Sojourn Time Analysis of a Single-Server Queue with Single- and Batch-Service Customers

Abstract

There are various types of sharing economy services, such as ride-sharing and shared-taxi rides. Motivated by these services, we consider a single-server queue in which customers probabilistically select the type of service, that is, the single service or batch service, or other services (e.g., train). In the proposed model, which is denoted by the M+M(K)/M/1 queue, we assume that the arrival process of all the customers follows a Poisson distribution, the batch size is constant, and the common service time (for the single- and batch-service customers) follows an exponential distribution. In this model, the derivation of the sojourn time distribution is challenging because the sojourn time of a batch-service customer is not determined upon arrival but depends on single customers who arrive later. This results in a two-dimensional recursion, which is not generally solvable, but we made it possible by utilizing a special structure of our model. We present an analysis using a quasi-birth-and-death process, deriving the exact and approximated sojourn time distributions (for the single-service customers, batch-service customers, and all the customers). Through numerical experiments, we demonstrate that the approximated sojourn time distribution is sufficiently accurate compared to the exact sojourn time distributions. We also present a reasonable approximation for the distribution of the total number of customers in the system, which would be challenging with a direct-conventional method. Furthermore, we presented an accurate approximation method for a more general model where the service time of single-service customers and that of batch-service customers follow two distinct distributions, based on our original model.

1. Introduction

In recent years, various studies have been conducted on the sharing economy. In the field of mobility, shared taxi rides and ride sharing services such as Uber and Lyft exist. Other services that allow people rent rooms and holiday homes, such as Airbnb and Vrbo, are also attracting considerable attention. A variety of service formats have shown promise and will continue to be developed in the future with the expansion of the CASE and MaaS concepts, especially in mobility platforms [1].

In queueing theory, various models have been developed to analyze sharing economy services such as shared-taxi rides and ride-sharing. Queueing models for shared mobilities have been extensively studied, as shown by the studies conducted by Afeche et al. [2], Banerjee et al. [3], Bai et al. [4], Wang et al. [5], Duan et al. [6], and Nakamura et al. [7]. In addition, an infinite server batch-service queue, i.e., the M/M(b)/∞ queue model designed for ride sharing applications, has been studied [8].

Batch-service queueing models, pioneered by Bailey [9], have been studied extensively. We refer to several the survey by Sasikala et al. [10], which provides further insights into this area. Batch-service queues having a batch size that follows an arbitrary random variable have been studied by Nakamura et al. [11], Banerjee et al. [12], Ho Chang et al. [13], and Cordeau et al. [14].

Several queueing models classifying arrivals or services into multiple classes have been considered. Baruah et al. [15] analyzed a model that classified services into two types with varying service rates and divided jobs into two classes: those requiring only one server and those requiring multiple servers. Olliaro et al. [16] considered a model in which jobs were classified into two classes: those requiring only one server and those requiring multiple servers. Lucantoni [17] and Chakravarthy et al. [18] studied models in which the jobs followed a batch Markov arrival process. Kempa [19] analyzed a model in which job arrivals are distinguished into two types: single arrivals and batch arrivals. Kato et al. [20] also studied a model that distinguished between these two types of arrivals. They divided the input arrivals into a general renewal input process and a Poisson input process. However, few studies have focused on models in which arriving customers/jobs probabilistically or strategically choose between a single or batch service.

The most relevant studies are those conducted by Zhou et al. [21] and Nakamura et al. [22]. In a previous study [21], a model was considered in which arriving strategic customers probabilistically select a single or batch service with a constant batch size by analyzing the equilibrium strategies of customers. Ref. [21] investigated the equilibrium strategies by introducing a simple approximation for the expected waiting times for both types of customers, and proposed the optimization of the system via two methods: revenue maximization and welfare maximization. Ref. [22] also discussed an unobservable model in which strategic customers choose single or batch services. They assumed that the service commences periodically according to a Poisson process and demonstrated a linear relationship for the socially optimal fees of both options. Similar research has been conducteded on models for the strategic selection of multiple capacity services by Nakamura et al. [23], Calvert [24], Afimeimounga et al. [25,26], Chen et al. [27], and Wang et al. [28].

In this study, we analyze and discuss a queueing model in which arriving customers probabilistically choose to receive single or batch services, or uses other services (see the detailed application in the transportation field in Section 2). The analysis is organized as follows. First, we present a stationary analysis of our proposed model using a matrix geometric method. Next, a rigorous derivation of the sojourn time distribution for both single- and batch-service customers and a proposal for its efficient approximations are presented. Notably, the exact expected waiting time for customers is not considered in the aforementioned research [21]. However, we provide the exact derivation for the Laplace–Stieltjes transform (LST) of the sojourn time distribution using the first-step analysis. The distribution of sojourn time based on batch size and selection probability has not been considered in existing research. In applications, such as transportation traffic, understanding not only the average but also the distribution is important. Previous studies [29,30,31] have focused solely on the average. Therefore, in this paper, we present an exact and approximate calculation method for the sojourn time distribution. Additionally, we propose an approximation formula for the total number of customers in the system that is computationally efficient. Regarding the exact formulation, it is necessary to record the sequence of both batch and single customers in the systems, resulting in an exceedingly complex and high-dimensional model. Therefore, conducting an analysis using an approximate formula, in comparison with simulation results, holds significant value.

The remainder of this paper is organized as follows. The definition and application of the model are described in Section 2. Section 3 presents the analysis of the model. In Section 4, we present several numerical results for analytical performance measures, such as exact and approximate sojourn time distributions. Section 5 examines further extensions and disucussions, including the approximated distributions of the number of customers for each type and the mean sojourn time when the service distributions differ for each type. Finally, we conclude this study by summarizing the research findings and prospects in Section 6.

2. Definition of the M+M(K)/M/1 Queue and Its Application

This section defines the model in detail. We consider a queueing model in which customers probabilistically select among three options: receiving a single or batch service, or using other services. Customers arrive at the system according to a Poisson process with the arrival rate $\lambda$. Let $p_s$, $p_b$, and $(1-p_s-p_b)$ denote the joining probabilities for the single service, batch service, and the probabilty that a customer uses other services, respectively (see Figure 1). The service time with a single server follows an exponential distribution with parameter $\mu$ regardless of the type of service (single or batch) selected. The constant K represents the batch size.

Let $N\left(t\right)$ and $S\left(t\right)$ denote the sum of the number of complete batches and customers who choose the single service, and the number of customers required to complete the collecting batch (the number of remaining customers) in the system at time t, respectively. Then, $\left\{\right(N\left(t\right),S\left(t\right));t\ge 0\}$ becomes a two-dimensional continuous-time Markov chain in the state space ${\mathbb{N}}_{+}\times \{K,K-1,\cdots ,1\}$, where ${\mathbb{N}}_{+}=\{0,1,2,\dots \}$. Figure 2 presents the state-transition diagram of this model. Let this model be the M+M$\left(K\right)$/M/1 queue using the notation proposed by Kendall [32].

