# A Kronecker Algebra Formulation for Markov Activity Networks with Phase-Type Distributions

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## Abstract

**:**

## 1. Introduction

## 2. State of the Art

## 3. Markov Activity Networks

- ${\mathcal{A}}_{{v}_{1}}^{-}$ contains those activities without dependencies that can start as soon as the execution of the network begins;
- The set of activities that departs from a vertex corresponds to ${\mathcal{A}}_{v}^{-}$, and they directly depend on ${\mathcal{A}}_{v}^{+}$;
- Activities start as soon as all of the preceding activities are completed; this means that there is no time span between the end of an activity and the start of its successors;
- Activities are never preempted and there is no limit to the number of activities that can be executed in parallel, that is, no resource constraint is enforced;
- The duration of each activity i is modeled as a continuous distribution ${\mathcal{X}}_{i}$.

## 4. Phase-Type Distributions

## 5. Markov Activity Networks Enhanced with PH Distributions

- The completion of the activity that caused the transition from ${\mathbf{s}}^{\prime}$ to $\mathbf{s}$;
- The memory of the states within the PH for the activities that were running in state ${\mathbf{s}}^{\prime}$ and will continue to run in $\mathbf{s}$;
- The start of activities triggered by the completion of activity i defined by precedence relations.

## 6. A Kroncker Algebra Approach for a Markov Activity Network with PH Distributions

## 7. Testing

#### 7.1. Activity Network with a Single PH Distributed Activity Duration

- A PH distribution is fitted to approximate the general distribution;
- The resulting PH distribution is plugged into the activity network.

- Normal: It represents the ideal scenario for PH approximation because it is continuous and light-tailed on both sides;
- Log-Normal: It is a more challenging scenario because of heavy tails;
- Uniform: It represents the more difficult case, because PH distributions cannot model a distribution with finite support. Hence, a higher number of phases will be needed to reach a reasonable approximation.

#### 7.1.1. PH Approximation for a Normally Distributed Activity

#### 7.1.2. PH Approximation for a Log-Normal-Distributed Activity

#### 7.1.3. PH Approximation for a Uniform Distribution

#### 7.2. PH Approximation for All the Activities in the Network

#### 7.3. Test on a Set of Activity Networks

- Large networks requiring a considerable amount of data to be stored in the RAM and slowing down the computation due to the swapping between primary and secondary memory;
- The calculation of the desired percentiles (performed with the bisection method) that increases the computational effort.

- The CTMC can be solved using more advanced techniques able to reduce the computation time;
- The CTMC defines a model for the execution of the network of activities that can support decomposition approaches, for example, calculating subnets and incorporating the obtained solution or estimation in the comprehensive network.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**State transition graph of the stochastic process for the network of activities in Figure 1.

**Figure 3.**Graphical representation of an order of three PH distributions (

**left**) and its probability density function (

**right**).

**Figure 4.**Comparison between an empirical distribution fitted by using both moment-based approaches and cluster-based fitting: cdf (

**left**) and tail of the cdf (

**right**).

**Figure 5.**Scatter plots of the data-points used for the fitting of the Normal distribution (

**left**), Log-Normal distribution (

**center**), and Uniform distribution (

**right**).

**Figure 6.**Comparison between the normal distribution with mean 5 and coefficient of variation equal to 0.2 and its fitting by means of PH distributions: pdf (

**left**), cdf (

**center**), and tail of the cdf (

**right**).

**Figure 7.**Makespan of the example activity network with the duration of activity 3 following a normal distribution with mean 5 and coefficient of variation equal to 0.2; full (

**left**), only the tail (

**center**), and the tail approximation error (

**right**).

**Figure 8.**Comparison between the log-normal distribution with parameters $a=-0.804$ and $b=1.268$ and its fitting by means of PH distributions: pdf (

**left**), cdf (

**center**), and tail of the cdf (

**right**).

**Figure 9.**Makespan of the example activity network with the duration of activity 3 following a log-normal distribution with parameters $a=-0.804$ and $b=1.268$; full (

**left**), only the tail (

**center**), and the tail approximation error (

**right**).

**Figure 10.**Comparison between the uniform distribution in the interval $\left(\right)$ and its fitting by means of PH distributions: pdf (

**left**), cdf (

**center**), and tail of the cdf (

**right**).

**Figure 11.**Makespan of the example activity network with the duration of activity 3 following a uniform distribution in the interval $\left(\right)$; full (

**left**), only the tail (

**center**), and the tail approximation error (

**right**).

**Figure 12.**Makespan of the five-activity subset within the network; full makespan (

**left**), tail (

**center**), and approximation error on the tail (

**right**).

**Figure 13.**Makespan of the 10-activity subset within the network; full makespan (

**left**), tail (

**center**), and approximation error on the tail (

**right**).

**Figure 14.**Makespan of the 20-activity subset within the network; full makespan (

**left**), tail (

**center**), and approximation error on the tail (

**right**).

**Figure 15.**Makespan of the full 32-activity network; full makespan (

**left**), tail (

**center**), and approximation error on the tail (

**right**).

**Figure 16.**Scatter plot of the number of states against the number of transitions composing the CTMC of the networks.

**Figure 17.**Time required to solve the networks with PH distributions as a function of the number of states (

**left**) and the number of transitions (

**right**).

**Table 1.**Infinitesimal generator of the CTMC describing the execution of the activity network in Figure 1. Due to space constraints, row labels have not been included (the order of the rows is the same as that of the columns). For the same reason, commas have not been used to separate the entries of each state.

