# Queuing-Inventory Models with MAP Demands and Random Replenishment Opportunities

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. General Description

- The subscript $i,i=1,2$ refers to the model under consideration.
- The symbol ${}^{\prime}$ stands for the transpose notation.
- ${e}^{\prime}=(1,1,\cdots ,1)$, whose dimension should be clear in the context. Where more clarity is needed, the dimension will be mentioned, e.g., $e\left(m\right)$ is a column vector of 1s of dimension m.
- ${{e}^{\prime}}_{i}=(0,0,\cdots ,1,0,\cdots ,0)$, where 1 is in the ith position.
- I denotes an identity matrix, whose dimension is dictated by the context.
- $\Delta ({E}_{1},\cdots ,{E}_{r})$ denotes a diagonal matrix with diagonal (possibly block) entries given by ${E}_{1}$ through ${E}_{r}$. In the context where this notation is used, it will be clear whether the entities are scalars or vectors or matrices.

## 3. Opportunistic Model 1

- ${J}_{1}\left(t\right)$ to be the number of customers in the system (including one in service),
- ${J}_{2}\left(t\right)$ to be the level of the inventory,
- ${J}_{3}\left(t\right)$ to be the phase of the service (if the server is idle, this will not be defined),
- ${J}_{4}\left(t\right)$ to be the phase of the arrival process,

- ${\widehat{p}}_{i}$, probabilities of demands greater than i, are computed as$${\widehat{p}}_{i}=\sum _{k=i+1}^{N}{p}_{k},\phantom{\rule{3.33333pt}{0ex}}1\le i\le N-1.$$
- P, a square matrix of dimension $K+1$ is defined as$$P=\left(\right)open="["\; close="]">\begin{array}{ccccccc}0& 0& 0& \cdots & 0& 0& 0\\ 1& 0& 0& \cdots & 0& 0& 0\\ {\widehat{p}}_{1}& {p}_{1}& 0& \cdots & 0& 0& 0\\ {\widehat{p}}_{2}& {p}_{2}& {p}_{1}& \cdots & 0& 0& 0\\ \vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\ {\widehat{p}}_{N-1}& {p}_{N-1}& {p}_{N-2}& \cdots & 0& 0& 0\\ 0& {p}_{N}& {p}_{N-1}& \cdots & 0& 0& 0\\ 0& 0& {p}_{N}& \cdots & 0& 0& 0\\ \vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\ 0& 0& 0& \cdots & {p}_{2}& {p}_{1}& 0\end{array}$$A quick look at P indicates that the customers’ demands are taken into consideration at the time of their arrivals. The first column of P justifies that the structure due to the customers’ demands are met partially. It is easy to verify that$$Pe={(0,1,\cdots ,1)}^{\prime}.$$

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

#### 3.1. Computation of R

**Lemma**

**3.**

**Proof.**

**Remark**

**1.**

#### 3.2. Selected System Measures in Steady-State

- 1.
- Server idle probability: ${\nu}_{1}={x}_{0}e$.
- 2.
- Probability of idle server with positive inventory: ${\nu}_{1I}={x}_{0}e-{x}_{0,0}e.$
- 3.
- Percent of server idle time with positive inventory: ${\nu}_{1}^{*}={\nu}_{1I}/{\nu}_{1}$
- 4.
- Mean number of customers in the system: ${\mu}_{1}={x}_{1}{(I-R)}^{-2}e$.
- 5.
- Variance of the number of customers in the system:$${\sigma}_{1}^{2}=2{x}_{1}{(I-R)}^{-3}e-{\mu}_{1}(1+{\mu}_{1}).$$
- 6.
- Probability of customer loss (at arrivals): ${\vartheta}_{1}=\frac{1}{\lambda}\left(\right)open="["\; close="]">{\sum}_{i=0}^{\infty}{x}_{i,0}(e\otimes {E}_{1}e)$
- 7.
- Mean inventory level: ${\widehat{\mu}}_{1}={\sum}_{j=1}^{K}j\left(\right)open="["\; close="]">{x}_{0,j}e+{\sum}_{i=1}^{\infty}{x}_{i,j}e$
- 8.
- Variance of the inventory level: ${\widehat{\sigma}}_{1}^{2}={\sum}_{j=1}^{K}{j}^{2}\left(\right)open="["\; close="]">{x}_{0,j}e+{\sum}_{i=1}^{\infty}{x}_{i,j}e$
- 9.
- Mean cycle time of replenishment:$${\kappa}_{1}={\left(\right)}^{\gamma}$$
- 10.
- Mean replenishment quantity:$${\Gamma}_{1}=\gamma \left(\right)open="["\; close="]">\sum _{j=0}^{L}(K-j)\left(\right)open="("\; close=")">{x}_{0,j}e+\sum _{i=1}^{\infty}{x}_{i,j}e$$
- 11.
- Probability of procuring an order when a replenishment opportunity arises:$${\xi}_{1}=\sum _{j=0}^{L}\left(\right)open="("\; close=")">{x}_{0,j}e+\sum _{i=1}^{\infty}{x}_{i,j}e.$$

## 4. Opportunistic Model 2

- ${\widehat{J}}_{1}\left(t\right)$ to be the number of customers in the system,
- ${\widehat{J}}_{2}\left(t\right)$ to be the level of the inventory,
- ${\widehat{J}}_{3}\left(t\right)$ to be the phase of the service (if the server is idle, this will not be defined),
- ${\widehat{J}}_{4}\left(t\right)$ to be the phase of the arrival process,