As stated in the introduction, this model can be applied to the mode-selection problem between individual trips (e.g., private cars) or shared mobility (e.g., ride-sharing or shared-taxis). Customers probabilistically choose between individual trips or shared mobility, or neither of the two options (i.e., train). In this context, $p_s,p_b$ and $(1-p_s-p_b)$ correspond to the probability that a customer uses a private car, a shared mobility, and other public services (such as train or airplane that does not use the common road as cars or shared mobility). When a certain number of customers who choose shared mobility gather, they share a trip as a group. Here, it can be considered that the proposed single-server queue corresponds to the “service station”, which is a segment on the road (e.g., refer to the detail in [33]). In [33], a method was proposed to model traffic congestion on roads in terms of queueing theory utilizing this service station. In brief, the service rate $\mu$ represents the processing power of the road in the proposed model.

The assumption of common service time distribution for different size of batches is often introduced for the analyzes of batch-service queues [21,22,29,30,31]. From the perspective of the transportation system, as mentioned before, it is natural to assume the common service time distribution. For example, the same service time distribution for both single- and batch-service customers is assumed in [22] because all types of vehicle, for example, buses and private cars, receive the same service by the service station on the same road. Additionally, Zhou et al. [21] suggested applications to services such as tourism platforms and online tutoring. Generally speaking, in services such as a one-to-many tour guide service, one-to-many tutoring courses as mentioned in [21], and rideshare services like the one from a train station to an airport, where the service time is the same for one or multiple people. In these cases, it is also reasonable to assume the same service time distribution.

3. Analysis

This section presents an analysis of the M+M(K)/M/1 queue.

3.1. Quasi-Birth-and-Death and Performance Analysis

In the M+M$\left(K\right)$/M/1 queue proposed in Section 2, $N\left(t\right)$ is considered to be the level while $S\left(t\right)$ is the phase, and it is apparent that the analysis can be performed using the quasi-birth-and-death process (QBD) [34,35]. A detailed presentation of the QBD is presented in Appendix A.

Next, we derive the performance measures for the system. N, S, B, L, W, $L_s$, and $L_b$ denote the random variables for the sum of the number of complete batches and the number of single-service customers in the system, the number of customers required for the collecting batch (the number of remaining customers), the batch size of an arbitrary complete batch (here, a single-service customer is also considered as a complete batch), the total number of customers, the sojourn time for all customers, and the number of single- and batch-service customers in the system, at the steady state, respectively. The mean of the sum of the complete batches and the number of single-service customers in the system is given by:

$$E\left[N\right]=\sum _{i=1}^{\infty }i{\mathit{\pi }}_i\mathit{e}={\mathit{\pi }}_1{(\mathit{I}-\mathit{R})}^{-2}\mathit{e}.$$

where $\mathit{e}$, ${\mathit{\pi }}_n$, and $\mathit{R}$ denote the K-dimensional column vector, whose elements are all one, a row vector of steady-state probabilities with level n and phase i$(i=K,K-1,\cdots ,1)$, and the rate matrix, respectively, (see Appendix A).

Here, we recall that the arrival rates of the customers who receive the single service and batch service, respectively, are $p_s\lambda$ and $p_b\lambda$. Therefore, the mean batch size is expressed as follows:

$$E\left[B\right]={\displaystyle \frac{p_s\lambda \times 1}{p_s\lambda +{\displaystyle \frac{p_b\lambda }{K}}}}+{\displaystyle \frac{{\displaystyle \frac{p_{b\lambda }}{K}}\times K}{p_s\lambda +{\displaystyle \frac{p_b\lambda }{K}}}}={\displaystyle \frac{p_s+p_b}{p_s+{\displaystyle \frac{p_b}{K}}}}.$$

As the mean of the total number of customers in the system, $E\left[L\right]$ can be expressed as the sum of the product, $E\left[N\right]E\left[B\right]$, and the number of batch-service customers in an incomplete collecting batch (which clearly follows a discrete uniform distribution with interval [0, $K-1$]), we obtain (1):

$$E\left[L\right]=E\left[N\right]E\left[B\right]+{\displaystyle \frac{K-1}{2}}.$$

(1)

From Little’s formula [36], the mean sojourn time for all customers in the system $E\left[W\right]$ is given as follows:

$$E\left[W\right]={\displaystyle \frac{E\left[L\right]}{(p_s+p_b)\lambda }}.$$

The mean number of customers for each type of system are as follows:

$$E\left[L_s\right]={\displaystyle \frac{p_s}{p_s+{\displaystyle \frac{p_b}{K}}}}E\left[N\right],$$

(2)

$$E\left[L_b\right]={\displaystyle \frac{p_b}{p_s+{\displaystyle \frac{p_b}{K}}}}E\left[N\right]+{\displaystyle \frac{K-1}{2}}.$$

(3)

3.2. Exact Analysis of Sojourn Time Distributions

In this subsection, we derive the sojourn time distributions for the single- and batch-service customers and those for all customers.

First, we consider the sojourn time distribution of customers who receive the single service. Let n denote the sum of the number of complete batches and customers who choose the single service in the system, excluding a tagged customer who receives the single service upon arrival. In this case, the sojourn time of the tagged customer is considered to be the sum of $(n+1)$ service times, including its own service time. Let ${\overline{f}}_s^{\left(n\right)}\left(t\right)$ be the conditional probability density function for the sojourn time distribution of a tagged single-service customer who observes n complete batches immediately before their arrival. In addition, let $Er(\beta ,\gamma )$ denote a random variable that follows an Erlang distribution with the mean $\beta /\gamma$ (the sum of $\beta$ exponential distributions with rate $\gamma$). It is apparent that ${\overline{f}}_s^{\left(n\right)}\left(t\right)$ is a probability density function of $Er(n+1,\mu )$, that is:

$${\overline{f}}_s^{\left(n\right)}\left(t\right)={\displaystyle \frac{{\mu }^{n+1}}{n!}}{t}^n{e}^{-\mu t}.$$

Owing to PASTA (Poisson arrival see time averages), the probability density function of the sojourn time distribution for the single-service customers $f_s\left(t\right)$ is given by:

$$\begin{array}{cc}\hfill f_s\left(t\right)& =\sum _{n=0}^{\infty }\sum _{i=1}^K{\pi }_{n,i}{\overline{f}}_s^{\left(n\right)}\left(t\right).\hfill \end{array}$$

(4)

In the numerical example section (Section 4), we truncate the infinite sum in (4) by a parameter $n^{*}$, which is given by:

$$n^{*}=\inf \left\{m\in {\mathbb{N}}_{+};1-\sum _{n=0}^m\sum _{i=1}^K{\pi }_{n,i}<{ϵ}_2\right\}.$$