RRPPP | TRRRP | TRRTP | TRTTP | RTPPP | TRTRP | TTRRP | TTRTP | TTTTR | TTTRR | TTTRT | TTTTT |
---|---|---|---|---|---|---|---|---|---|---|---|

$-({\lambda}_{1}+{\lambda}_{2})$ | ${\lambda}_{1}$ | 0 | 0 | ${\lambda}_{2}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0 | $-({\lambda}_{2}+{\lambda}_{3}+{\lambda}_{4})$ | ${\lambda}_{4}$ | 0 | 0 | ${\lambda}_{3}$ | ${\lambda}_{2}$ | 0 | 0 | 0 | 0 | 0 |

0 | 0 | $-({\lambda}_{2}+{\lambda}_{3})$ | ${\lambda}_{3}$ | 0 | 0 | 0 | ${\lambda}_{2}$ | 0 | 0 | 0 | 0 |

0 | 0 | 0 | $-{\lambda}_{2}$ | 0 | 0 | 0 | 0 | ${\lambda}_{3}$ | 0 | 0 | 0 |

0 | 0 | 0 | 0 | $-{\lambda}_{1}$ | 0 | ${\lambda}_{1}$ | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | ${\lambda}_{4}$ | 0 | $-({\lambda}_{2}+{\lambda}_{4})$ | 0 | 0 | 0 | ${\lambda}_{2}$ | 0 | 0 |

0 | 0 | 0 | 0 | 0 | 0 | $-({\lambda}_{3}+{\lambda}_{4})$ | ${\lambda}_{4}$ | 0 | ${\lambda}_{3}$ | 0 | 0 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | $-{\lambda}_{3}$ | ${\lambda}_{3}$ | 0 | 0 | 0 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-{\lambda}_{5}$ | 0 | 0 | ${\lambda}_{5}$ |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ${t}_{4}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{5}$ | $-({\lambda}_{4}+{\lambda}_{5})$ | ${\lambda}_{5}$ | 0 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-{\lambda}_{4}$ | ${\lambda}_{4}$ |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 2.**Infinitesimal generator of the CTMC enhanced with PH distributions describing the model in Figure 1.

RRPPP | TRRRP | TRRTP | TRTTP | RTPPP | TRTRP | TTRRP | TTRTP | TTTTR | TTTRR | TTTRT | TTTTT |
---|---|---|---|---|---|---|---|---|---|---|---|

${T}_{1}\phantom{\rule{-0.166667em}{0ex}}\oplus \phantom{\rule{-0.166667em}{0ex}}{T}_{2}$ | ${t}_{1}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{2}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{\beta}_{3}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{\beta}_{4}$ | 0 | 0 | ${I}_{1}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{t}_{2}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0 | ${T}_{2}\phantom{\rule{-0.166667em}{0ex}}\oplus \phantom{\rule{-0.166667em}{0ex}}{T}_{3}\phantom{\rule{-0.166667em}{0ex}}\oplus \phantom{\rule{-0.166667em}{0ex}}{T}_{4}$ | ${I}_{2}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{3}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{t}_{4}$ | 0 | 0 | ${I}_{2}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{t}_{3}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{4}$ | ${t}_{2}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{3}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{4}$ | 0 | 0 | 0 | 0 | 0 |

0 | 0 | ${T}_{2}\phantom{\rule{-0.166667em}{0ex}}\oplus \phantom{\rule{-0.166667em}{0ex}}{T}_{3}$ | ${I}_{2}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{t}_{3}$ | 0 | 0 | 0 | ${t}_{2}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{3}$ | 0 | 0 | 0 | 0 |

0 | 0 | 0 | ${T}_{2}$ | 0 | 0 | 0 | 0 | ${t}_{2}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{\beta}_{5}$ | 0 | 0 | 0 |

0 | 0 | 0 | 0 | ${T}_{1}$ | 0 | ${t}_{1}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{\beta}_{3}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{\beta}_{4}$ | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | ${I}_{2}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{t}_{4}$ | 0 | ${T}_{2}\phantom{\rule{-0.166667em}{0ex}}\oplus \phantom{\rule{-0.166667em}{0ex}}{T}_{4}$ | 0 | 0 | 0 | ${t}_{2}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{4}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{\beta}_{5}$ | 0 | 0 |

