# Optimal Decision Making for Customer-Intensive Services Based on Queuing System Considering the Heterogeneity of Customer Advertising Perception

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model Establishment

#### 3.1. Customer Utility

#### 3.2. Decision Making and Revenue of Service Providers

- (1)
- There are L customers in the system (Situation ${S}_{L}$)
- ①
- Situation ${S}_{L1}$. All L customers join the system. Since ${V}_{NH}\left(\mu ,P\right)>{V}_{NL}\left(\mu ,P\right)$, this effectively means that all H customers also join the service system.
- ②
- Situation ${S}_{L2}$. A part of L customers join the queue, and ${V}_{NL}\left(\mu ,P\right)=0$. Since ${V}_{NH}\left(\mu ,P\right)>{V}_{NL}\left(\mu ,P\right)$, it means that the utility of H customers is positive, and they will all join the service system.

- (2)
- There is no L customer in the system (Situation ${S}_{H}$)
- ①
- Situation ${S}_{H1}$. All H customers join the service system, but no L customers join. ${V}_{NH}\left(\mu ,P\right)>0$, ${V}_{NL}\left(\mu ,P\right)<0$.
- ②
- Situation ${S}_{H2}$. Only some H customers join the service system, and ${V}_{NH}\left(\mu ,P\right)=0$. This means that the net utility of L customers must be negative, and they will not join the service system.

#### 3.3. Service Rate Range

## 4. Decision Making with Low-Sensitivity Customers

#### 4.1. Optimal Decision Making

**Proposition**

**1.**

- i.
- From $0<\mathsf{\Lambda}\le {\widehat{\lambda}}_{L2}$, we know that the effective arrival rate of ${S}_{L1}$ is less than the effective customer arrival rate of ${S}_{L2}$.
- ii.
- When $\mathsf{\Lambda}$ satisfies $0<\mathsf{\Lambda}\le {\widehat{\lambda}}_{L2}$, that is, situation ${S}_{L1}$, the optimal decision is: ${\mu}_{L1}^{*}=\mathsf{\Lambda}+\sqrt{\frac{c}{\beta}}$, ${P}_{L1}^{*}\left({\mu}^{*}\right)=m+{k}_{L}a-\beta \mathsf{\Lambda}-2\sqrt{\beta c}$; the optimal effective arrival rate is ${\lambda}_{L1}^{*}\left({\mu}^{*},{P}^{*}\left({\mu}^{*}\right)\right)=\mathsf{\Lambda}$.
- iii.
- When ${\widehat{\lambda}}_{L2}<\mathsf{\Lambda}<\frac{{\widehat{\lambda}}_{L2}}{q}$, that is, situation ${S}_{L2}$, a part of L customers can enter the system, and the optimal decision is: ${\mu}_{L2}^{*}=\frac{m+{k}_{L}a}{2\beta}$, ${P}_{L2}^{*}\left({\mu}^{*}\right)=\frac{m+{k}_{L}a-2\sqrt{\beta c}}{2}$; the optimal effective arrival rate is ${\lambda}_{L2}^{*}\left({\mu}^{*},{P}^{*}\left({\mu}^{*}\right)\right)=\frac{m+{k}_{L}a-2\sqrt{\beta c}}{2\beta}$. (Proof in Appendix A).