(5)

Therefore, the probability density function $f_s^{*}\left(t\right)$ can be calculated as follows:

$$\begin{array}{c}\hfill f_s^{*}\left(t\right)=\sum _{n=0}^{n^{*}}\sum _{i=1}^K{\pi }_{n,i}{\overline{f}}_s^{\left(n\right)}\left(t\right).\end{array}$$

(6)

Formula (4) can be further written in closed-form as follows. Let ${\mathcal{L}}_s\left(s\right)$ denote the LST of the sojourn time distribution of a single-service customer. We then obtain:

$$\begin{array}{cc}\hfill {\mathcal{L}}_s\left(s\right)& =\sum _{n=0}^{\infty }{q}_n{\left({\displaystyle \frac{\mu }{\mu +s}}\right)}^{n+1}\hfill \\ \hfill & =q_0{\displaystyle \frac{\mu }{\mu +s}}+\sum _{n=1}^{\infty }{q}_n{\left({\displaystyle \frac{\mu }{\mu +s}}\right)}^{n+1}\hfill \\ \hfill & =q_0{\displaystyle \frac{\mu }{\mu +s}}+{\left({\displaystyle \frac{\mu }{\mu +s}}\right)}^2{\mathit{\pi }}_{\mathbf{1}}{\left(\mathit{I}-{\displaystyle \frac{\mu \mathit{R}}{\mu +s}}\right)}^{-1}\mathit{e},\hfill \end{array}$$

(7)

where $q_n$ represents the probability that there are n complete batches and single-service customers in the system, i.e., $q_n={\mathit{\pi }}_n\mathit{e}$ (due to PASTA). Letting $f_{ss}\left(t\right)$ be the inverse transform of ${\mathcal{L}}_s\left(s\right)$, the following is then obtained:

$$\begin{array}{cc}\hfill f_{ss}\left(t\right)& =q_0\mu e^{-\mu t}+{\mathit{\pi }}_{\mathbf{1}}\sum _{n=1}^{\infty }{\mathit{R}}^{n-1}{\displaystyle \frac{{\mu }^{n+1}}{n!}}{t}^n{e}^{-\mu t}\mathit{e}\hfill \\ \hfill & =q_0\mu e^{-\mu t}+\mu e^{-\mu t}{\mathit{\pi }}_{\mathbf{1}}{\mathit{R}}^{-1}\left(\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\mu \mathit{R}t\right)}^n}{n!}}-\mathit{I}\right)\mathit{e}\hfill \\ \hfill & =q_0\mu e^{-\mu t}+\mu e^{-\mu t}{\mathit{\pi }}_{\mathbf{1}}{\mathit{R}}^{-1}(e^{\mu \mathit{R}t}-\mathit{I})\mathit{e}.\hfill \end{array}$$

(8)

Similarly, we consider the waiting time distribution. Let ${\mathcal{I}}_s\left(s\right)$ be denoted as the LST of the waiting time distribution for a single customer service. In addition, let $f_{s^{*}s}\left(t\right)$ be denoted as its inverse transform. Thus, the following holds:

$$\begin{array}{cc}\hfill {\mathcal{I}}_s\left(s\right)& =\sum _{n=0}^{\infty }{q}_n{\left({\displaystyle \frac{\mu }{\mu +s}}\right)}^n\hfill \\ \hfill & =q_0+\sum _{n=1}^{\infty }{q}_n{\left({\displaystyle \frac{\mu }{\mu +s}}\right)}^n\hfill \\ \hfill & =q_0+{\displaystyle \frac{\mu }{\mu +s}}{\mathit{\pi }}_{\mathbf{1}}{\left(\mathit{I}-{\displaystyle \frac{\mu \mathit{R}}{\mu +s}}\right)}^{-1}\mathit{e},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill f_{s^{*}s}\left(t\right)& =q_0\delta \left(t\right)+\mu {\mathit{\pi }}_{\mathbf{1}}\sum _{n=1}^{\infty }{\displaystyle \frac{{\left(\mu \mathit{R}t\right)}^{n-1}}{(n-1)!}}{e}^{-\mu t}\mathit{e}\hfill \\ \hfill & =q_0\delta \left(t\right)+\mu {\mathit{\pi }}_{\mathbf{1}}{e}^{\mu \mathit{R}t-\mu \mathit{I}t}\mathit{e}.\hfill \end{array}$$

(10)

where (8) and (10), respectively, correspond to the sojourn time and waiting time distributions in the M/M/1 queueing model and $\delta \left(t\right)$ is the Delta function. In this system, the arrival rate is $\lambda$, the service rate is $\mu$, and the traffic density is $\rho =\lambda /\mu$, where $\rho$ corresponds to $\mathit{R}$ in Equations (8) and (10). For the case $p_s=0$ and $p_b=1$, the sojourn and waiting time distributions are the same as the sojourn time and waiting time distributions of the M/M(K)/1 queueing model [29,30,31].

Subsequently, we consider the sojourn time distribution for customers who receive the batch service. The residual sojourn time of a batch-service customer depends on the ongoing service progression with one server as well as the arrival process of the subsequent single- and batch-service customers. Let ${\delta }_b(n,i)$ denote the random variable representing the conditional sojourn time distribution provided $(N,S)=(n,i)$ is observed just before the arrival of a tagged batch-service customer, and let ${\overline{f}}_b^{(n,i)}\left(t\right)$ denote the probability density function. Subsequently, ${\delta }_b(n,i)$ is obtained by considering the cases of $(n,i)$.

Because ${\delta }_b(n,1)$ is a random variable that represents the sojourn time distribution for the tagged batch-service customers who observe the state $(N,S)=(n,1)$ and become the final customer for the collecting batch, ${\delta }_b(n,1)$ can be expressed as the sum of $(n+1)$ service times, including the service time of the batch to which the customer belongs. Therefore, as in the study conducted by Yajima et al. [37], on applying the first-step analysis, ${\delta }_b(n,i)$ for $n=0,1,\dots$, and, $i=1,2,\dots ,K$ can be expressed based on ${\delta }_b(n,1)$ recursively as follows:

$${\delta }_b(n,1)=Er(n+1,\mu ),n=0,1,2,\cdots ,$$

(11)

$${\delta }_b(0,i)=X\left((p_s+p_b)\lambda \right)+\left\{\begin{array}{cc}{\delta }_b(1,i),\hfill & \mathrm{w}.\mathrm{p}.{\displaystyle \frac{p_s\lambda }{(p_s+p_b)\lambda }},\hfill \\ {\delta }_b(0,i-1),\hfill & \mathrm{w}.\mathrm{p}.{\displaystyle \frac{p_b\lambda }{(p_s+p_b)\lambda }},\hfill \end{array}\right.i=2,3,\cdots ,K,$$

