Limiting the oscillations in queues with delayed information through a novel type of delay announcement

Abstract

Many service systems use technology to notify customers about their expected waiting times or queue lengths via delay announcements. However, in many cases, either the information might be delayed or customers might require time to travel to the queue of their choice, thus causing a lag in information. In this paper, we construct a neutral delay differential equation model for the queue length process and explore the use of velocity information in our delay announcement. Our results illustrate that using velocity information can have either a beneficial or detrimental impact on the system. Thus, it is important to understand how much velocity information a manager should use. In some parameter settings, we show that velocity information can eliminate oscillations created by delays in information. We derive a fixed point equation for determining the optimal amount of velocity information that should be used and find closed-form upper and lower bounds on its value. When the oscillations cannot be eliminated altogether, we identify the amount of velocity information that minimizes the amplitude of the oscillations. However, we also find that using too much velocity information can create oscillations in the queue lengths that would otherwise be stable.

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Acknowledgements

Funding was provided by Division of Civil, Mechanical and Manufacturing Innovation (Grant No. 1751975).

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Correspondence to Jamol Pender.

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Appendix

Appendix

Uniqueness and existence of the equilibrium

Proof of Theorem 3.2: To check that \(q_i(t)= \frac{\lambda }{N\mu }\) is an equilibrium, plug into Eq. (3.9) to get

(7.1)

To show uniqueness, we will argue by contradiction. Suppose there is another equilibrium given by \({\bar{q}}_i\), \(1\le i \le N\), and for some i we have \(q^*_i \ne {\bar{q}}_i\). The following condition must hold:

(7.2)

Hence, the mean of \({\bar{q}}_i\) is \(\frac{\lambda }{N\mu } \) and, since \({\bar{q}}_i\) cannot all be equal to each other, there must exist some \({\bar{q}}_s\) that is smaller than the mean, and some \({\bar{q}}_g\) that is greater than the mean:

$$\begin{aligned} {\bar{q}}_s = \frac{\lambda }{\mu N} - \gamma , \qquad {\bar{q}}_b = \frac{\lambda }{N\mu } + \epsilon , \qquad \gamma , \epsilon > 0. \end{aligned}$$
(7.3)

This leads to a contradiction:

(7.4)
(7.5)
(7.6)
(7.7)
(7.8)

Since , then \({\bar{q}}_i(t)\) is not an equilibrium, and the equilibrium (3.12) is unique.

Approximation to amplitude of oscillations in queues

To see how the velocity information affects the behavior of the queues after a Hopf bifurcation occurs, we need to develop approximations for the amplitude of oscillations. In Sect. 5, we find a first-order approximation to the amplitude but observe that it is not sufficiently accurate. Hence, we require a second-order approximation. The steps to determine the second-order approximation are outlined below.

This process is very closely related to the steps taken in Theorem 5.1. We begin with Eq. (5.7), and expand the time \(\tau = \omega t\). Then expand our functions of interest in \(\epsilon \) to the second order:

$$\begin{aligned}&x(\tau ) = x_0(\tau ) + \epsilon x_1 (\tau ) + + \epsilon ^2 x_2 (\tau ),\\&\quad \Delta = \Delta _0 + \epsilon \Delta _1 + \epsilon ^2 \Delta _2, \quad \omega = \omega _0 + \epsilon \omega _1 + \epsilon ^2 \omega _2, \end{aligned}$$

where \(\Delta _0\) and \(\omega _0\) are the delay and frequency at bifurcation, so \(\Delta _0 = \Delta _\mathrm{cr}\) and \(\omega _\mathrm{cr}\). By collecting all the terms with the like powers of \(\epsilon \) into separate equations, we get equations from which we can solve for \(x_0\) and \(x_1\). From the equation for \(\epsilon ^0\), we find that \(x_0(\tau ) = A \cos (\tau )\) is a solution. Next, we use the equation for \(\epsilon ^1\) terms to solve for A, which has the expression given by Eq. (5.35). We can now find \(x_1\) that has a solution of the form \(x_1(\tau ) = a_1 \sin (\tau ) + a_2 \cos (\tau ) + a_3 \sin (3 \tau ) + a_4 \cos (3 \tau )\). The coefficients \(a_3\) and \(a_4\) are determined from the equation for \(\epsilon ^1\) terms. We impose the initial condition \(x'(0) = 0\) to ensure that the maximum amplitude is at 0, which implies \(a_1 = - 3 a_3\). Lastly, we determine \(a_2\) by eliminating the secular terms from the equation for \(\epsilon ^2\) terms. Therefore, the second-order approximation of the amplitude of oscillations can be deduced from