**Lemma**

**1.**

#### 4.2. Optimal Revenue

**Lemma**

**2.**

## 5. Decision Making without Low-Sensitivity Customers

#### 5.1. Optimal Decision Making

**Proposition**

**2.**

- i.
- As the potential arrival rate increases further, the L customers’ utility continues to decrease, and they exit the system completely.
- ii.
- At ${\widehat{\lambda}}_{L2}<q\mathsf{\Lambda}\le {\widehat{\lambda}}_{H2}(\frac{{\widehat{\lambda}}_{L2}}{q}<\mathsf{\Lambda}\le \frac{{\widehat{\lambda}}_{H2}}{q}$), all H customers are served in ${S}_{H1}$. The optimal service rate and price are ${\mu}_{H1}^{*}=q\mathsf{\Lambda}+\sqrt{\frac{c}{\beta}},{P}_{H1}^{*}\left({\mu}^{*}\right)=m+{k}_{H}a-q\beta \mathsf{\Lambda}-2\sqrt{\beta c}$; the optimal effective arrival rate is ${\lambda}_{H1}^{*}\left({\mu}^{*},{P}^{*}\left({\mu}^{*}\right)\right)=q\mathsf{\Lambda}$.
- iii.
- At $q\mathsf{\Lambda}>{\widehat{\lambda}}_{H2}$($\mathsf{\Lambda}>\frac{{\widehat{\lambda}}_{H2}}{q}$), only some customers in ${S}_{H2}$ can obtain positive net utility. The optimal service rate and price are ${\mu}_{H2}^{*}=\frac{m+{k}_{H}a}{2\beta}$, ${P}_{H2}^{*}\left({\mu}^{*}\right)=\frac{m+{k}_{H}a-2\sqrt{\beta c}}{2}$; the optimal effective arrival rate is ${\lambda}_{H2}^{*}\left({\mu}^{*},{P}^{*}\left({\mu}^{*}\right)\right)=\frac{m+{k}_{H}a-2\sqrt{\beta c}}{2\beta}$. (Proof in Appendix A).

**Lemma**

**3.**

#### 5.2. Optimal Revenue

**Lemma**

**4.**

#### 5.3. Comparison of Revenue

**Lemma**

**5.**

#### 5.4. Simulation

## 6. Extensions: The Impact of Ignoring Heterogeneity

#### 6.1. All Homogeneous L Customers

**Homogeneous L decision ①:**when $0\le \mathsf{\Lambda}\le {\widehat{\lambda}}_{L}$, the optimal decisions are ${\mu}_{L}^{*}=\mathsf{\Lambda}+\sqrt{\frac{c}{\beta}},{P}_{L}^{*}\left({\mu}^{*}\right)=m+{k}_{L}a-\beta \mathsf{\Lambda}-2\sqrt{\beta c}$, and the service provider can serve all customers.

**Homogeneous L decision ②:**when $\mathsf{\Lambda}>{\widehat{\lambda}}_{L}$, the optimal decisions are ${\mu}_{L}^{*}=\frac{m+{k}_{L}a}{2\beta},{P}_{L}^{*}\left({\mu}^{*}\right)=\frac{m+{k}_{L}a-2\sqrt{\beta c}}{2}$, and the service provider serves only a part of the customers.

**Proposition**

**3.**

#### 6.2. All Homogeneous M Customers

**Homogeneous M decision**$\u2460$

**:**when $\mathsf{\Lambda}\le {\widehat{\lambda}}_{M}$, the optimal decisions are ${\mu}_{M}^{*}=\mathsf{\Lambda}+\sqrt{\frac{c}{\beta}},\text{}{P}_{M}^{*}\left({\mu}^{*}\right)=m+{k}_{M}a-\beta \mathsf{\Lambda}-2\sqrt{\beta c}$, and the service provider can serve all customers.

**Homogeneous M decision ②:**when $\mathsf{\Lambda}>{\widehat{\lambda}}_{M}$, the optimal decisions are ${\mu}_{M}^{*}=\frac{m+{k}_{M}a}{2\beta},\text{}{P}_{M}^{*}\left({\mu}^{*}\right)=\frac{m+{k}_{M}a-2\sqrt{\beta c}}{2}$.

**Proposition**

**4.**

#### 6.3. All Homogeneous H customers

**Homogeneous H decision ①:**when $\mathsf{\Lambda}\le {\widehat{\lambda}}_{H}$, the optimal decisions are ${\mu}_{H}^{*}=\mathsf{\Lambda}+\sqrt{\frac{c}{\beta}},\text{}{P}_{H}^{*}\left({\mu}^{*}\right)=m+{k}_{H}a-\beta \mathsf{\Lambda}-2\sqrt{\beta c}$, and the service provider can serve all customers.

**Homogeneous H decision ②:**when $\mathsf{\Lambda}>{\widehat{\lambda}}_{H}$, the optimal decisions are ${\mu}_{H}^{*}=\frac{m+{k}_{H}a}{2\beta},\text{}{P}_{H}^{*}\left({\mu}^{*}\right)=\frac{m+{k}_{H}a-2\sqrt{\beta c}}{2}$, and the service provider can serve only a part of the customers.