(12)

$$\begin{array}{cc}\hfill & {\delta }_b(n,i)=X((p_s+p_b)\lambda +\mu )\hfill \\ \hfill & +\left\{\begin{array}{cc}{\delta }_b(n-1,i),\hfill & \mathrm{w}.\mathrm{p}.{\displaystyle \frac{\mu }{(p_s+p_b)\lambda +\mu }},\hfill \\ {\delta }_b(n+1,i),\hfill & \mathrm{w}.\mathrm{p}.{\displaystyle \frac{p_s\lambda }{(p_s+p_b)\lambda +\mu }},\hfill \\ {\delta }_b(n,i-1),\hfill & \mathrm{w}.\mathrm{p}.{\displaystyle \frac{p_b\lambda }{(p_s+p_b)\lambda +\mu }},\hfill \end{array}\right.n=1,2,\cdots ,i=2,3,\cdots ,K,\hfill \end{array}$$

(13)

where $X\left(\alpha \right)$ denotes a random variable that follows an exponential distribution with a mean $1/\alpha$. It should be noted that (11), (12), and (13) represent equal signs in terms of the distribution. Let ${\varphi }_b(n,i,s)$ be the LST of ${\delta }_b(n,i)$, that is:

$${\varphi }_b(n,i,s)={\int }_{t=0}^{\infty }{e}^{-st}{\overline{f}}_b^{(n,i)}\left(t\right)dt.$$

On taking the LST on both sides of (11), (12), and (13), we obtain (14), (15), and (16), respectively. It is apparent that ${\varphi }_b(n,i,s)$ in (14) becomes the LST of $Er(n+1,\mu )$, as follows:

$${\varphi }_b(n,1,s)={\left({\displaystyle \frac{\mu }{s+\mu }}\right)}^{n+1},n=0,1,2,\cdots .$$

(14)

In addition, we can consider ${\varphi }_b(0,i,s)$, which is the LST for the case wherein the tagged customer observes $(N,S)=(0,i),(i=2,3,\cdots ,K)$ as in (15). It should be noted that ${\varphi }_b(0,i,s)$ depends on the arrival processes of subsequent single- and batch-service customers, in contrast to (14). Therefore, the following equation is obtained:

$$\begin{array}{cc}\hfill {\varphi }_b(0,i,s)=& {\displaystyle \frac{p_s\lambda }{s+(p_s+p_b)\lambda }}{\varphi }_b(1,i,s)\hfill \\ \hfill & +{\displaystyle \frac{p_b\lambda }{s+(p_s+p_b)\lambda }}{\varphi }_b(0,i-1,s),i=2,3,\cdots ,K.\hfill \end{array}$$

(15)

We can also derive the LST for the case wherein the tagged customer observes $(N,S)=(n,i)$ for $n=1,2,\dots$ and $i=1,2,\dots K$ as in the case of (15). The LST in this case depends not only on the subsequent single- and batch-service customers, but also on the ongoing service progression with one server. Thus, the following recursive relationships can be obtained:

$$\begin{array}{cc}\hfill & {\varphi }_b(n,i,s)={\displaystyle \frac{\mu }{s+(p_s+p_b)\lambda +\mu }}{\varphi }_b(n-1,i,s)+{\displaystyle \frac{p_s\lambda }{s+(p_s+p_b)\lambda +\mu }}{\varphi }_b(n+1,i,s)\hfill \\ \hfill & +{\displaystyle \frac{p_b\lambda }{s+(p_s+p_b)\lambda +\mu }}{\varphi }_b(n,i-1,s),n=1,2,\cdots ,\mathrm{and}i=2,3,\cdots ,K.\hfill \end{array}$$

(16)

Let ${\mathit{\varphi }}_b(i,s)$ be a column vector as follows:

$${\mathit{\varphi }}_b(i,s)={\left(\begin{array}{cccc}{\varphi }_b(0,i,s)& {\varphi }_b(1,i,s)& {\varphi }_b(2,i,s)& \cdots \end{array}\right)}^{\top },i=1,2,3,\cdots ,K,$$

where ${\mathit{a}}^{\top }$ denotes the transpose vector of $\mathit{a}$. From (14), (15), and (16), we obtain:

$${\mathit{\varphi }}_b(1,s)={\left(\begin{array}{cccc}{\displaystyle \frac{\mu }{s+\mu }}& {\left({\displaystyle \frac{\mu }{s+\mu }}\right)}^2& {\left({\displaystyle \frac{\mu }{s+\mu }}\right)}^3& \cdots \end{array}\right)}^{\top },$$

and

$${\mathit{\varphi }}_b(i,s)=\mathit{D}{\mathit{\varphi }}_b(i,s)+\mathit{C}{\mathit{\varphi }}_b(i-1,s),i=2,3,\cdots ,K,$$

where $\mathit{C}$ and $\mathit{D}$ are matrices of infinite size that can be expressed as follows:

$$\mathit{C}={\left(C_{j,k}\right)}_{(j,k)\in {\mathbb{N}}_{+}\times {\mathbb{N}}_{+}},$$

$$C_{j,k}=\left\{\begin{array}{cc}{\displaystyle \frac{p_b\lambda }{s+(p_s+p_b)\lambda }},\hfill & j=k=0,\hfill \\ {\displaystyle \frac{p_b\lambda }{s+(p_s+p_b)\lambda +\mu }},\hfill & j=k,j\ge 1,\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

$$\mathit{D}={\left(D_{j,k}\right)}_{(j,k)\in {\mathbb{N}}_{+}\times {\mathbb{N}}_{+}},$$

$$D_{j,k}=\left\{\begin{array}{cc}{\displaystyle \frac{p_s\lambda }{s+(p_s+p_b)\lambda }},\hfill & j=1,k=0,\hfill \\ {\displaystyle \frac{p_s\lambda }{s+(p_s+p_b)\lambda +\mu }},\hfill & j=k-1,j\ge 2,\hfill \\ {\displaystyle \frac{\mu }{s+(p_s+p_b)\lambda +\mu }},\hfill & j=k+1,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

As we can prove that $(\mathit{I}-\mathit{D})$ has an inverse map as in the study of Yajima et al. [37], we obtain:

$${\mathit{\varphi }}_b(i,s)={(\mathit{I}-\mathit{D})}^{-1}\mathit{C}{\mathit{\varphi }}_b(i-1,s),i=2,3,\cdots ,K,$$

and therefore, the following holds:

$${\mathit{\varphi }}_b(i,s)={\left\{{(\mathit{I}-\mathit{D})}^{-1}\mathit{C}\right\}}^{i-1}{\mathit{\varphi }}_b(1,s),i=2,3,\cdots ,K.$$