**Theorem**

**2.**

#### 4.1. Computation of ${y}_{0}$ and ${y}_{1}$

#### 4.2. Selected System Measures in Steady-State

- 1.
- Server idle probability: ${\nu}_{2}={y}_{0}e.$
- 2.
- Probability of idle server with positive inventory: ${\nu}_{2I}={y}_{0}e-{y}_{0,0}e.$
- 3.
- Percent of server idle time with positive inventory: ${\nu}_{2}^{*}={\nu}_{2I}/{\nu}_{2}.$
- 4.
- Mean number of customers in the system: ${\mu}_{2}={y}_{1}{(I-\widehat{R})}^{-2}e$.
- 5.
- Variance of the number of customers in the system:$${\sigma}_{2}^{2}=2{y}_{1}{(I-\widehat{R})}^{-3}e-{\mu}_{2}(1+{\mu}_{2}).$$
- 6.
- Probability of customer loss at arrivals:$${\vartheta}_{2}=\frac{1}{\lambda}\left(\right)open="["\; close="]">{y}_{0,0}(e\otimes {E}_{1}e)+\sum _{i=1}^{\infty}(i-1){y}_{i,0}({S}^{0}\otimes e)$$
- 7.
- Mean inventory level: ${\widehat{\mu}}_{2}={\sum}_{j=1}^{K}j\left(\right)open="["\; close="]">{y}_{0,j}e+{\sum}_{i=1}^{\infty}{y}_{i,j}e$
- 8.
- Variance of the inventory level: ${\widehat{\sigma}}_{2}^{2}={\sum}_{j=1}^{K}{j}^{2}\left(\right)open="["\; close="]">{y}_{0,j}e+{\sum}_{i=1}^{\infty}{y}_{i,j}e$
- 9.
- Mean cycle time of replenishment:$${\kappa}_{2}={\left(\right)}^{\gamma}.$$
- 10.
- Mean replenishment quantity:$${\Gamma}_{2}=\gamma \left(\right)open="["\; close="]">\sum _{j=0}^{L}(K-j)\left(\right)open="("\; close=")">{y}_{0,j}e+\sum _{i=1}^{\infty}{y}_{i,j}e$$
- 11.
- Probability of procuring an order when a replenishment opportunity arises:$${\xi}_{2}=\sum _{j=0}^{L}\left(\right)open="("\; close=")">{y}_{0,j}e+\sum _{i=1}^{\infty}{y}_{i,j}e.$$

## 5. Illustrative Numerical Examples

- ERA
- Erlang distributed inter-arrival times with density ${\tilde{\lambda}}^{k}{x}^{k-1}{e}^{-\tilde{\lambda}x}/(k-1)!$, $\tilde{\lambda}=4$, and $k=4$ (standard deviation $\approx 0.5774$).
- HEA
- Hyperexponential inter-arrival times with density ${\sum}_{i=1}^{k}{p}_{i}{\tilde{\lambda}}_{i}{e}^{-{\tilde{\lambda}}_{i}x}$, $k=4$, ${\tilde{\lambda}}_{i}=\left(63.1\right)\xb7{10}^{-i},i=1,2,3$, and $p=(0.6,0.25,0.10,0.05)$ (standard deviation $\approx 4.9629$).
- PCA
- Markov arrival process $({E}_{0},{E}_{1})$ with positive correlation $\approx 0.4637$ (standard deviation $\approx 1.3153$), where$${E}_{0}=\left(\right)open="("\; close=")">\begin{array}{ccc}-1.05& 1.05& 0\\ 0& -1.05& 0\\ 0& 0& -10.5\end{array}.$$

- ERS
- Erlang distributed service times having density ${\tilde{\mu}}^{k}{x}^{k-1}{e}^{-\tilde{\mu}x}/(k-1)!$ with $k=3$.
- EXS
- Exponentially distributed service times of rate $\mu $.
- HES
- Hyperexponentially distributed service times with density ${\sum}_{i=1}^{k}{p}_{i}{\tilde{\mu}}_{i}{e}^{-{\tilde{\mu}}_{i}x}$, $k=3$, $p=(0.7,0.25,0.05)$ and ${\tilde{\mu}}_{i}=\left(8.2\phantom{\rule{0.166667em}{0ex}}\mu \right)\phantom{\rule{0.166667em}{0ex}}\xb7{10}^{-i},i=1,\cdots ,3$.

- In ISS, inventory is consumed and replenished as needed and is characterized by the parameters $K,L$, $\gamma $ and the distribution of the time between two opportunities for replenishment. The arrival process to CSS impacts ISS through the demand for items. ${\widehat{\mu}}_{1}[{\widehat{\mu}}_{2}$], ${\widehat{\sigma}}_{1}[{\widehat{\sigma}}_{2}$], ${\Gamma}_{1}[{\Gamma}_{2}$], ${\xi}_{1}[{\xi}_{2}$], ${\kappa}_{1}[{\kappa}_{2}$] represent the measures of performance for ISS for Model 1 [Model 2].
- In CSS, customers arrive, receive service (plus items from inventory), and depart and are characterized by the arrival and service processes. Some arrivals are lost due to lack of inventory at the time of arrival. In Model 2, customers may also be lost at a service completion epoch due to lack of inventory at that moment. Customer loss is impacted by the availability of inventory in ISS. ${\mu}_{1}\left[{\mu}_{2}\right]$, ${\sigma}_{1}\left[{\sigma}_{2}\right]$, ${\nu}_{1}\left[{\nu}_{2}\right]$, ${\nu}_{1}^{*}\left[{\nu}_{2}^{*}\right]$, ${\vartheta}_{1}[{\vartheta}_{2a},{\vartheta}_{2d}]$ represent measures of performance for CSS for Model 1 [Model 2].

