Game of arrivals at a two queue network with heterogeneous customer routes

TL;DR

The paper studies a two-queue queuing network with two customer classes and class-specific routes, modeled as a fluid game where arrivals are strategic and costs are linear in waiting and departure times. It establishes a rigorous characterization of equilibrium arrival profiles (EAPs) under both heterogeneous and equal-cost preferences, showing unique EAPs for unequal preferences and convex sets of equilibria for equal preferences. By partitioning parameter space into multiple regimes, the authors uncover up to eight distinct EAP structures in HAS/HDS, including cases with disjoint, contiguous, or overlapping arrival intervals, and even multiple intervals for a single class. The work reveals rich structural properties of equilibria in simple networks and argues that equilibrium learning or learning-approximation will become increasingly challenging as networks grow. These insights provide a foundation for understanding strategic arrivals in more complex networks and offer avenues for further exploration of non-fluid extensions and learning dynamics.

Abstract

We consider a queuing network that opens at a specified time, where customers are non-atomic and belong to different classes. Each class has its own route, and as is typical in the literature, the costs are a linear function of waiting and service completion time. We restrict ourselves to a two class, two queue network: this simplification is well motivated as the diversity in solution structure as a function of problem parameters is substantial even in this simple setting (e.g., a specific routing structure involves eight different regimes), suggesting a combinatorial blow up as the number of queues, routes and customer classes increase. We identify the unique Nash equilibrium customer arrival profile when the customer linear cost preferences are different. This profile is a function of problem parameters including the size of each class, service rates at each queue, and customer cost preferences. When customer cost preferences match, under certain parametric settings, the equilibrium arrival profiles may not be unique and may lie in a convex set. We further make a surprising observation that in some parametric settings, customers in one class may arrive in disjoint intervals. Further, the two classes may arrive in contiguous intervals or in overlapping intervals, and at varying rates within an interval, depending upon the problem parameters.

Figures (17)

