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Strategic arrivals to a queue with service rate uncertainty

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Abstract

We study the problem of strategic choice of arrival time to a single-server queue with opening and closing times when there is uncertainty regarding service speed. A Poisson population of customers choose their arrival time with the goal of minimizing their expected waiting times and are served on a first-come first-served basis. There are two types of customers that differ in their beliefs regarding the service-time distribution. The inconsistent beliefs may arise from randomness in the server state along with noisy signals that customers observe. Customers are aware of the two types of populations with differing beliefs. We characterize the Nash equilibrium dynamics for exponentially distributed service times and show how they substantially differ from the model with homogeneous customers. We further provide an explicit solution for a fluid approximation of the game. For general service-time distributions we provide an algorithm for computing the equilibrium in a discrete-time setting. We find that in equilibrium customers with different beliefs arrive during different (and often disjoint) time intervals. Numerical analysis further shows that the mean waiting time increases with the coefficient of variation of the service time. Furthermore, we present a learning agent-based model (ABM) in which customers make joining decisions based solely on their signals and past experience. We numerically compare the long-term average outcome of the ABM with that of the equilibrium and find that the arrival distributions are quite close if we assume (for the equilibrium solution) that customers are fully rational and have knowledge of the system parameters, while they may greatly differ if customers have limited information or computing abilities.

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Acknowledgements

The authors wish to thank Refael Hassin for his helpful comments on a preliminary draft of this manuscript. The first author’s work was supported by the NWO Gravitation project Networks, Grant Number 024.002.003. The second author’s work was supported by JSPS KAKENHI Grant No. JP18K11186.

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Correspondence to Liron Ravner.

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Appendices

Appendices

1.1 Detailed proof of Lemma 1

To prove Lemma 1 we construct a coupling of the virtual waiting time and queueing processes for both types of customers and show that for every sample path type a customers face a longer queue and waiting time than type b customers for any arrival time \(t\in \mathcal {T}\).

Lemma 6

Let \(X_{i,k}\sim \exp (\mu _i)\) denote the job size of the k’th arrival when the service distribution is of type \(i\in \mathcal {C}\) customers. If \(\mu _a<\mu _b\), then the virtual waiting time \(V_a\) and the queue length \(Q_a\) of type a customers is stochastically larger than the virtual waiting time \(V_b\) and queue length \(Q_b\) of type b customers for any given strategy profile \((F_a,F_b)\):

$$\begin{aligned} V_b(t)<_\mathrm{st} V_a(t),\ Q_b(t)\le _\mathrm{st} Q_a(t),\ \forall t\in [0,T], \end{aligned}$$

which further implies that

$$\begin{aligned} \mathrm {E}V_b(t)\le \mathrm {E}V_a(t),\ \mathrm {E}Q_b(t)\le \mathrm {E}Q_a(t),\ \forall t\in [0,T] . \end{aligned}$$

Proof

The virtual waiting time, or workload, for type \(i\in \mathcal {C}\) customers can be constructed as follows: The input process of work is a nonhomogeneous Poisson process defined by the arrival strategies, each job has a size of \(X_i\), and work is continuously processed at a rate of one per unit of time. The following arguments are for a sample path constructed of the identical arrival process and coupled sequences of job sizes \(\{X_{i,k}\}_{k=1}^\infty \) such that \(X_{a,k}\ge X_{b,k}\) for all \(k\ge 1\). Specifically, let \(U_k\sim \)Unif[0, 1] and denote

$$\begin{aligned} X_{ki}=-\frac{1}{\mu _i}\log U_k, \end{aligned}$$

then clearly \(X_{a,k}=\frac{\mu _b}{\mu _a}X_{b,k}>X_{b,k}\) for all \(k\ge 1\). For any strategy profile \((F_a,F_b)\), the virtual waiting time for a type \(i\in \mathcal {C}\) arriving at \(t\in \mathcal {T}\) is given by

$$\begin{aligned} V_i(t):=A_i(t)- t+L_i(t), \end{aligned}$$

where \(A_i(t)\) is a nonhomogeneous compound Poisson process with cumulative rate \(\lambda _a F_a(t)+\lambda _b F_b(t)\) and jump sizes \(X_i\), and

$$\begin{aligned} L_i(t):=\left( -\inf _{s\in [0-,t]}\{A_i(s)- s\}-V_0\right) ^{+} . \end{aligned}$$