#### 5.1. Impact of Service Rate and Service Time Distribution

#### 5.2. Effect of Arrival Process

#### 5.3. Impact of $\gamma $

#### 5.4. Impact of K and L

#### 5.5. Cost Analysis

#### 5.6. Comparing Model 1 and Model 2

#### 5.7. Comparing $(K,L)$ System with $(s,S)$ System

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Model 1 | Model 2 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Arr. | Ser. | $\mathbf{\gamma}$ | $\mathit{K}$ | $\mathit{L}$ | ${\mathbf{\mu}}_{\mathbf{1}}$ | ${\mathbf{\sigma}}_{\mathbf{1}}$ | ${\mathbf{\nu}}_{\mathbf{1}}$ | ${\mathbf{\nu}}_{\mathbf{1}}^{*}$ | ${\mathbf{\vartheta}}_{\mathbf{1}}$ | ${\mathbf{\mu}}_{\mathbf{2}}$ | ${\mathbf{\sigma}}_{\mathbf{2}}$ | ${\mathbf{\nu}}_{\mathbf{2}}$ | ${\mathbf{\nu}}_{\mathbf{2}}^{*}$ | ${\mathbf{\vartheta}}_{\mathbf{2}\mathbf{a}}$ | ${\mathbf{\vartheta}}_{\mathbf{2}\mathbf{d}}$ |