• Figure 1: EAP structure (left) and resulting queue length process (right) when two classes of users with cost preferences $\gamma^{(1)}$ and $\gamma^{(2)}$ (assuming $\gamma^{(1)}<\gamma^{(2)}$) are arriving at a queue of capacity $\mu$. The support boundaries are $T^{(2)}_f=\frac{\Lambda^{(1)}+\Lambda^{(2)}}{\mu},~T^{(2)}_a=\frac{\Lambda^{(1)}}{\mu}-\left(\frac{1}{\gamma^{(2)}}-1\right)\frac{\Lambda^{(2)}}{\mu}$ and $T^{(1)}_a=-\left(\frac{1}{\gamma^{(1)}}-1\right)\frac{\Lambda^{(1)}}{\mu}-\left(\frac{1}{\gamma^{(2)}}-1\right)\frac{\Lambda^{(2)}}{\mu}$. The queue length process is illustrated only for the situation $T^{(2)}_a>0$, or equivalently $\Lambda^{(1)}>\left(\frac{1}{\gamma^{(2)}}-1\right)\Lambda^{(2)}$. Red and blue, respectively, represents class 1 and 2 populations. The black dashed line represents the total waiting mass of the two classes in the plot for queue length.
• Figure 2: Illustrative EAP (left) and resulting queue length process (right) of HDS with $\mu_1>\mu_2$ under case 1: $\gamma^{(1)}\leq\frac{\mu_2}{\mu_1}\cdot\gamma^{(2)}$ and the condition $T^{(1)}_f>0$ in (\ref{['eq_bdary_inst1_uneqpref_case1']}). $T\in(T^{(1)}_f,T^{(2)}_f)$ is the time at which queue 1 empties after time zero. In queue 1, since class 1 users stop arriving after $T^{(1)}_f$, class 1 waiting mass decreases faster in $[T^{(1)}_f,\tau_1(T^{(1)}_f)]$ than in $[0,T^{(1)}_f]$.
• Figure 3: Illustrative EAP (left) and resulting queue length process (right) of HDS with $\mu_1>\mu_2$ under case 2a of Theorem \ref{['mainthm_inst1']} with the assumption $T^{(2)}_a<0$ in (\ref{['eq_bdary_inst1_uneqpref_case2a']}). Queue 1 divides its capacity between the two classes from $\tau_1(T^{(2)}_a)$ proportionally to their arrival rate. As a result, queue 1 serves class 1 users at a slower rate in $[\tau_1(T^{(2)}_a),T^{(1)}_f]$ than in $[0,\tau_1(T^{(2)}_a)]$, causing class 1 mass in queue 1 to decrease at a slower rate in the former interval than in the latter one.
• Figure 4: Illustrative EAP (left) of HDS with $\mu_1>\mu_2$ under case 2b and 3b of Theorem \ref{['mainthm_inst1']}. The resulting queue length process (right) is for case 2b with the assumption $T^{(2)}_a<0$ in (\ref{['eq_bdary_inst1_uneqpref_case2b']}). Queue 1 divides its capacity between the two classes from $\tau_1(T^{(1)}_a)$ proportionally to their service rates. As a result, queue 1 serves class 2 users at a lesser rate in $[\tau_1(T^{(1)}_a),T^{(1)}_f]$ than in $[0,\tau_1(T^{(1)}_a)]$, causing class 2 waiting mass to decrease at a lesser rate in the former interval than in the latter one. In queue 2, waiting mass increases in $[\tau_1(T^{(1)}_a),T^{(1)}_f]$ because class 2 arrival rate to queue 2 is $\mu_2\gamma^{(1)}/\gamma^{(2)} >\mu_2$. However the waiting mass increases in $[\tau_1(T^{(1)}_a),T^{(1)}_f]$ at a rate slower than in $[0,\tau_1(T^{(1)}_a)]$, since the arrival rate in the formal interval is $\mu_2\gamma^{(2)}/\gamma^{(1)}<\mu_1=$ arrival rate in the latter interval. For case 3b of Theorem \ref{['mainthm_inst1']}, with the assumption $T^{(1)}_a<0$ on (\ref{['eq_bdary_inst1_uneqpref_case2b']}), queue 1 length process has the same structure. Queue 2 length process of case 3b is also same as case 2b, except for one region: queue 2 length decreases in $[\tau_1(T^{(1)}_a),T^{(1)}_f]$, since class 2 arrival rate to queue 2 is $\mu_2\gamma^{(2)}/\gamma^{(1)}<\mu_2$ in case 3b.
• Figure 5: Illustrative EAP (left) and resulting queue length process (right) of HDS with $\mu_1>\mu_2$ under case 3a of Theorem \ref{['mainthm_inst1']} with the assumption $T^{(1)}_a<0$ and $\tau_1(T^{(1)}_a)=\frac{\mu_2\gamma^{(2)}}{\mu_1}(T^{(1)}_a-T^{(2)}_a)<T^{(2)}_f$ in (\ref{['eq_bdary_inst1_uneqpref_case3a']}). In $[\tau_1(T^{(1)}_a),T^{(1)}_f]$, queue 1 divides its capacity between the two classes proportionally to their arrival rates. As a result, class 2 mass in queue 1 decreases at a lesser rate in $[\tau_1(T^{(1)}_a),T^{(2)}_f]$ than in $[0,\tau_1(T^{(1)}_a)]$. After $T^{(2)}_f$, as class 2 users stop arriving, the class 2 mass in queue 1 decreases to zero at a higher rate in $[T^{(2)}_f,\tau_1(T^{(2)}_f)]$ than in $[\tau_1(T^{(1)}_a),T^{(2)}_f]$. Class 1 mass in queue 1 initially decreases in $[\tau_1(T^{(1)}_a),T^{(2)}_f]$. Since class 1 users increase their arrival rate after $T^{(2)}_f$, slope of the red line increases and the class 1 mass in queue 1 can even increase in $[T^{(2)}_f,\tau_1(T^{(2)}_f)]$ if $\mu_1\gamma^{(1)}>\mu_1-\mu_2\gamma^{(2)}/\gamma^{(1)}$. After $\tau_1(T^{(2)}_f)$, queue 1 allocates the entire capacity to class 1 and the class 1 mass in queue 1 decreases to zero in $[\tau_1(T^{(2)}_f),T^{(1)}_f]$.

Theorems & Definitions (100)

• Definition 2.1
• Definition 2.2: Support of arrival profile
• Definition 2.3: Equilibrium Arrival Profile (EAP)
• Remark 1
• Lemma 2.1
• Theorem 3.1
• Lemma 3.1: Threshold Behavior
• Lemma 3.2: Structure of supports
• Lemma 3.3
• Lemma 3.4: Rates of arrival