Let ${\varphi }_b\left(s\right)$ be the LST of the sojourn time distribution for the batch-service customers. Owing to the PASTA property, we have:

$$\begin{array}{cc}\hfill {\varphi }_b\left(s\right)& =\sum _{n=0}^{\infty }\sum _{i=1}^K{\pi }_{n,i}{\varphi }_b(n,i,s).\hfill \end{array}$$

(17)

In the numerical example (in Section 4), the following formula (18) is used, where the infinite sum in (17) is truncated by the parameter $n^{*}$, which is defined as in (5), and $(\mathit{I}-\mathit{D})$ is truncated to its $(n^{*}+1)\times (n^{*}+1)$ north-west corner matrix, where $n^{*}$ is determined in (5):

$$\begin{array}{cc}\hfill {\varphi }_b^{*}\left(s\right)& =\sum _{n=0}^{n^{*}}\sum _{i=1}^K{\pi }_{n,i}{\varphi }_b(n,i,s).\hfill \end{array}$$

(18)

Let $t_j$ be $t_j=j\Delta t=jT/N(j=0,1,2,\cdots ,N-1)$. Based on the fast Laplace inverse transform (FLIT) in the study of Durbin [38], the inverse Laplace transform of (18) can be calculated as $f_b^{*}\left(t_j\right)$ in (19):

$$f_b^{*}\left(t_j\right)={\displaystyle \frac{2}{T}}\exp \left(aj\Delta t\right)\left\{-{\displaystyle \frac{1}{2}}\mathrm{Re}\left[{\varphi }^{*}\left(a\right)\right]+\mathrm{Re}\left[\sum _{k=0}^{N-1}\left(A\left(k\right)+iB\left(k\right)\right)\exp \left({\displaystyle \frac{ijk2\pi }{N}}\right)\right]\right\},$$

(19)

$$A\left(k\right)=\sum _{l=0}^L\mathrm{Re}\left\{{\varphi }^{*}\left(a+i(k+lN){\displaystyle \frac{2\pi }{T}}\right)\right\},$$

$$B\left(k\right)=\sum _{l=0}^L\mathrm{Im}\left\{{\varphi }^{*}\left(a+i(k+lN){\displaystyle \frac{2\pi }{T}}\right)\right\},$$

where $i=\sqrt{-1}$, Re$\{x+iy\}=x$, and, Im$\{x+iy\}=y$. Here $N=100$ and $T=5$ were selected.

Finally, we consider the overall sojourn time distribution (for all single- and batch-service customers). Let $f^{*}\left(t\right)$ be the probability density function of the overall sojourn time, which can be expressed using (6) and (19) as follows:

$$f^{*}\left(t\right)={\displaystyle \frac{p_s}{p_s+p_b}}{f}_s^{*}\left(t\right)+{\displaystyle \frac{p_b}{p_s+p_b}}{f}_b^{*}\left(t\right),$$

(20)

3.3. Approximation Method for Sojourn Time Distributions

Next, we propose approximation formulas of the sojourn time distributions for the single- and batch-service customers, and for all the customers. The numerical experiments for the probability density functions $f_s^{*}\left(t\right)$, $f_b^{*}\left(t_j\right)$, and $f^{*}\left(t\right)$, obtained in Section 3.2, provide accurate results, but they are computationally intensive and gave long output times. Therefore, we consider efficient approximations for the sojourn time distributions.

For the M+M$\left(K\right)$/M/1 queue, we also consider an approximation method in which the behavior of the system is considered as an M/M/1 queue with the arrival rate ${\lambda }^{\prime }=p_s\lambda +p_b\lambda /K$, service rate $\mu$, and traffic density $\rho ={\lambda }^{\prime }/\mu$. As mentioned in Section 1, a similar idea for the approximation of the expected waiting time is adopted in [21]. It should be emphasized that we use this approximation method for the sojourn time distributions in this subsection.

First, we consider the approximation of the sojourn time distribution of customers who receive the single service. Let $W_s\left(s\right)$ be the LST of the sojourn time distribution of a single-service customer. Using the stationary distribution of the number of customers in the system for the M/M/1 queue, i.e., $(1-\rho ){\rho }^n$, and the LST of the sum of $(n+1)$ service times, i.e., $Er(n+1,\mu )$, we have:

$$W_s\left(s\right)=\sum _{n=0}^{\infty }(1-\rho ){\rho }^n{\left({\displaystyle \frac{\mu }{\mu +s}}\right)}^{n+1}={\displaystyle \frac{(1-\rho )\mu }{(1-\rho )\mu +s}}={\displaystyle \frac{\mu -{\lambda }^{\prime }}{\mu -{\lambda }^{\prime }+s}}.$$

(21)

We assume $f_{ws}\left(t\right)$ to be the inverse Laplace transform of (21), we obtain the following:

$$f_{ws}\left(t\right)=(1-\rho )\mu e^{-(1-\rho )\mu t}.$$

(22)

Next, we consider the approximation of the sojourn time distribution of customers who receive batch services. If $G\left(s\right)$ denote the LST of the residual time required to complete a collecting batch for a tagged arriving batch-service customer, G(s) can be expressed as follows:

$$G\left(s\right)={\displaystyle \frac{1}{K}}\sum _{i=0}^{K-1}{\left({\displaystyle \frac{p_b\lambda }{p_b\lambda +s}}\right)}^i.$$

Consequently, the LST for the sojourn time distribution of customers who receive batch services can be expressed as follows:

$$W_s\left(s\right)G\left(s\right)=\left({\displaystyle \frac{\mu -{\lambda }^{\prime }}{\mu -{\lambda }^{\prime }+s}}\right){\displaystyle \frac{1}{K}}\sum _{i=0}^{K-1}{\left({\displaystyle \frac{p_b\lambda }{p_b\lambda +s}}\right)}^i.$$

(23)

Here, the term:

$$\left({\displaystyle \frac{\mu -{\lambda }^{\prime }}{\mu -{\lambda }^{\prime }+s}}\right){\left({\displaystyle \frac{p_b\lambda }{p_b\lambda +s}}\right)}^i$$

in (23) is the exact LST for the sum of $X(\mu -{\lambda }^{\prime })$ and $Er(i,p_b\lambda )$. Therefore, on considering the convolution of the exponential and Erlang distributions and performing an inverse transformation, the following is obtained:

$${\int }_0^{t}f(x;i,p_b\lambda )(\mu -{\lambda }^{\prime })e^{-(\mu -{\lambda }^{\prime })(t-x)}dx=(\mu -{\lambda }^{\prime })e^{-(\mu -{\lambda }^{\prime })t}{\int }_0^{t}f(x;i,p_b\lambda )e^{(\mu -{\lambda }^{\prime })x}dx,$$

(24)

where $f(x;i,p_b\lambda )$ in (24) represents the probability density function of $Er(i,p_b\lambda )$. We then obtain the following:

$$f(x;i,p_b\lambda )={\displaystyle \frac{{\left(p_b\lambda \right)}^i}{(i-1)!}}{x}^{(i-1)}{e}^{-p_b\lambda t}.$$