ERA | ERS | 0.05 | 50 | 20 | 0.800 | 1.243 | 0.594 | 0.146 | 0.553 | 0.851 | 1.387 | 0.605 | 0.140 | 0.520 | 0.046 |

ERA | ERS | 0.05 | 50 | 30 | 0.841 | 1.284 | 0.581 | 0.151 | 0.539 | 0.896 | 1.433 | 0.591 | 0.145 | 0.505 | 0.045 |

ERA | ERS | 0.05 | 60 | 20 | 0.929 | 1.362 | 0.552 | 0.162 | 0.507 | 0.978 | 1.500 | 0.564 | 0.156 | 0.476 | 0.044 |

ERA | ERS | 0.05 | 60 | 30 | 0.982 | 1.412 | 0.536 | 0.169 | 0.490 | 1.033 | 1.554 | 0.548 | 0.161 | 0.460 | 0.043 |

ERA | ERS | 0.1 | 50 | 20 | 1.307 | 1.536 | 0.394 | 0.295 | 0.333 | 1.356 | 1.683 | 0.414 | 0.275 | 0.300 | 0.056 |

ERA | ERS | 0.1 | 50 | 30 | 1.412 | 1.618 | 0.368 | 0.314 | 0.305 | 1.465 | 1.770 | 0.388 | 0.292 | 0.275 | 0.052 |

ERA | ERS | 0.1 | 60 | 20 | 1.467 | 1.657 | 0.355 | 0.325 | 0.290 | 1.504 | 1.789 | 0.377 | 0.302 | 0.263 | 0.051 |

ERA | ERS | 0.1 | 60 | 30 | 1.590 | 1.754 | 0.328 | 0.348 | 0.261 | 1.632 | 1.889 | 0.349 | 0.323 | 0.236 | 0.048 |

ERA | HES | 0.05 | 50 | 20 | 3.598 | 7.554 | 0.594 | 0.301 | 0.554 | 4.493 | 10.916 | 0.646 | 0.282 | 0.461 | 0.149 |

ERA | HES | 0.05 | 50 | 30 | 3.899 | 8.040 | 0.581 | 0.310 | 0.539 | 4.827 | 11.457 | 0.633 | 0.291 | 0.447 | 0.150 |

ERA | HES | 0.05 | 60 | 20 | 4.553 | 9.000 | 0.551 | 0.335 | 0.506 | 5.160 | 11.827 | 0.613 | 0.311 | 0.420 | 0.154 |

ERA | HES | 0.05 | 60 | 30 | 4.976 | 9.648 | 0.536 | 0.346 | 0.490 | 5.580 | 12.476 | 0.598 | 0.321 | 0.404 | 0.154 |

ERA | HES | 0.1 | 50 | 20 | 8.942 | 14.632 | 0.394 | 0.522 | 0.333 | 6.785 | 13.328 | 0.504 | 0.474 | 0.263 | 0.192 |

ERA | HES | 0.1 | 50 | 30 | 10.233 | 16.246 | 0.368 | 0.549 | 0.305 | 7.428 | 14.150 | 0.483 | 0.494 | 0.242 | 0.189 |

ERA | HES | 0.1 | 60 | 20 | 10.935 | 17.085 | 0.355 | 0.565 | 0.290 | 7.590 | 14.288 | 0.474 | 0.509 | 0.231 | 0.191 |

ERA | HES | 0.1 | 60 | 30 | 12.650 | 19.147 | 0.328 | 0.596 | 0.261 | 8.368 | 15.260 | 0.452 | 0.532 | 0.210 | 0.187 |

HEA | ERS | 0.05 | 50 | 20 | 1.854 | 3.755 | 0.702 | 0.591 | 0.672 | 4.103 | 9.868 | 0.729 | 0.527 | 0.368 | 0.334 |

HEA | ERS | 0.05 | 50 | 30 | 1.940 | 3.866 | 0.695 | 0.597 | 0.664 | 4.411 | 10.366 | 0.717 | 0.534 | 0.355 | 0.334 |

HEA | ERS | 0.05 | 60 | 20 | 2.433 | 4.633 | 0.664 | 0.618 | 0.630 | 4.898 | 11.088 | 0.699 | 0.544 | 0.337 | 0.332 |

HEA | ERS | 0.05 | 60 | 30 | 2.560 | 4.790 | 0.655 | 0.627 | 0.620 | 5.305 | 11.715 | 0.685 | 0.552 | 0.323 | 0.330 |

HEA | ERS | 0.1 | 50 | 20 | 3.140 | 5.378 | 0.589 | 0.811 | 0.548 | 6.310 | 12.724 | 0.630 | 0.707 | 0.202 | 0.391 |

HEA | ERS | 0.1 | 50 | 30 | 3.363 | 5.637 | 0.576 | 0.821 | 0.533 | 7.104 | 13.825 | 0.606 | 0.718 | 0.186 | 0.381 |

HEA | ERS | 0.1 | 60 | 20 | 4.135 | 6.737 | 0.546 | 0.836 | 0.500 | 7.416 | 14.206 | 0.597 | 0.723 | 0.179 | 0.378 |

HEA | ERS | 0.1 | 60 | 30 | 4.469 | 7.121 | 0.530 | 0.848 | 0.483 | 8.441 | 15.577 | 0.570 | 0.736 | 0.162 | 0.365 |

HEA | HES | 0.05 | 50 | 20 | 2.724 | 6.166 | 0.702 | 0.584 | 0.672 | 6.331 | 17.011 | 0.738 | 0.530 | 0.396 | 0.315 |

HEA | HES | 0.05 | 50 | 30 | 2.863 | 6.385 | 0.695 | 0.590 | 0.664 | 6.756 | 17.756 | 0.728 | 0.536 | 0.385 | 0.316 |