**Proposition**

**5.**

- i.
- In homogeneous H decision ①, if $\sqrt{\beta c}+{k}_{L}a-{k}_{H}a\ge 0$, the revenue is ${R}_{HL}^{*}={P}_{H}^{*}{\lambda}_{HL}^{*}=\left(m+{k}_{H}a-\beta \mathsf{\Lambda}-2\sqrt{\beta c}\right)\left(\mathsf{\Lambda}+\sqrt{\frac{c}{\beta}}-\frac{c}{\sqrt{\beta c}+{k}_{L}a-{k}_{H}a}\right)$; once $\sqrt{\beta c}+{k}_{L}a-{k}_{H}a<0$, the revenue is ${R}_{HH1}^{*}={P}_{H}^{*}{\lambda}_{HH1}^{*}=\left(m+{k}_{H}a-\beta \mathsf{\Lambda}-2\sqrt{\beta c}\right)q\mathsf{\Lambda}$.
- ii.
- In homogeneous H strategy ②, if $\mathsf{\Lambda}>{\widehat{\lambda}}_{H}$, the revenue in the homogeneous H decision is the same as that in ${S}_{H2}$. When ${\widehat{\lambda}}_{H}<\mathsf{\Lambda}<\frac{{\widehat{\lambda}}_{H}}{q}$, the revenue of the service provider is ${R}_{HH2}^{*}=\frac{m+{k}_{H}a-2\sqrt{\beta c}}{2}q\mathsf{\Lambda}$.

#### 6.4. Simulation

## 7. Conclusions

#### 7.1. Conclusions and Remarks

- (1)
- With the increase in the potential arrival rate, the service provider’s revenue from serving a part of customers is higher than that from serving all customers regardless of whether there are L customers in the service system.
- (2)
- The service provider has more dominance in the market when the potential arrival rate exceeds the effective arrival rate threshold (${\widehat{\lambda}}_{L2}<\mathsf{\Lambda}<\frac{{\widehat{\lambda}}_{L2}}{q}$ and $\mathsf{\Lambda}>\frac{{\widehat{\lambda}}_{H2}}{q}$); this is manifested in the fact that the decision making of the service provider is no longer affected by the potential customer arrival rate $\mathsf{\Lambda}$.
- (3)
- The revenue is further improved with the increase in the number of highly sensitive customers, which reflects the importance of advertising and market segmentation. Advertising can effectively increase the number of highly sensitive customers in the market, thereby bringing more revenue to the service provider.
- (4)
- Once the service provider adopts a homogeneous decision, it reduces the revenue in a large range, which shows that it is extremely necessary for the service provider to conduct market segmentation.

#### 7.2. Management Implication

- (1)
- When there are L customers in the market, the potential arrival rate $\mathsf{\Lambda}$ in the market must be low. In $0<\mathsf{\Lambda}\le {\widehat{\lambda}}_{L2}$, the decision making of service providers must be made according to the scale of $\mathsf{\Lambda}$. The potential arrival rate is low and advertising does not cause system congestion, so there is no need to change the service rate. However, in the case of ${\widehat{\lambda}}_{L2}<\mathsf{\Lambda}<\frac{{\widehat{\lambda}}_{L2}}{q}$, due to the increase in the potential arrival rate, the system congestion caused by advertising requires the service provider to increase the service rate.
- (2)
- When $\mathsf{\Lambda}$ is high, H customers have a crowding-out effect on L customers. In ${\widehat{\lambda}}_{L2}<q\mathsf{\Lambda}\le {\widehat{\lambda}}_{H2}$, the decision of service providers should be made according to the scale of $\mathsf{\Lambda}$ and the proportion $q$. When there are few H customers, the service provider only needs to control the crowding degree of the system through price. In $q\mathsf{\Lambda}>{\widehat{\lambda}}_{H2}$, since there are many H customers, the service provider not only needs to control the congestion of the system through higher prices but also needs to formulate a higher service rate.
- (3)
- Customer-intensive service providers should segment the target market, determine the general number of various customers through market research, and make optimal decisions accordingly. Meanwhile, service providers should tap more high-sensitivity customers through advertising so as to further improve service providers’ revenue. It also shows that forgoing some low-sensitivity customers often leads to higher revenue.