(25)

On substituting (25) into (24), the following transformation is obtained:

$$\begin{array}{c}{\int }_0^{t}f(x;i,p_b\lambda )e^{(\mu -{\lambda }^{\prime })x}dx={\displaystyle \frac{{\left(p_b\lambda \right)}^i}{{(p_b\lambda +{\lambda }^{\prime }-\mu )}^i}}{\int }_0^{t}f(x;i,p_b\lambda +{\lambda }^{\prime }-\mu )dx\hfill \\ \hfill ={\displaystyle \frac{{\left(p_b\lambda \right)}^i}{{(p_b\lambda +{\lambda }^{\prime }-\mu )}^i}}\left(1-e^{-(p_b\lambda +{\lambda }^{\prime }-\mu )t}\sum _{n=0}^{i-1}{\displaystyle \frac{{\left((p_b\lambda +{\lambda }^{\prime }-\mu )t\right)}^n}{n!}}\right).\end{array}$$

(26)

By substituting (26) into the left-hand side of (24), the following is obtained.

$$\begin{array}{c}{\int }_0^{t}f(x;i,p_b\lambda )(\mu -{\lambda }^{\prime })e^{-(\mu -{\lambda }^{\prime })(t-x)}dx\hfill \\ \hfill =(\mu -{\lambda }^{\prime })e^{-(\mu -{\lambda }^{\prime })t}{\displaystyle \frac{{\left(p_b\lambda \right)}^i}{{(p_b\lambda +{\lambda }^{\prime }-\mu )}^i}}\left(1-e^{-(p_b\lambda +{\lambda }^{\prime }-\mu )t}\sum _{n=0}^{i-1}{\displaystyle \frac{{\left((p_b\lambda +{\lambda }^{\prime }-\mu )t\right)}^n}{n!}}\right).\end{array}$$

(27)

We then obtain the approximated probability density function of the sojourn time for the batch-service customers, $f_{wsbs}\left(t\right)$, as follows:

$$\begin{array}{c}{f}_{wsbs}\left(t\right)={\displaystyle \frac{1}{K}}\sum _{i=0}^{K-1}(\mu -{\lambda }^{\prime })e^{-(\mu -{\lambda }^{\prime })t}{\displaystyle \frac{{\left(p_b\lambda \right)}^i}{{(p_b\lambda +{\lambda }^{\prime }-\mu )}^i}}\hfill \\ \hfill -{\displaystyle \frac{1}{K}}\sum _{i=0}^{K-1}(\mu -{\lambda }^{\prime })e^{-(\mu -{\lambda }^{\prime })t}{\displaystyle \frac{{\left(p_b\lambda \right)}^i}{{(p_b\lambda +{\lambda }^{\prime }-\mu )}^i}}{e}^{-(p_b\lambda +{\lambda }^{\prime }-\mu )t}\sum _{l=0}^{i-1}{\displaystyle \frac{{\left((p_b\lambda +{\lambda }^{\prime }-\mu )t\right)}^l}{l!}}.\end{array}$$

(28)

Finally, we consider the approximation formula for the sojourn time distribution of all the customers. Let $f_{all}\left(t\right)$ be the probability density function of the sojourn time distribution of all customers. We can calculate $f_{all}\left(t\right)$ using (22) and (28) as follows:

$$f_{all}\left(t\right)={\displaystyle \frac{p_s}{p_s+p_b}}{f}_{ws}\left(t\right)+{\displaystyle \frac{p_b}{p_s+p_b}}{f}_{wsbs}\left(t\right).$$

(29)

4. Numerical Experiments

This section presents numerical results of the performance measures for the M+M(K)/M/1 queue that are derived in Section 3.1, and the exact and approximated sojourn time distributions (Section 3.2 and Section 3.3). Numerical experiments are conducted using Python 3.11.7, with computations utilizing the Python library SciPy and NumPy. In the simulation, Monte Carlo simulations are conducted, generating one million events. The first 100,000 events are discarded to account for the impact of initial conditions. We developed our original simulation program for this model based on Python. Random numbers are generated using Python random module. Here, the events refer to customer arrivals or service completions. In the following sections, we compare experiments based on exact and approximate methods. QBD and inversion of LST are not exact analytical solutions but numerical computational methods. However, increasing the truncation point brings the results closer to exact solutions, making the comparison with approximate methods meaningful. The numerical methods are time consuming while approximation allows simple closed form solution whose results can be easily obtained (see

Table 1. Specific differences in computation times of sojourn time distribution for batch-service customers.

${\mathit{ϵ}}_2$	Computation Time of (19)	Computation Time of (28)
${10}^{-2}$	11,078 s	$1.7\times {10}^{-3}$ s
${10}^{-5}$	11,032 s	$1.7\times {10}^{-3}$ s
${10}^{-7}$	11,117 s	$1.7\times {10}^{-3}$ s
${10}^{-10}$	11,074 s	$1.7\times {10}^{-3}$ s

).

4.1. Numerical Results for Performance Measures of the M+M(K)/M/1 Queue

First, we present the results of the numerical experiments on the theoretical performance measures in Section 3.1 while we conduct the simulation experiments for the same parameter settings. For the mean number of customers in the system, $E\left[L\right]$, we set the parameters as $\mu =100,p_s=0.4,p_b=0.5$, and, $K\in \{3,5\}$. Figure 3 and Figure 4 present the results of $E\left[L\right]$ for various $\lambda$, and $K\in \{3,5\}$. The solid lines represent the theoretical values of $E\left[L\right]$ (denoted as ’theoretical’), $E\left[L_s\right]$ (denoted as ’theoretical_single’), and $E\left[L_b\right]$ (denoted as ’theoretical_batch’), while the markers represent those for the simulation results. It is confirmed that the theoretical and simulation results match well, i.e., our theoretical analysis is valid. Moreover, it is apparent that $E\left[L\right]$ increases as $\lambda$ increases.

Furthermore, we present the theoretical and simulation results of $E\left[W\right]$ for $K\in \{3,5,8\}$ in Figure 5 provided that the other parameter settings are equivalent to those in Figure 3, Figure 4, Figure 5 and Figure 6. It is also confirmed that the theoretical and simulated results are in good agreement. We observe that $E\left[W\right]$ gradually decreases as $\lambda$ increases in the range wherein $\lambda$ is small (approximately 50 or less). In addition, $E\left[W\right]$ decreases as the batch size K decreases within the aforementioned range of $\lambda$. These results are attributed to the fact that the larger $\lambda$ is, the shorter the batch formation time and the larger the batch size, the longer the batch formation time. However, in the range wherein $\lambda$ is large (approximately 150 or more), $E\left[W\right]$ gradually increases as $\lambda$ increases. In addition, we find that $E\left[W\right]$ decreases as K increases because the time required to complete a batch when the arrival rate is high is marginal, and a larger batch size causes a decrease in the total number of batches in the system.