HEA | HES | 0.05 | 60 | 20 | 3.608 | 7.659 | 0.664 | 0.611 | 0.630 | 7.351 | 18.528 | 0.709 | 0.549 | 0.361 | 0.319 |

HEA | HES | 0.05 | 60 | 30 | 3.818 | 7.975 | 0.655 | 0.619 | 0.620 | 7.897 | 19.437 | 0.697 | 0.556 | 0.348 | 0.318 |

HEA | HES | 0.1 | 50 | 20 | 5.208 | 10.094 | 0.589 | 0.781 | 0.548 | 8.912 | 20.301 | 0.654 | 0.702 | 0.240 | 0.379 |

HEA | HES | 0.1 | 50 | 30 | 5.610 | 10.656 | 0.576 | 0.790 | 0.533 | 9.735 | 21.506 | 0.636 | 0.710 | 0.226 | 0.374 |

HEA | HES | 0.1 | 60 | 20 | 6.863 | 12.535 | 0.546 | 0.806 | 0.501 | 10.192 | 21.985 | 0.623 | 0.719 | 0.212 | 0.374 |

HEA | HES | 0.1 | 60 | 30 | 7.468 | 13.348 | 0.530 | 0.817 | 0.483 | 11.226 | 23.441 | 0.604 | 0.729 | 0.196 | 0.368 |

PCA | ERS | 0.05 | 50 | 20 | 0.762 | 2.337 | 0.673 | 0.473 | 0.640 | 2.372 | 12.737 | 0.682 | 0.462 | 0.419 | 0.230 |

PCA | ERS | 0.05 | 50 | 30 | 0.798 | 2.407 | 0.660 | 0.498 | 0.626 | 2.641 | 13.843 | 0.669 | 0.486 | 0.396 | 0.239 |

PCA | ERS | 0.05 | 60 | 20 | 0.978 | 3.094 | 0.645 | 0.517 | 0.610 | 3.020 | 15.335 | 0.658 | 0.503 | 0.375 | 0.248 |

PCA | ERS | 0.05 | 60 | 30 | 1.028 | 3.195 | 0.632 | 0.546 | 0.595 | 3.408 | 16.816 | 0.643 | 0.531 | 0.349 | 0.257 |

PCA | ERS | 0.1 | 50 | 20 | 1.553 | 4.613 | 0.553 | 0.727 | 0.508 | 4.362 | 19.515 | 0.574 | 0.701 | 0.218 | 0.311 |

PCA | ERS | 0.1 | 50 | 30 | 1.654 | 4.825 | 0.537 | 0.768 | 0.490 | 5.186 | 22.225 | 0.555 | 0.740 | 0.188 | 0.320 |

PCA | ERS | 0.1 | 60 | 20 | 2.092 | 6.237 | 0.529 | 0.765 | 0.481 | 5.427 | 23.006 | 0.555 | 0.736 | 0.187 | 0.321 |

PCA | ERS | 0.1 | 60 | 30 | 2.238 | 6.543 | 0.513 | 0.806 | 0.464 | 6.573 | 26.474 | 0.536 | 0.774 | 0.158 | 0.328 |

PCA | HES | 0.05 | 50 | 20 | 2.132 | 5.285 | 0.673 | 0.525 | 0.640 | 5.142 | 22.448 | 0.701 | 0.496 | 0.432 | 0.239 |

PCA | HES | 0.05 | 50 | 30 | 2.290 | 5.552 | 0.660 | 0.549 | 0.626 | 5.574 | 23.685 | 0.688 | 0.516 | 0.413 | 0.244 |

PCA | HES | 0.05 | 60 | 20 | 2.616 | 6.316 | 0.645 | 0.568 | 0.610 | 6.030 | 24.915 | 0.678 | 0.534 | 0.390 | 0.256 |

PCA | HES | 0.05 | 60 | 30 | 2.811 | 6.634 | 0.632 | 0.594 | 0.595 | 6.589 | 26.449 | 0.664 | 0.556 | 0.369 | 0.261 |

PCA | HES | 0.1 | 50 | 20 | 4.248 | 9.239 | 0.553 | 0.762 | 0.508 | 7.712 | 28.419 | 0.607 | 0.712 | 0.254 | 0.314 |

PCA | HES | 0.1 | 50 | 30 | 4.606 | 9.777 | 0.537 | 0.797 | 0.490 | 8.545 | 30.479 | 0.590 | 0.742 | 0.230 | 0.319 |

PCA | HES | 0.1 | 60 | 20 | 5.195 | 11.193 | 0.529 | 0.796 | 0.481 | 8.870 | 31.287 | 0.588 | 0.742 | 0.222 | 0.325 |

PCA | HES | 0.1 | 60 | 30 | 5.625 | 11.838 | 0.513 | 0.830 | 0.464 | 9.930 | 33.825 | 0.571 | 0.772 | 0.199 | 0.329 |

Model 1 | Model 2 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Arr. | $\mathbf{\gamma}$ | $\mathit{K}$ | $\mathit{L}$ | ${\widehat{\mathbf{\mu}}}_{\mathbf{1}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{1}}$ | ${\Gamma}_{\mathbf{1}}$ | ${\mathbf{\xi}}_{\mathbf{1}}$ | ${\mathbf{\kappa}}_{\mathbf{1}}$ | ${\widehat{\mathbf{\mu}}}_{\mathbf{2}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{2}}$ | ${\Gamma}_{\mathbf{2}}$ | ${\mathbf{\xi}}_{\mathbf{2}}$ | ${\mathbf{\kappa}}_{\mathbf{2}}$ |