0 | 0 | 0 | 0 | 0 | 0 | ${T}_{3}\phantom{\rule{-0.166667em}{0ex}}\oplus \phantom{\rule{-0.166667em}{0ex}}{T}_{4}$ | ${I}_{3}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{t}_{4}$ | 0 | ${t}_{3}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{4}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{\beta}_{5}$ | 0 | 0 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | ${T}_{3}$ | ${t}_{3}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{\beta}_{5}$ | 0 | 0 | 0 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ${T}_{5}$ | 0 | 0 | ${t}_{5}$ |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ${t}_{4}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{I}_{5}$ | ${T}_{4}\phantom{\rule{-0.166667em}{0ex}}\oplus \phantom{\rule{-0.166667em}{0ex}}{T}_{5}$ | ${I}_{4}\phantom{\rule{-0.166667em}{0ex}}\otimes \phantom{\rule{-0.166667em}{0ex}}{t}_{5}$ | 0 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ${T}_{4}$ | ${t}_{4}$ |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 3.**Activities in the network with the associated distributions. Horizontal lines indicate the subsets that have been incrementally taken into consideration.

Activity | Distribution | Average | CV | Dependencies |
---|---|---|---|---|

1 | L | 1 | 0.5 | - |

2 | H | 8 | 1 | 1 |

3 | H | 9 | 1.5 | 1 |

4 | H | 1 | 1.5 | 1 |

5 | L | 4 | 0.5 | 3 |

6 | H | 4 | 1 | 2 |

7 | L | 8 | 1 | 3 |

8 | L | 3 | 0.5 | 7 |

9 | H | 3 | 1.5 | 6 |

10 | L | 9 | 0.5 | 8 |

11 | H | 6 | 1 | 9 |

12 | H | 3 | 1 | 5 |

13 | H | 5 | 1.5 | 9 |

14 | H | 4 | 1 | 12 |

15 | H | 9 | 1 | 8 |

16 | H | 5 | 1.5 | 7, 11, 13 |

17 | H | 9 | 1.5 | 6 |

18 | H | 9 | 1.5 | 4 |

19 | H | 7 | 1.5 | 12, 17 |

20 | L | 7 | 0.5 | 10, 14 |

21 | L | 8 | 1 | 16, 17, 20 |

22 | H | 6 | 1.5 | 10 |

23 | H | 10 | 1 | 13, 22 |

24 | H | 2 | 1.5 | 15, 21 |

25 | H | 1 | 1.5 | 13 |

26 | L | 9 | 0.5 | 19, 20, 23 |

27 | H | 3 | 1.5 | 14, 18 |

28 | H | 7 | 1.5 | 16 |

29 | L | 10 | 0.5 | 18 |

30 | H | 7 | 1.5 | 18, 24, 26 |

31 | L | 9 | 1 | 25, 27, 28 |

32 | L | 1 | 0.5 | 29, 30, 31 |

# Activities | # States | #Transitions | Time Numerical | Time Monte Carlo |
---|---|---|---|---|

5 | 13 | 77 | 0 s | 15 s |

10 | 73 | 1559 | 0 s | 20 s |

20 | 1351 | 60,710 | 6 s | 22 s |

32 | 6836 | 579,341 | 124 s | 25 s |

Average | St. Dev. | Conf. Interval | Min | Max | |
---|---|---|---|---|---|

States | 2763.73 | 2037.00 | (2335.28, 3192.16) | 779 | 11,858 |

Transitions | 207,343.59 | 237,726.36 | (157,342.71, 257,344.45) | 17,298 | 935,440 |

Entropy | 0.0359 | 0.001 | (0.03575, 0.0362) | 0.0321 | 0.037 |

**Table 6.**Computation time for the $99\%$ quantile of the distribution of the makespan for the experiments.

Approach | # Average | St. Dev. | Conf. Interval | Min | Max |
---|---|---|---|---|---|

PH distributions | 85.10 | 138.035 | (56.07, 114.135) | 2 | 610 |

Monte Carlo | 20.81 | 4.0 | (27.2, 28.9) | 20 | 30 |

Percentile | # Average | St. Dev. | Conf. Interval | Min | Max |
---|---|---|---|---|---|

90 | 1.16 | 0.80 | (0.99,1.33) | 0.017 | 3.46 |

95 | 1.72 | 1.16 | (1.47,1.96) | 0.006 | 4.55 |

97 | 2.49 | 1.49 | (2.17,2.79) | 0.08 | 5.66 |

99 | 2.72 | 2.09 | (2.28,3.15) | 0.04 | 9.60 |

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**MDPI and ACS Style**

Angius, A.; Horváth, A.; Urgo, M.
A Kronecker Algebra Formulation for Markov Activity Networks with Phase-Type Distributions. *Mathematics* **2021**, *9*, 1404.
https://doi.org/10.3390/math9121404

**AMA Style**

Angius A, Horváth A, Urgo M.
A Kronecker Algebra Formulation for Markov Activity Networks with Phase-Type Distributions. *Mathematics*. 2021; 9(12):1404.
https://doi.org/10.3390/math9121404

**Chicago/Turabian Style**

Angius, Alessio, András Horváth, and Marcello Urgo.
2021. "A Kronecker Algebra Formulation for Markov Activity Networks with Phase-Type Distributions" *Mathematics* 9, no. 12: 1404.
https://doi.org/10.3390/math9121404