Observe that \(L_i(t)\) is a non-decreasing process that increases only when \(V_i(t)=0\) and that \(A_a(0)>0\) if and only if \(A_b(0)>0\). Suppose that \(A_a(0),A_b(0)>0\) and denote \(\tau _i=\inf \{t\ge 0:\ V_i(t)=0\}\), then as \(A_a(t)\ge A_b(t)\) we have that

$$\begin{aligned} V_a(t)=A_a(t)- t \ge A_b(t)- t= V_b(t),\ t\in [0,\tau _b] . \end{aligned}$$

Furthermore, as \(X_{a,k}>X_{b,k}\) for all jobs, the number of departures from queue a is at most equal to the number of departures from queue b during \([0,\tau _b]\) because both servers are working continuously. The common arrival time of jobs to both systems further implies that

$$\begin{aligned} Q_a(t) \ge Q_b(t),\ t\in [0,\tau _b] . \end{aligned}$$

The process \(L_b(t)\) increases after \(\tau _b\) as long as \(V_b(t)=0\), but clearly \(0=V_b(t)\le V_a(t)\) and \(0=Q_b(t)\le Q_a(t)\) during any such period. If a jump \(A_b(t)-A_b(t-)\) occurs at \(t<\tau _a\), then \(V_a(t)\) has a bigger jump than \(V_b(t)\) both and the dominance remains until the next time s such that \(V_b(s)=0\). This continues until \(\tau _a\) for which

$$\begin{aligned} V_a(\tau _a)=V_b(\tau _a)=Q_a(\tau _a)=Q_b(\tau _a)=0, \end{aligned}$$

and both \(V_i\) stay at zero until the next arrival time and the same process starts again with a new jump for both processes. The same argument is valid when \(A_a(0)=A_b(0)=0\) and the process starts at the time of the first arrival. We conclude that for every sample path \(V_a(t)\ge V_b(t)\) for all \(t\in \mathcal {T}\) and \(V_a(s)> V_b(s)\) for all \(\mathcal {S}\subset \mathcal {T}\) such that \(\mathcal {S}\) is a non-empty union of positive intervals, i.e., the set \(\mathcal {S}\) has nonzero measure. Similarly, the same holds for \(Q_a(t)\) and \(Q_b(t)\). \(\square \)

1.2 Best response algorithms

This appendix details Algorithm 1 and a necessary subroutine that we denote Algorithm 1-1. The purpose of the algorithm is to find the symmetric within type best response arrival distribution \(\varvec{p}_{i}\) that satisfies the equilibrium condition (21), when the customers of the other type arrive according to \(\varvec{p}_{-i}\). The outline is as follows: (1) finding the first time slot \(\theta \in \mathcal {T}\) such that \(p_{i,\theta }>0\), (2) a bisection search for the value of \(p_{i,\theta }\) which uniquely defines all probabilities \(p_{i,u}\) for \(u>\theta \), (3) stopping the procedure when \(\sum _{u=\theta }^Tp_{i,u}=1\). Algorithm 1 is designed for step (1) and checking the equilibrium condition of (3), and Algorithm 1-1 performs the bisection search of part (2).

Note that the implementation of the algorithm may be refined to make it more efficient, but the purpose of the description here is to provide a concise description. For example, in line 4 of Algorithm 1-1, we may choose \(p_{i,\theta }^{(R)}\) in a slightly better way, e.g., \(p_{i,\theta }^{(R)} = \min \left\{ 1, \mathrm{Eq.}\,(25) \right\} \), but we have chosen \(p_{i,\theta }^{(R)} := 1\) for the simplicity of the presentation.

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Ravner, L., Sakuma, Y. Strategic arrivals to a queue with service rate uncertainty. Queueing Syst 97, 303–341 (2021). https://doi.org/10.1007/s11134-020-09683-7

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