ERA | 0.05 | 50 | 20 | 11.692 | 16.321 | 47.055 | 0.735 | 27.228 | 12.398 | 16.642 | 46.897 | 0.718 | 27.861 |

ERA | 0.05 | 50 | 30 | 12.410 | 16.727 | 44.536 | 0.804 | 24.878 | 13.155 | 17.034 | 44.260 | 0.789 | 25.342 |

ERA | 0.05 | 60 | 20 | 15.421 | 19.918 | 56.746 | 0.677 | 29.537 | 16.253 | 20.247 | 56.578 | 0.660 | 30.296 |

ERA | 0.05 | 60 | 30 | 16.403 | 20.333 | 54.240 | 0.735 | 27.215 | 17.223 | 20.632 | 54.003 | 0.720 | 27.791 |

ERA | 0.10 | 50 | 20 | 18.142 | 17.416 | 44.610 | 0.583 | 17.158 | 18.849 | 17.486 | 44.379 | 0.566 | 17.678 |

ERA | 0.10 | 50 | 30 | 19.864 | 17.752 | 40.356 | 0.673 | 14.855 | 20.583 | 17.778 | 40.015 | 0.658 | 15.206 |

ERA | 0.10 | 60 | 20 | 22.995 | 20.644 | 54.013 | 0.515 | 19.433 | 23.727 | 20.715 | 53.788 | 0.500 | 20.003 |

ERA | 0.10 | 60 | 30 | 25.101 | 20.820 | 49.822 | 0.583 | 17.155 | 25.839 | 20.818 | 49.509 | 0.569 | 17.592 |

HEA | 0.05 | 50 | 20 | 22.059 | 22.423 | 47.842 | 0.529 | 37.781 | 23.096 | 21.213 | 46.314 | 0.499 | 40.081 |

HEA | 0.05 | 50 | 30 | 22.786 | 22.726 | 46.149 | 0.563 | 35.502 | 23.943 | 21.331 | 43.373 | 0.557 | 35.894 |

HEA | 0.05 | 60 | 20 | 26.855 | 26.413 | 57.532 | 0.499 | 40.049 | 28.058 | 24.908 | 55.905 | 0.462 | 43.287 |

HEA | 0.05 | 60 | 30 | 27.820 | 26.714 | 55.848 | 0.529 | 37.782 | 29.090 | 24.932 | 53.056 | 0.511 | 39.182 |

HEA | 0.10 | 50 | 20 | 28.909 | 21.539 | 46.540 | 0.376 | 26.572 | 28.910 | 19.603 | 43.757 | 0.363 | 27.541 |

HEA | 0.10 | 50 | 30 | 30.178 | 21.720 | 44.134 | 0.410 | 24.368 | 30.311 | 19.383 | 39.322 | 0.431 | 23.199 |

HEA | 0.10 | 60 | 20 | 34.649 | 25.150 | 55.954 | 0.348 | 28.752 | 34.484 | 22.873 | 52.992 | 0.328 | 30.487 |

HEA | 0.10 | 60 | 30 | 36.260 | 25.219 | 53.575 | 0.376 | 26.573 | 36.080 | 22.419 | 48.708 | 0.382 | 26.208 |

PCA | 0.05 | 50 | 20 | 16.747 | 17.684 | 45.521 | 0.615 | 32.508 | 17.322 | 17.644 | 45.190 | 0.603 | 33.193 |

PCA | 0.05 | 50 | 30 | 18.184 | 18.117 | 41.838 | 0.697 | 28.701 | 18.788 | 18.022 | 41.335 | 0.688 | 29.058 |

PCA | 0.05 | 60 | 20 | 21.269 | 21.103 | 55.111 | 0.553 | 36.149 | 21.979 | 21.000 | 54.703 | 0.539 | 37.141 |

PCA | 0.05 | 60 | 30 | 23.030 | 21.436 | 51.562 | 0.615 | 32.510 | 23.778 | 21.255 | 50.943 | 0.604 | 33.102 |

PCA | 0.10 | 50 | 20 | 23.346 | 17.188 | 42.458 | 0.454 | 22.055 | 23.965 | 16.855 | 41.808 | 0.440 | 22.751 |

PCA | 0.10 | 50 | 30 | 25.980 | 17.206 | 36.985 | 0.541 | 18.480 | 26.618 | 16.736 | 36.101 | 0.534 | 18.762 |

PCA | 0.10 | 60 | 20 | 28.547 | 20.167 | 51.733 | 0.394 | 25.408 | 29.262 | 19.730 | 50.926 | 0.379 | 26.462 |

PCA | 0.10 | 60 | 30 | 31.492 | 19.905 | 46.540 | 0.454 | 22.055 | 32.211 | 19.311 | 45.446 | 0.444 | 22.564 |

Model 1 ($\mathit{\mu}$ = 1.1, $\mathit{\gamma}$ = 0.1, K = 50) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Arr. | Ser. | $\mathit{L}$ | ${\mathbf{\mu}}_{\mathbf{1}}$ | ${\mathbf{\sigma}}_{\mathbf{1}}$ | ${\widehat{\mathbf{\mu}}}_{\mathbf{1}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{1}}$ | ${\Gamma}_{\mathbf{1}}$ | ${\mathbf{\kappa}}_{\mathbf{1}}$ | ${\mathbf{\nu}}_{\mathbf{1}}$ | ${\mathbf{\nu}}_{\mathbf{1}}^{*}$ | ${\mathbf{\vartheta}}_{\mathbf{1}}$ | ${\mathbf{\vartheta}}_{\mathbf{1}}^{*}$ |

