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Queues with inspection cost: To see or not to see?

Jake Clarkson, Konstantin Avrachenkov, Eitan Altman

Abstract

Consider an M/M/1-type queue where joining attains a known reward, but a known waiting cost is paid per time unit spent queueing. In the 1960s, Naor showed that any arrival optimally joins the queue if its length is less than a known threshold. Yet acquiring knowledge of the queue length often brings an additional cost, e.g., website loading time or data roaming charge. Therefore, our model presents any arrival with three options: join blindly, balk blindly, or pay a known inspection cost to make the optimal joining decision by comparing the queue length to Naor's threshold. In a recent paper, Hassin and Roet-Green prove that a unique Nash equilibrium always exists and classify regions where the equilibrium probabilities are non-zero. We complement these findings with new closed-form expressions for the equilibrium probabilities in the majority of cases. Further, Hassin and Roet-Green show that minimizing inspection cost maximises social welfare. Envisaging a queue operator choosing where to invest, we compare the effects of lowering inspection cost and increasing the queue-joining reward on social welfare. We prove that the former dominates and that the latter can even have a detrimental effect on social welfare.

Queues with inspection cost: To see or not to see?

Abstract

Consider an M/M/1-type queue where joining attains a known reward, but a known waiting cost is paid per time unit spent queueing. In the 1960s, Naor showed that any arrival optimally joins the queue if its length is less than a known threshold. Yet acquiring knowledge of the queue length often brings an additional cost, e.g., website loading time or data roaming charge. Therefore, our model presents any arrival with three options: join blindly, balk blindly, or pay a known inspection cost to make the optimal joining decision by comparing the queue length to Naor's threshold. In a recent paper, Hassin and Roet-Green prove that a unique Nash equilibrium always exists and classify regions where the equilibrium probabilities are non-zero. We complement these findings with new closed-form expressions for the equilibrium probabilities in the majority of cases. Further, Hassin and Roet-Green show that minimizing inspection cost maximises social welfare. Envisaging a queue operator choosing where to invest, we compare the effects of lowering inspection cost and increasing the queue-joining reward on social welfare. We prove that the former dominates and that the latter can even have a detrimental effect on social welfare.

Paper Structure

This paper contains 34 sections, 13 theorems, 80 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and Model
  2. Utility Functions
  3. No Customers Inspect the Queue
  4. All Customers Inspect the Queue
  5. Properties of Utility Functions
  6. Equilibrium Structure as $R$ and $C_I$ Vary
  7. Conditions Where No Customers Inspect
  8. Properties of $C_{I,0}^1(R)$:
  9. Properties of $C_{I,0}^{2,3}(R)$:
  10. Conditions Where All Customers Inspect
  11. Properties of $C_{I,1}^{3}(R)$:
  12. Properties of $C_{I,1}^{1,2}(R)$:
  13. Equilibria Where Some Customers Inspect
  14. Scenario 1
  15. Scenario 2
  16. ...and 19 more sections

Key Result

Lemma 1

Suppose both $\rho_L$ and $\rho_U$ are increased, at least one by a non-zero amount. Then we have the following effects:

Figures (2)

  • Figure 1: (Best viewed in colour) For $\lambda=0.5, \mu=0.8$ and $c_w=1$, equilibria as $R$ and $C_I$ vary. The scenarios 1, 2 and 3 are labelled at the top of the plot and separated by the black, vertical, dashed lines. The increments of $c_w/\mu$ on the $R$-axis demonstrate the regions where $n_e$ is constant. The colour represents a different equilibrium form in that region. The lines represent various functions from Section \ref{['sec:find_eq']} which define the coloured regions.
  • Figure 2: (Best viewed in colour) For the example in Figure \ref{['fig:all']} with $\lambda=0.5, \mu=0.8$ and $c_w=1$, a contour plot showing changes to social welfare as $R$ and $C_I$ vary. As in Figure \ref{['fig:all']}, the increments of $c_w/\mu$ on the $R$-axis demonstrate the regions where $n_e$ is constant. Only those coloured regions where the social welfare is non zero (namely green, blue and cyan) are retained from Figure \ref{['fig:all']}. The darker the contour lines, the greater the social welfare.

Theorems & Definitions (14)

  • Lemma 1
  • Lemma 2
  • Corollary 3
  • Remark 4
  • Lemma 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Corollary 9
  • Proposition 10
  • ...and 4 more