\mathrm{diag} \left( \mathbf{P }^{t+1}-\mathbf{P }\right) \end{aligned}$$

(77)

The vector $W_1$ can be thus obtained by solving the following system of equations:

$$\begin{aligned} W_1(I-P+A)&= w\nonumber \\ W_1 u&= 0 \end{aligned}$$

(78)

Finally, the vector $W$ can be obtained by solving this second system of equations:

$$\begin{aligned} W(I-P+A)&= W_1\nonumber \\ W u&= 0 \end{aligned}$$

(79)

Therefore, the variance formula reduces to

$$\begin{aligned} var \left[ Z_{t}\right] = t e(1-3e)+2t Q {\varvec{\mu }}+ 2W {\varvec{\mu }}\end{aligned}$$

(80)

1.3 C.3 Exact formula for the variance of the inter-departure time

The same procedure can be adopted for increasing the computational efficiency in the calculation of the term $(\mathbf{I }-\mathbf{P }_{D,D})^{-3}$ in Eq. 66.