4.2. Numerical Results for Sojourn Time Distribution

In this subsection, we present the results for the exact and approximated values for the sojourn time distributions (which are derived in Section 3.2 and Section 3.3). In this subsection, we set the parameters as $\lambda =4, \mu =5, p_s=0.4$, and $p_b=0.4$, and for FLIT, $a=1, N=100, L=50$, and $t=5$.

4.2.1. Numerical Results for the Exact Sojourn Time Distribution

Figure 7, Figure 8 and Figure 9 present the exact sojourn time distributions according to (6), (19), and (20), which are derived in Section 3.2 for the single and batch services and all the customers, respectively, when $K=3$. In these figures, the value of ${ϵ}_2$ in (5) is set to ${10}^{-5}$. The simulation results verify the validity of our theoretical analysis for the sojourn time distributions.

Figure 10, Figure 11 and Figure 12 present the numerical results of exact methods based (8) when $K=3,5$, and 8, respectively, and the other parameters are set as follows: $\lambda =4$, $\mu =5,$ $p_s=0.4$, and $p_b=0.4$. It can be observed that the results agree with the simulation results. Furthermore, the calculation time is less than that for the sojourn time distribution according to (6), and the graph can be output in a short time. The computation times for (6), (8), and (22) are summarized in

Table 2. Specific differences in computation times of sojourn time distribution for single-service customers.

${\mathit{ϵ}}_2$	Computation Time of (6)	Computation Time of (8)	Computation Time of (22)
${10}^{-2}$	2.25 s	0.43 s	$6.9\times {10}^{-5}$ s
${10}^{-5}$	2.31 s	0.43 s	$6.9\times {10}^{-5}$ s
${10}^{-7}$	2.50 s	0.43 s	$6.9\times {10}^{-5}$ s
${10}^{-10}$	2.66 s	0.43 s	$6.9\times {10}^{-5}$ s

. Here, the computation time does not include the output time of the graph, but represents the total computation time for a given range of t. It can be observed that the computation time of (8) is significantly shorter compared to (6). In Figure 7, a threshold condition, i.e., ${ϵ}_2={10}^{-5}$ is adopted. Under these conditions, the computational speed of (8) is approximately six times faster than that of (6), as shown in Table 2. Considering (22), the computation time is the shortest for any threshold conditions. It can be observed that the calculation time is approximately 35,000 times faster than (6) and approximately 6231 times faster than (8). As can be seen from Table 1, the computation time with (19) is extremely long and 6.51 million times longer than that in (28). This implies the usefulness of the approximate formula.

4.2.2. Numerical Results for the Approximated Sojourn Time Distributions

Next, we present the numerical results of the approximations for the sojourn time distributions in Section 3.3. Figure 13, Figure 14 and Figure 15 present the approximated sojourn time distributions according to (22), (28), and (29) for the single and batch services and all the customers, respectively, when $\lambda =4, \mu =5, p_s=0.4, p_b=0.4$, and $K=3$. On taking into consideration the results of (Figure 13, Figure 14 and Figure 15), it is presumed that our proposed approximation method in Section 3.3 has a high accuracy. It should also be emphasized that the computation times for the approximation formulas are significantly shorter than those of the FLIT for identical distributions.

Figure 16 and Figure 17 show the numerical results for a batch size $K=20$ and 50. Figure 16 has the other parameters set to $\lambda =7, \mu =5, p_s=0.4$, and $p_b=0.4$, while in Figure 17, the parameters are $\lambda =12, \mu =5, p_s=0.4$, and $p_b=0.4$. In these two numerical experiments, the number of events is set to 10 million, with the first 100,000 events being discarded. As can be seen from this, even when the batch size increases, the approximation formula matches the simulation well. However, slight deviations can be observed in regions with high frequency within the graphs and when the batch size is 50 (around $t=0$ to $7.5$ in Figure 16 and around $t=15$, and 35 to 55 in Figure 17). The decrease in the accuracy of the approximation as the batch size increases can be attributed to the stronger influence of subsequent customers. As the number of subsequent customers increases, it becomes more difficult to approximate the inter-arrival of batches using an exponential distribution, which in turn leads to greater discrepancies between the approximation and the simulation results. Therefore, it can be said that one of the shortcomings of the approximation formula is that the deviations become more pronounced as the batch size increases.

4.3. Comparison with M/M(K)/1 Queue [29,30,31]

In this section, We compare the inverse LST transformation with the existing studies [29,30,31] on the M/M(K)/1 queue by setting $p_s=0$ and $p_b=1$.

Figure 18 shows the results of comparing the inverse LST transformation with the simulation for $p_s=0$ and $p_b=1$. As can be seen, the inverse LST transformation also matches the simulation with high accuracy for the case $p_s=0, p_b=1$.

Figure 19 shows the results from previous studies [30,31], with the simulation, approximation, and exact method (theoretical) for the case of $p_s=0$, and the other parameters are set as follows: $\lambda =3, \mu =1, p_b=1$, and $K=4$. Regarding the approximation, it does not perform well when $p_s=0$ or when $p_s$ is close to zero. However, it functions adequately when $p_s$ takes on relatively large values. Additionally, even for $p_s=0$ or when $p_s$ is close to zero, the approximation works well when $\lambda$ is small, but it fails when $\lambda$ is large (see Figure 20, Figure 21 and Figure 22 with the parameters set as follows: $\mu =5, p_s=0, p_b=1, K=5$, and $\lambda$ is $4, 6$, and 8, respectively). The incorrectness of the approximation when $p_s=0$ or when $p_s$ is close to zero is due to the fact that when $p_s$ is close to zero, the arrival process is far from Poisson as in the approximation. However, since the approximation works well for the purpose of this study, which focuses on a model that probabilistically chooses between single and batch service, it can be concluded that the performance of the approximation is sufficiently adequate. As seen in Figure 19, in the case of the exact method, using the inverse LST, the computational cost is extremely high, resulting in long processing times. However, the results obtained are significantly more accurate compared to [31] and our approximation, especially when t is close to 0. However, when t is larger, the inverse LST is less accurate. This is because in the algorithm for the inversion of LST, there are several turning parameters, such as a and T, which need to be determined by try and error. Our simulation well matches the exact one from [30], showing the accuracy of our simulation.

Table 3. Simulation results and approximations.