ERA | ERS | 20 | 1.307 | 1.536 | 18.142 | 17.416 | 44.610 | 17.158 | 0.394 | 0.295 | 0.333 | 0.000 |

ERA | ERS | 30 | 1.412 | 1.618 | 19.864 | 17.752 | 40.356 | 14.855 | 0.368 | 0.314 | 0.305 | 0.000 |

ERA | EXS | 20 | 1.844 | 2.391 | 18.143 | 17.415 | 44.609 | 17.158 | 0.394 | 0.345 | 0.333 | 0.000 |

ERA | EXS | 30 | 2.023 | 2.566 | 19.867 | 17.751 | 40.353 | 14.855 | 0.368 | 0.366 | 0.305 | 0.000 |

ERA | HES | 20 | 8.942 | 14.632 | 18.141 | 17.415 | 44.608 | 17.165 | 0.394 | 0.522 | 0.333 | 0.000 |

ERA | HES | 30 | 10.233 | 16.246 | 19.865 | 17.750 | 40.348 | 14.861 | 0.368 | 0.549 | 0.305 | 0.000 |

HEA | ERS | 20 | 3.140 | 5.378 | 28.909 | 21.539 | 46.540 | 26.572 | 0.589 | 0.811 | 0.548 | 0.000 |

HEA | ERS | 30 | 3.363 | 5.637 | 30.178 | 21.720 | 44.134 | 24.368 | 0.576 | 0.821 | 0.533 | 0.000 |

HEA | EXS | 20 | 3.244 | 5.605 | 28.909 | 21.539 | 46.540 | 26.570 | 0.589 | 0.809 | 0.548 | 0.000 |

HEA | EXS | 30 | 3.476 | 5.878 | 30.178 | 21.720 | 44.134 | 24.366 | 0.576 | 0.819 | 0.533 | 0.000 |

HEA | HES | 20 | 5.208 | 10.094 | 28.909 | 21.539 | 46.540 | 26.573 | 0.589 | 0.781 | 0.548 | 0.000 |

HEA | HES | 30 | 5.610 | 10.656 | 30.177 | 21.720 | 44.134 | 24.367 | 0.576 | 0.790 | 0.533 | 0.000 |

PCA | ERS | 20 | 1.553 | 4.613 | 23.346 | 17.188 | 42.458 | 22.055 | 0.553 | 0.727 | 0.508 | 0.000 |

PCA | ERS | 30 | 1.654 | 4.825 | 25.980 | 17.206 | 36.985 | 18.480 | 0.537 | 0.768 | 0.490 | 0.000 |

PCA | EXS | 20 | 1.681 | 4.771 | 23.345 | 17.190 | 42.460 | 22.050 | 0.553 | 0.732 | 0.508 | 0.000 |

PCA | EXS | 30 | 1.793 | 4.991 | 25.982 | 17.205 | 36.983 | 18.479 | 0.537 | 0.772 | 0.490 | 0.000 |

PCA | HES | 20 | 4.248 | 9.239 | 23.345 | 17.189 | 42.458 | 22.055 | 0.553 | 0.762 | 0.508 | 0.000 |

PCA | HES | 30 | 4.606 | 9.777 | 25.978 | 17.207 | 36.987 | 18.480 | 0.537 | 0.797 | 0.490 | 0.000 |

ERA | ERS | 20 | 1.356 | 1.683 | 18.849 | 17.486 | 44.379 | 17.678 | 0.414 | 0.275 | 0.356 | 0.156 |

ERA | ERS | 30 | 1.465 | 1.770 | 20.583 | 17.778 | 40.015 | 15.206 | 0.388 | 0.292 | 0.327 | 0.160 |

ERA | EXS | 20 | 1.916 | 2.678 | 19.334 | 17.625 | 44.312 | 18.046 | 0.428 | 0.314 | 0.371 | 0.210 |

ERA | EXS | 30 | 2.101 | 2.860 | 21.112 | 17.925 | 39.932 | 15.509 | 0.402 | 0.332 | 0.342 | 0.217 |

ERA | HES | 20 | 6.785 | 13.328 | 22.408 | 18.541 | 44.228 | 20.777 | 0.504 | 0.474 | 0.454 | 0.422 |

ERA | HES | 30 | 7.428 | 14.150 | 24.551 | 19.020 | 40.071 | 18.015 | 0.483 | 0.494 | 0.431 | 0.439 |

HEA | ERS | 20 | 6.310 | 12.724 | 28.910 | 19.603 | 43.757 | 27.541 | 0.630 | 0.707 | 0.593 | 0.659 |

HEA | ERS | 30 | 7.104 | 13.825 | 30.311 | 19.383 | 39.322 | 23.199 | 0.606 | 0.718 | 0.567 | 0.673 |

HEA | EXS | 20 | 6.480 | 13.190 | 29.025 | 19.635 | 43.826 | 27.733 | 0.632 | 0.706 | 0.595 | 0.654 |

HEA | EXS | 30 | 7.280 | 14.306 | 30.446 | 19.437 | 39.457 | 23.427 | 0.609 | 0.717 | 0.570 | 0.668 |

HEA | HES | 20 | 8.912 | 20.301 | 30.279 | 19.907 | 44.550 | 30.039 | 0.654 | 0.702 | 0.619 | 0.612 |

HEA | HES | 30 | 9.735 | 21.506 | 31.912 | 19.896 | 40.902 | 26.200 | 0.636 | 0.710 | 0.600 | 0.624 |

PCA | ERS | 20 | 4.362 | 19.515 | 23.965 | 16.855 | 41.808 | 22.751 | 0.574 | 0.701 | 0.529 | 0.587 |

PCA | ERS | 30 | 5.186 | 22.225 | 26.618 | 16.736 | 36.101 | 18.762 | 0.555 | 0.740 | 0.509 | 0.630 |

PCA | EXS | 20 | 4.568 | 20.087 | 24.067 | 16.891 | 41.823 | 22.856 | 0.576 | 0.702 | 0.533 | 0.584 |

PCA | EXS | 30 | 5.392 | 22.750 | 26.719 | 16.774 | 36.134 | 18.851 | 0.557 | 0.740 | 0.512 | 0.625 |

PCA | HES | 20 | 7.712 | 28.419 | 25.607 | 17.460 | 42.148 | 24.912 | 0.607 | 0.712 | 0.568 | 0.552 |

PCA | HES | 30 | 8.545 | 30.479 | 28.307 | 17.489 | 36.858 | 20.822 | 0.590 | 0.742 | 0.549 | 0.581 |

Service Dist. | ERS | EXS | HES | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Arr. | $\mathit{L}/\mathit{K}$ | 40 | 50 | 60 | 40 | 50 | 60 | 40 | 50 | 60 |

Model 1 | ||||||||||

ERA | 10 | 13.085 | 12.946 | 13.125 | 13.084 | 12.942 | 13.123 | 13.080 | 12.941 | 13.123 |

ERA | 20 | 13.982 | 13.695 | 13.799 | 13.981 | 13.695 | 13.797 | 13.979 | 13.691 | 13.796 |

ERA | 30 | 15.304 | 14.751 | 14.718 | 15.303 | 14.750 | 14.717 | 15.301 | 14.745 | 14.708 |

HEA | 10 | 15.453 | 15.999 | 16.667 | 15.453 | 16.000 | 16.669 | 15.453 | 16.000 | 16.668 |

HEA | 20 | 15.925 | 16.470 | 17.145 | 15.926 | 16.471 | 17.146 | 15.926 | 16.470 | 17.146 |

HEA | 30 | 16.444 | 16.982 | 17.662 | 16.445 | 16.983 | 17.663 | 16.444 | 16.983 | 17.663 |

PCA | 10 | 14.161 | 14.445 | 14.976 | 14.164 | 14.446 | 14.981 | 14.162 | 14.444 | 14.979 |

PCA | 20 | 15.355 | 15.448 | 15.886 | 15.356 | 15.451 | 15.888 | 15.355 | 15.449 | 15.885 |

PCA | 30 | 17.125 | 16.807 | 17.046 | 17.127 | 16.810 | 17.050 | 17.125 | 16.808 | 17.047 |

Model 2 | ||||||||||

ERA | 10 | 13.258 | 13.184 | 13.412 | 13.363 | 13.333 | 13.598 | 13.913 | 14.120 | 14.596 |

ERA | 20 | 14.169 | 13.928 | 14.077 | 14.281 | 14.084 | 14.260 | 14.871 | 14.957 | 15.364 |

ERA | 30 | 15.541 | 14.993 | 14.979 | 15.656 | 15.144 | 15.161 | 16.158 | 16.000 | 16.289 |

HEA | 10 | 15.622 | 16.262 | 17.034 | 15.635 | 16.283 | 17.062 | 15.778 | 16.507 | 17.358 |

HEA | 20 | 16.280 | 16.786 | 17.465 | 16.292 | 16.812 | 17.502 | 16.423 | 17.089 | 17.901 |

HEA | 30 | 17.263 | 17.557 | 18.106 | 17.258 | 17.577 | 18.143 | 17.229 | 17.795 | 18.550 |

PCA | 10 | 14.306 | 14.664 | 15.277 | 14.333 | 14.703 | 15.323 | 14.618 | 15.096 | 15.803 |

PCA | 20 | 15.533 | 15.681 | 16.174 | 15.556 | 15.720 | 16.228 | 15.787 | 16.093 | 16.710 |

PCA | 30 | 17.360 | 17.072 | 17.347 | 17.375 | 17.107 | 17.401 | 17.428 | 17.373 | 17.817 |

$(\mathit{K},\mathit{L})$ System ($\mathit{\gamma}$ = 0.1, $\mathit{\mu}$ = 1.1, K = 60, L = 20) | $(\mathit{s},\mathit{S})$ System ($\mathit{\gamma}$ = 0.1, $\mathit{\mu}$ = 1.1, S = 60, s = 20) | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Model 1 | |||||||||||||||

Arr. | Ser. | ${\mathbf{\vartheta}}_{\mathbf{1}}$ | ${\mathbf{\nu}}_{\mathbf{1}}$ | ${\mathbf{\mu}}_{\mathbf{1}}$ | ${\mathbf{\sigma}}_{\mathbf{1}}$ | ${\widehat{\mathbf{\mu}}}_{\mathbf{1}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{1}}$ | ${\mathbf{\kappa}}_{\mathbf{1}}$ | ${\mathbf{\vartheta}}_{\mathbf{1}}$ | ${\mathbf{\nu}}_{\mathbf{1}}$ | ${\mathbf{\mu}}_{\mathbf{2}}$ | ${\mathbf{\sigma}}_{\mathbf{2}}$ | ${\mathbf{\mu}}_{\mathbf{2}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{2}}$ | ${\mathbf{\kappa}}_{\mathbf{1}}$ |

ERA | ERS | 0.355 | 0.290 | 1.467 | 1.657 | 22.995 | 20.644 | 19.433 | 0.423 | 0.365 | 1.209 | 1.475 | 15.660 | 16.179 | 16.221 |

