Estimating customer delay and tardiness sensitivity from periodic queue length observations

Abstract

A single server commences its service at time zero every day. A random number of customers decide when to arrive to the system so as to minimize the waiting time and tardiness costs. The costs are proportional to the waiting time and the tardiness with rates $\alpha$ and $\beta$, respectively. Each customer’s optimal arrival time depends on the others’ decisions; thus, the resulting strategy is a Nash equilibrium. This work considers the estimation of the ratio $\displaystyle \theta \equiv \beta /(\alpha +\beta )$ from queue length data observed daily at discrete time points, given that customers use a Nash equilibrium arrival strategy. A method of moments estimator is constructed from the equilibrium conditions. Remarkably, the method does not require estimation of the Nash equilibrium arrival strategy itself, or even an accurate estimate of its support. The estimator is strongly consistent, and the estimation error is asymptotically normal. Moreover, the asymptotic variance of the estimation error as a function of the queue length covariance matrix (at sampling times) is derived. The estimator performance is demonstrated through simulations and is shown to be robust to the number of sampling instants each day.

Appendices

Appendix A: Discrete approximation

This section explains how $F_e$ can be numerically approximated. If we denote a general arrival distribution by F and its probability density function by f, then the arrival process is a non-homogeneous Poisson process with intensity measure $\lambda f(t)$ for all $t\ge 0$. The queue length dynamics satisfy the Kolmogorov forward equations

$$\begin{aligned}&P'_0(t) \, = \, P_1(t) \mu - P_0(t) \lambda f(t), \end{aligned}$$

(A.1)

$$\begin{aligned}&P'_k(t) \, = \, P_{k-1}(t) \lambda f(t) + P_{k+1}(t) \mu - P_k(t) \, \left( \lambda f(t)+\mu \right) , \qquad k = 1,2,\ldots . \end{aligned}$$

(A.2)

The equilibrium arrival distribution $F_e$ satisfies (2.1), (2.2), and a set of nonlinear differential equations, which do not admit an analytic expression. We adopt the finite difference method, which was also mentioned in Haviv and Ravner [12, Sect. 3.1, Algorithm 1] and was termed as a discrete approximation, to numerically obtain $F_e$ and the associated expected cost.

To make the calculation of the expected queue length feasible, we truncate the queue length at K. Specifically, we assume that customers can choose to arrive at a time on a discrete grid ${\mathcal {T}} \equiv \{0, \delta , 2\delta , \ldots \}$, and the queue has a buffer size of K. When the value of $\delta$ is very small, with high probability there is at most one event happening in $\delta$, thus for $r = 1,2,\ldots$ and $k = 0,1, \ldots , K$, the queue length dynamics $P_k$ on ${\mathcal {T}}$ satisfy

$$\begin{aligned} P_0((r+1)\delta ) \, \approx&\, P_0(r \delta ) + P_1(r\delta ) \mu -P_0(k\delta )\lambda f(r\delta ) + o(\delta ) \end{aligned}$$

(A.3)

$$\begin{aligned} P_k((r+1)\delta ) \, \approx&\, P_k(r\delta ) + P_{k-1}(r\delta ) \lambda f(r\delta ) + P_{k+1}(r\delta ) \mu \nonumber \\&- P_k(r\delta ) \, \left( \lambda f(r\delta )+\mu \right) + o(\delta ), \, 1 \le k \le K-1\end{aligned}$$

(A.4)

$$\begin{aligned} P_K((r+1)\delta ) \approx&\,1 - \sum _{k=0}^{K-1} \, P_k((r+1)\delta )+ o(\delta ) , \end{aligned}$$

(A.5)

which are the finite difference scheme applied to Eqs. (A.1) and (A.2). The expected cost

$$\begin{aligned} {\mathbb {E}}_F[C(r\delta )]\, \approx \, {\left\{ \begin{array}{ll} \frac{(\alpha +\beta )\lambda p_e}{2\mu } &{} r = 0\\ \frac{\alpha +\beta }{\mu } q(r\delta )+ \beta r \delta &{} r> 0 , \end{array}\right. } \end{aligned}$$

(A.6)

where $q(r\delta ) \equiv \sum _{k=1}^{K}k P_k(r\delta )$ is the approximated expected queue length at slot r. For convenience, we drop the subscript F, and let the expected value and dynamics be that under the given arrival distribution for the rest of the paper. Increasing K or decreasing $\delta$ clearly improves the accuracy of the approximation, but this is at the expense of calculation speed. We set $K = \min \{m: \sum _{k = 0}^{m} {\lambda ^k \, e^{-\lambda }}/k! \ge 1-10^{-6}\}$, and $\delta = 0.001$ throughout the paper.