Figure	Simulation (Mean)	Simulation (95% Confidence Interval)	Approximation (Mean)
13	0.3437	[−0.31886, 1.0040]	0.3432
14	0.9372	[−0.7546, 2.6381]	0.9736
15	0.6393	[−0.7693, 2.0506]	0.6584
16	3.8509	[−0.8146, 8.5166]	3.8782
17	14.3425	[−0.52315, 34.2430]	14.4756
19	3.0644	[−2.1851, 8.6216]	3.9885
20	0.7040	[−0.3513, 1.7582]	0.7381
21	0.5455	[−0.2215, 1.3152]	0.5965
22	0.4743	[−0.1707, 1.1226]	0.5441

compares the mean values from the simulation and the 95% confidence intervals from the simulation, with the mean values derived from the approximation method, in Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 and Figure 19, Figure 20, Figure 21 and Figure 22. As can be seen from these results, the expected values from the simulation and those from the approximation method are almost identical, demonstrating the usefulness of the approximation formula. However, for Figure 19, although the mean value from the approximation falls within the 95% confidence interval, there is a slight discrepancy compared to the simulated mean value.

5. Further Extensions and Discussions

This section provides some further extensions and discussion for the proposed model.

5.1. Approximated Distributions of the Number of Customers for Each Type

In addition to the exact mean number of customers for each type of system, i.e., (2) and (3), we further propose an approximation calculation method for the distribution of the number of customers for each type in this subsection. Let $N_s$ and $N_b$ denote the random variables for the number of single-service customers and batch-service customers, respectively.

In our proposed method, we define the probability that the type of a new complete batch (i.e., single-service customer or batch of size K) is a single-service customer by $p_{new}$:

$$p_{new}={\displaystyle \frac{p_s}{p_s+{\displaystyle \frac{p_b}{K}}}}.$$

Then, we can approximately calculate the distribution of the number of single-service customers as follows:

$$P(N_s=l)=\sum _{n=l}^{n^{*}}{q}_n{\displaystyle \frac{n!}{l!(n-l)!}}{p}_{new}^l{(1-p_{new})}^{n-l},$$

(30)

where $n^{*}$ is given by (5). Similarly, the distribution of the number of batch-service customers can be approximately calculated as follows:

$$P(N_b=Kl+i)=\sum _{n=l}^{n^{*}}{\pi }_{n,K-i}{\displaystyle \frac{n!}{l!(n-l)!}}{(1-p_{new})}^l{p}_{new}^{n-l}.$$

(31)

In what follows, we show the numerical results for $\lambda =4, \mu =5$ and $K=2$. Figure 23 and Figure 24 show the numerical results for the approximated number of single-service customers, i.e., (30), for $(p_s,p_b)=(0.5,0.5)$, and $(p_s,p_b)=(0.2,0.8)$, respectively. The comparisons with the simulation results confirm the accuracy of our approximation method. Subsequently, Figure 25 and Figure 26 show the numerical results for the approximated number of batch-service customers, i.e., (31), for $(p_s,p_b)=(0.5,0.5)$, and $(p_s,p_b)=(0.2,0.8)$, respectively. Although marginal discrepancies are observed, the approximation formula captured overall tendencies. It is also observed that the approximation formula takes accurate value as $p_b$ increases. This tendency is attributed to the assumption in which the portion of batches with K customers from within all batches (including single-service customers) at an arbitrary time point is calculated by the binary distribution with parameter $1-p_{new}$. In the proposed model, the positions of batches with K customers are mutually dependent because of the batch formation time, and the effect of batch formation time becomes larger as $p_b$ becomes smaller—which induces larger disparities. Overall, it is clear that the proposed approximation method is effective to obtain distributions of the number of customers for each type.

5.2. Mean Sojourn Time When the Service Distributions Differ for Each Type

Up to this point, we have assumed the same service time distribution; however, assuming different service time distributions allows for broader applications. Figure 27 and Figure 28 show the numerical results of the mean sojourn time of single and batch customers when the service rates of single and batch services differ.

Then, considering the service time distribution as hyperexponential and treating it as an M/G/1 queue model, we can obtain the following approximation formula for the mean sojourn time of single customer. Note that $W_a$ represents the sojourn time of an single customer according to this approximation:

$$E\left[W_a\right]={\displaystyle \frac{\tilde{\lambda }(V\left[X\right]+E{\left[X\right]}^2)}{2(1-\tilde{\lambda }E\left[X\right])}}+E\left[X\right].$$

(32)

where $\tilde{\lambda }$, $E\left[X\right]$, and $V\left[X\right]$ are the overall arrival rate, the mean service time, and the variance of the service time, respectively, and are given as follows:

$$\tilde{\lambda }=p_s\lambda +{\displaystyle \frac{p_b\lambda }{K}},$$

$$E\left[X\right]={\displaystyle \frac{p_{new}}{{\mu }_1}}+{\displaystyle \frac{1-p_{new}}{{\mu }_2}},$$

$$V\left[X\right]={\displaystyle \frac{2p_{new}}{{\mu }_1^2}}+{\displaystyle \frac{2(1-p_{new})}{{\mu }_2^2}}-E{\left[X\right]}^2.$$

Let $W_b$ be the sojourn time of batch customers. As $E\left[W_b\right]$ can be expressed as the equation in (32) plus the batch formation time, and we obtain (33):

$$E\left[W_b\right]=E\left[W_a\right]+\sum _{i=0}^{K-1}{\displaystyle \frac{i}{K{p}_b\lambda }}.$$

(33)

In this experiment, the service rate for single service is ${\mu }_1=5$, the service rate for batch service is ${\mu }_2=7$, the batch size is $K=3$, and $p_s=p_b=0.5$. There is a deviation from the simulation values, which can be attributed to the following reason. In this experiment, the service time distribution is approximated by a hyper-exponential distribution, which is a mix of two exponential distributions with parameters ${\mu }_1$ and ${\mu }_2$, respectively. The arrival rate to the queue is also approximated by a Poisson process with rate $\lambda p_s+\lambda p_b/K$. Despite of these approximations, we observe that the mean sojourn time of single customers and that of batch customers match.

6. Conclusions

In this study, we proposed a model in which arriving customers choose from three options: receiving the single or batch service, or other services. We then analyzed the performance measures, exact sojourn time distributions (for the single-service customers, batch-service customers, and all the customers), and an approximation method for the sojourn time distributions. Using numerical experiments, we presented the results of the performance measures, exact sojourn time distributions, and approximated sojourn time distributions. For the sojourn time distribution of single-service customers, it was found that the direct calculation using the rate matrix was significantly less computationally intensive than the calculation comprising the use of the formula with a cutting condition on an infinite sum. Furthermore, the approximations of the sojourn time distributions were verified as highly accurate.

In future research, it is conceivable to delve deeper into the differences in service distributions between single and batch services. This would enable the application of this model to a broader range of social phenomena. Numerical experiments on game-theoretical analysis, e.g., [21,22], should be considered. Through game-theoretical analysis, we can consider about maximizing the social utility and monopolist’s profit. In addition, a model in which the batch size follows an arbitrary random variable can be considered.