#### 7.3. Innovation and Contribution

#### 7.4. Future Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of Proposition 1.**

**Proof**

**of Lemma 1.**

**Proof**

**of Lemma 2.**

**Proof**

**of Proposition 2.**

**Proof**

**of Lemma 3.**

**Proof**

**of Lemma 4.**

**Proof**

**of Proposition 3.**

**Proof**

**of Proposition 4.**

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**Figure 1.**The left panel illustrates the optimal service rate with respect to $\mathsf{\Lambda}$. The right panel illustrates the optimal price with respect to $\mathsf{\Lambda}$.

**Figure 2.**The left panel illustrates the optimal effective customer arrival rate with respect to $\mathsf{\Lambda}$. The right panel illustrates the optimal revenue with respect to $\mathsf{\Lambda}$.

**Figure 3.**The revenue between the homogeneous L decision and the heterogeneous decision (parameters: $a=5,{V}_{b}=10,{\mu}_{b}=2,\beta =1,c=1,{k}_{L}=0.5,\text{}{k}_{H}=1,q=0.5$.)

**Figure 4.**The left panel illustrates the revenue between the homogeneous M decision and the heterogeneous decision at ${k}_{\mathrm{M}}=0.6$. The right panel illustrates the revenue between the homogeneous M decision and the heterogeneous decision at ${k}_{\mathrm{M}}=0.65$. (Parameters: $a=5,{V}_{b}=10,{\mu}_{b}=2,\beta =1,c=1,{k}_{L}=0.5,{k}_{\mathrm{M}}=0.6or0.65,\text{}{k}_{H}=1,q=0.5$.)

**Figure 5.**The revenue between the homogeneous H decision and the heterogeneous decision. (Parameters: $a=5,{V}_{b}=10,{\mu}_{b}=2,\beta =1,c=1,{k}_{L}=0.5,\text{}{k}_{H}=1,q=0.5$.)

**Figure 6.**The left panel illustrates the revenue in the homogeneous H decision and that in ${S}_{L1}$. The middle panel illustrates the revenue in the homogeneous H decision and that in ${S}_{L2}$ and ${S}_{H1}$ at ${k}_{H}=0.55$. The right panel illustrates the revenue in the homogeneous H decision and that in ${S}_{L2}$ and ${S}_{H1}$ at ${k}_{H}=0.6$.

Parameters | Description |
---|---|

${k}_{i}$ | $i$ customers’ perception of the advertising effect ($i=H,L$) |

$\mathsf{\Lambda}$ | The total potential arrival rate of customers (potential market demand) |

$\lambda $ | The effective arrival rate of customers (effective market demand) |

$q$ | The percentage of H customers in the system |

$\mu $ | Service rate |

$t$ | The average service time |

$V\left(\mu \right)$ | The utility perceived by customer |

$W\left(\mu ,\lambda \right)$ | The waiting time in the system |

${V}_{b}$ | The baseline service utility |

${\mu}_{b}$ | The baseline service rate |

$\beta $ | The degree of customers’ perception of the service rate |

${V}_{Ni}$ | The net utility of $i$ customers ($i=H,L$) |

$R\left(\mu ,P\right)$ | The revenue of the service provider |

$P$ | The price that the service provider charges |

$a$ | The advertising intensity |

$m$ | The maximum service utility |

$c$ | The waiting cost per unit time. |

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**MDPI and ACS Style**

Fu, G.; Dong, L.; Zhan, W.; Jiang, M.
Optimal Decision Making for Customer-Intensive Services Based on Queuing System Considering the Heterogeneity of Customer Advertising Perception. *Systems* **2022**, *10*, 261.
https://doi.org/10.3390/systems10060261

**AMA Style**

Fu G, Dong L, Zhan W, Jiang M.
Optimal Decision Making for Customer-Intensive Services Based on Queuing System Considering the Heterogeneity of Customer Advertising Perception. *Systems*. 2022; 10(6):261.
https://doi.org/10.3390/systems10060261

**Chicago/Turabian Style**

Fu, Gang, Linxiao Dong, Wentao Zhan, and Minghui Jiang.
2022. "Optimal Decision Making for Customer-Intensive Services Based on Queuing System Considering the Heterogeneity of Customer Advertising Perception" *Systems* 10, no. 6: 261.
https://doi.org/10.3390/systems10060261