ERA | EXS | 0.355 | 0.290 | 2.116 | 2.644 | 22.999 | 20.642 | 19.439 | 0.423 | 0.365 | 1.686 | 2.267 | 15.663 | 16.178 | 16.222 |

ERA | HES | 0.355 | 0.290 | 10.935 | 17.085 | 22.986 | 20.646 | 19.437 | 0.423 | 0.365 | 7.784 | 13.214 | 15.657 | 16.179 | 16.223 |

HEA | ERS | 0.546 | 0.500 | 4.135 | 6.737 | 34.649 | 25.150 | 28.752 | 0.629 | 0.592 | 2.438 | 4.414 | 24.481 | 19.098 | 25.437 |

HEA | EXS | 0.546 | 0.501 | 4.272 | 7.014 | 34.650 | 25.149 | 28.749 | 0.629 | 0.592 | 2.520 | 4.601 | 24.481 | 19.098 | 25.436 |

HEA | HES | 0.546 | 0.501 | 6.863 | 12.535 | 34.649 | 25.150 | 28.751 | 0.629 | 0.592 | 4.018 | 8.259 | 24.481 | 19.098 | 25.437 |

PCA | ERS | 0.529 | 0.481 | 2.092 | 6.237 | 28.547 | 20.167 | 25.408 | 0.568 | 0.525 | 1.220 | 3.468 | 21.748 | 16.972 | 21.571 |

PCA | EXS | 0.529 | 0.482 | 2.239 | 6.409 | 28.550 | 20.163 | 25.411 | 0.568 | 0.525 | 1.338 | 3.620 | 21.749 | 16.970 | 21.571 |

PCA | HES | 0.529 | 0.481 | 5.195 | 11.193 | 28.547 | 20.163 | 25.415 | 0.568 | 0.525 | 3.677 | 7.968 | 21.747 | 16.972 | 21.572 |

Model 2 | |||||||||||||||

Arr. | Ser. | ${\mathbf{\vartheta}}_{\mathbf{2}}$ | ${\mathbf{\nu}}_{\mathbf{2}}$ | ${\mathbf{\mu}}_{\mathbf{2}}$ | ${\mathbf{\sigma}}_{\mathbf{2}}$ | ${\widehat{\mathbf{\mu}}}_{\mathbf{2}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{2}}$ | ${\mathbf{\kappa}}_{\mathbf{2}}$ | ${\mathbf{\vartheta}}_{\mathbf{2}}$ | ${\mathbf{\nu}}_{\mathbf{2}}$ | ${\mathbf{\mu}}_{\mathbf{2}}$ | ${\mathbf{\sigma}}_{\mathbf{2}}$ | ${\mathbf{\mu}}_{\mathbf{2}}$ | ${\widehat{\mathbf{\sigma}}}_{\mathbf{2}}$ | ${\mathbf{\kappa}}_{\mathbf{2}}$ |

ERA | ERS | 0.377 | 0.315 | 1.504 | 1.789 | 23.727 | 20.715 | 20.003 | 0.441 | 0.385 | 1.271 | 1.637 | 16.357 | 16.730 | 16.748 |

ERA | EXS | 0.391 | 0.330 | 2.151 | 2.883 | 24.273 | 20.838 | 20.445 | 0.453 | 0.398 | 1.796 | 2.600 | 16.843 | 16.946 | 17.113 |

ERA | HES | 0.474 | 0.421 | 7.590 | 14.288 | 27.673 | 21.708 | 23.619 | 0.524 | 0.476 | 6.418 | 12.959 | 19.846 | 18.270 | 19.642 |

HEA | ERS | 0.597 | 0.556 | 7.416 | 14.206 | 34.484 | 22.873 | 30.487 | 0.651 | 0.616 | 5.766 | 12.099 | 25.218 | 18.109 | 26.769 |

HEA | EXS | 0.599 | 0.559 | 7.594 | 14.674 | 34.623 | 22.909 | 30.690 | 0.653 | 0.618 | 5.931 | 12.567 | 25.307 | 18.153 | 26.928 |

HEA | HES | 0.623 | 0.586 | 10.192 | 21.985 | 36.118 | 23.210 | 33.174 | 0.675 | 0.643 | 8.199 | 19.488 | 26.326 | 18.629 | 28.909 |

PCA | ERS | 0.555 | 0.508 | 5.427 | 23.006 | 29.262 | 19.730 | 26.462 | 0.583 | 0.540 | 4.024 | 18.494 | 22.363 | 17.226 | 22.283 |

PCA | EXS | 0.557 | 0.512 | 5.641 | 23.527 | 29.380 | 19.768 | 26.565 | 0.585 | 0.543 | 4.220 | 19.058 | 22.436 | 17.207 | 22.384 |

PCA | HES | 0.588 | 0.547 | 8.870 | 31.287 | 31.069 | 20.387 | 28.799 | 0.617 | 0.578 | 7.246 | 27.332 | 23.755 | 17.800 | 24.279 |

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## Share and Cite

**MDPI and ACS Style**

Chakravarthy, S.R.; Rao, B.M.
Queuing-Inventory Models with *MAP* Demands and Random Replenishment Opportunities. *Mathematics* **2021**, *9*, 1092.
https://doi.org/10.3390/math9101092

**AMA Style**

Chakravarthy SR, Rao BM.
Queuing-Inventory Models with *MAP* Demands and Random Replenishment Opportunities. *Mathematics*. 2021; 9(10):1092.
https://doi.org/10.3390/math9101092

**Chicago/Turabian Style**

Chakravarthy, Srinivas R., and B. Madhu Rao.
2021. "Queuing-Inventory Models with *MAP* Demands and Random Replenishment Opportunities" *Mathematics* 9, no. 10: 1092.
https://doi.org/10.3390/math9101092