The values of $t_a$ and $t_b$ are approximated by $r_a \delta$ and $r_b\delta$. In the following, we explain how to find $p_e$, $r_a$, $r_b$, and $f_e$ on ${\mathcal {T}}\cap [r_a\delta , r_b\delta ]$. In each iteration, when the value of $p_e$ is given, the expected cost ${{\,\mathrm{{\mathbb {E}}}\,}}[C(0)]$ faced by customers arriving at time zero can be calculated. It follows from [10] that $F_e$ has a zero density along the interval $(0,t_a)$, which means $f(r \delta ) = 0$ until $r \ge r_a$. For $r = 1,2, \ldots , r_a$, since $f(r\delta ) = 0$, the queue length dynamics at time $r\delta$ can be calculated using Eqs. (A.3)–(A.5), the expected cost faced by a customer arriving at $r\delta$ can then be determined. The reason for $f_e(t) = 0, t \in (0,t_a)$ is that the expected cost faced by customers arriving at anytime in $(0,r_a \delta )$ is greater than ${{\,\mathrm{{\mathbb {E}}}\,}}[C(0)]$, which can also be inferred from Eq. (2.8). Hence, to determine the value of $r_a$, we keep computing the queue dynamics, and then the expected cost for $t = r \delta$ from $r =1$ until ${{\,\mathrm{{\mathbb {E}}}\,}}[C(t)] \le {{\,\mathrm{{\mathbb {E}}}\,}}[C(0)]$, then $r_a = \inf \{r: {{\,\mathrm{{\mathbb {E}}}\,}}[C(r \delta )] \le {{\,\mathrm{{\mathbb {E}}}\,}}[C(0)], r \ge 1\}$. In Haviv [10], the author calculated $t_a$ by working out the expression of the expected cost at time $t \in (0,t_a)$. Here, we use an alternative way and provide a more detailed explanation of the method in Haviv [10] and its comparison with our method in Remark 4.

For $r \ge r_a$, the arrival density $f(r\delta )$ is defined by Eq. (2.1), then the queue length dynamics can be obtained using Eqs. (A.3)–(A.5). We keep calculating $f(r\delta )$ and the queue length dynamics until $f(r\delta ) \le 0$, and $r_b = \inf \{r: f(r\delta ) \le 0, r > r_a \}$. Thus, given the value of $p_e$, the values of $r_a$, $r_b$, and $f(r\delta )$ in $[r_a\delta , r_b \delta ]$ can be determined. Another condition that $p_e$, $f_e$, $r_a$ and $r_b$ need to satisfy is

$$\begin{aligned} p_e+ \int _{t = t_a}^{t_b} f_e(t)\, \mathrm{{d}}t = 1. \end{aligned}$$

(A.7)

Hence, we can initially guess a value for $p_e$, and then adjust it iteratively using the bisection method until Eq. (A.7) is satisfied. Specifically, we start with $p_1 = 0, p_2 = 1$, and always set $p_e=\displaystyle \frac{p_1+p_2}{2}$. At the end of each iteration, we set $p_2 = p_e$ if the total probability is greater than one, and $p_1 = p_e$ otherwise. This calculation process is summarized in Algorithm 2.

Remark 4

The arrival distribution has zero density in $(0,t_a)$, so given $p_e$ at time zero, the expected waiting time faced by a customer if she arrives at any time $t \le t_a$ has an analytic expression. This expression was derived in Haviv [10, Lemma 3.3], where the author proposed two methods to calculate its quantity. One method is computing it with the assistance of Bessel’s functions, and the other method is estimating it using a Monte Carlo simulation procedure. The goal of working out the expression is to find the time at which if a customer arrives, her expected cost will be the same as the expected cost if she arrives at time 0. In our work, we do not adopt the expression of $t_a$, but keep calculating the expected cost until it is no longer greater than ${\mathbb {E}}[C(0)]$ and note down the time $\inf \{r: {{\,\mathrm{{\mathbb {E}}}\,}}[C(r \delta )] \le {{\,\mathrm{{\mathbb {E}}}\,}}[C(0)], r \ge 1\}$. Although our method to estimate $t_a$ does not use the analytical properties of $t_a$, it performs very well. In fact, in all the numerical examples we tried, it calculated $t_a$ faster.

Appendix B: Algorithms

1.1 Appendix B.1 The approximated expected queue length

1.2 Appendix B.2 The case with no arrivals before time 0

1.3 Appendix B.3 Finite closing time

1.4 Appendix B.4 The case with arrivals before time 0