$$\begin{aligned} x(\tau )\approx & {} x_0(\tau ) + \epsilon x_1(\tau ) \end{aligned}$$
(7.9)
$$\begin{aligned}= & {} A \cos (\tau ) + \epsilon \big ( a_1 \sin (\tau ) + a_2 \cos (\tau ) + a_3 \sin (3 \tau ) + a_4 \cos (3 \tau ) \big ),\qquad \end{aligned}$$
(7.10)

where the coefficients are given below:

$$\begin{aligned} A= & {} \sqrt{ \frac{4 \Delta _1 (\lambda ^2\theta ^2 - 4 \mu ^2)(4 - \delta ^2 \lambda ^2\theta ^2)^2}{\theta ^2(1 - \delta ^2\mu ^2)(16\mu + \lambda ^2\theta ^2(4\Delta _0 -4\delta + \delta ^3\lambda ^2\theta ^2 - 4\delta ^2\mu -4\delta ^2 \Delta _0\mu ^2))}}, \\ \omega _1= & {} \frac{4 \Delta _1 \theta ^2 \lambda ^2 \left( \delta ^2 \mu ^2-1\right) \sqrt{\theta ^2 \lambda ^2-4 \mu ^2}}{\sqrt{4-\delta ^2 \theta ^2 \lambda ^2} \left( \theta ^2 \lambda ^2 \left( \delta \left( \delta ^2 \theta ^2 \lambda ^2-4 \delta \mu ({\Delta 0} \mu +1)-4\right) +4 {\Delta 0}\right) +16 \mu \right) }, \\ a_1= & {} - 3 a_3 = -\Big (2 A^3 \theta ^2 \omega _0^3 \left( \theta ^2 \lambda ^2 \mu \left( \delta ^2 \omega _0^2+1\right) ^3-4 \delta ^3 \left( \mu ^2+\omega _0^2\right) ^3\right) \Big )\\&/\Big (\theta ^4 \lambda ^4 \left( \delta ^2 \omega _0^2+1\right) ^3 \left( \mu ^2+9 \omega _0^2\right) +16 \left( 9 \delta ^2 \omega _0^2+1\right) \left( \mu ^2+\omega _0^2\right) ^3 \\&+8 \theta ^2 \lambda ^2 \big (-9 \delta ^4 \omega _0^8-6 \mu ^2 \omega _0^2 \left( \delta ^2 \mu ^2+1\right) +2 \delta ^2 \omega _0^6 (\delta \mu (9 \delta \mu -32)+9) \\&+3 \omega _0^4 \left( \delta ^4 \mu ^4-12 \delta ^2 \mu ^2+1\right) -\mu ^4\big )\Big ), \\ a_4= & {} -\frac{1}{12}\Big (A^3 \theta ^2 \left( \theta ^2 \lambda ^2 \left( \delta ^2 \omega _0^2+1\right) ^3 \left( \mu ^4+6 \mu ^2 \omega _0^2-3 \omega _0^4\right) \right. \\&\left. +4 \left( 3 \delta ^4 \omega _0^4-6 \delta ^2 \omega _0^2-1\right) \left( \mu ^2+\omega _0^2\right) ^3\right) \Big ) \\&/\Big (\theta ^4 \lambda ^4 \left( \delta ^2 \omega _0^2+1\right) ^3 \left( \mu ^2+9 \omega _0^2\right) +16 \left( 9 \delta ^2 \omega _0^2+1\right) \left( \mu ^2+\omega _0^2\right) ^3 \\&+8 \theta ^2 \lambda ^2 \big (-9 \delta ^4 \omega _0^8-6 \mu ^2 \omega _0^2 \left( \delta ^2 \mu ^2+1\right) +2 \delta ^2 \omega _0^6 (\delta \mu (9 \delta \mu -32)+9) \\&+3 \omega _0^4 \left( \delta ^4 \mu ^4-12 \delta ^2 \mu ^2+1\right) -\mu ^4\big )\Big ), \\ a_2= & {} \frac{1}{12}\Big (A^5 \theta ^4 \left( \delta ^2 \omega _0^2+1\right) ^2 \left( \delta ^2 \mu \omega _0^2+\mu ^2 \left( \delta \left( \delta \Delta _0 \omega _0^2, -1\right) +\Delta _0\right) \right. \\&\left. +\omega _0^2 \left( \delta \left( \delta \Delta _0 \omega _0^2-1\right) +\Delta _0\right) +\mu \right) \\&-12 A^3 \theta ^2 \omega _0 \big (\omega _1 \left( \delta \left( 3 \delta ^3 \Delta _0 \omega _0^4+\delta \omega _0^2 (\delta (\delta \mu (2 \Delta _0 \mu +3)-3)\right. \right. \\&\left. \left. +4 \Delta _0)+\delta \mu (-2 \delta \mu +2 \Delta _0 \mu +3)-1\right) +\Delta _0\right) \\&-\Delta _1 \omega _0 \left( \delta ^2 \mu ^2-1\right) \left( \delta ^2 \omega _0^2+1\right) \big ) +12 A^2 \theta ^2 \big (a_1 \omega _0 \left( \delta ^2 \mu ^2-1\right) \left( \delta ^2 \omega _0^2+1\right) \\&+a_3 \omega _0 \left( \delta ^2 \left( \omega _0^2 (\delta \mu (-3 \delta \mu +8 \Delta _0 \mu +8)-5)+8 \delta \Delta _0 \omega _0^4+\mu ^2\right) -1\right) \\&+a_4 (3 \delta ^4 \Delta _0 \omega _0^6+3 \delta ^2 \omega _0^4 (\delta (\delta \mu (\Delta _0 \mu +1)-1)-2 \Delta _0)\\&+\omega _0^2 (\delta (\delta \mu (5 \delta \mu -6 \Delta _0 \mu -6)+1)-\Delta _0) \\&+\mu (\delta \mu -\Delta _0 \mu -1))\big ) -96 A \big (2 \Delta _1 \omega _0 \omega _1 (\mu \left( \mu \left( 2 \delta ^2\right. \right. \\&\left. \left. -2 \delta \Delta _0+\Delta _0^2\right) -\delta +\Delta _0\right) +\delta ^2 \Delta _0^2 \omega _0^4 \\&+\Delta _0 \omega _0^2 (\delta (\delta \mu (\Delta _0 \mu +1)-2)+\Delta _0)-1) +\Delta _0 \omega _1^2 (\delta ^2 \Delta _0^2 \omega _0^4\\&+\Delta _0 \omega _0^2 (\delta (\delta \mu (\Delta _0 \mu +1)-3)+\Delta _0) \\&+\mu (2 \delta -\Delta _0) (\delta \mu -\Delta _0 \mu -1))+\Delta _1^2 \omega _0^2 (\delta ^2 \Delta _0 \omega _0^4 \end{aligned}$$
$$\begin{aligned}&+\omega _0^2 (\delta (\delta \mu (\Delta _0 \mu +1)-1)+\Delta _0) \\&+\mu (-\delta \mu +\Delta _0 \mu +1))\big )-192 a_1 \big (\omega _1 (\delta ^2 \Delta _0^2 \omega _0^4\\&+\Delta _0 \omega _0^2 (\delta (\delta \mu (\Delta _0 \mu +2)-2)+\Delta _0)\\&+(-\delta \mu +\Delta _0 \mu +1)^2)+\Delta _1 \omega _0 \left( \delta ^2 \Delta _0 \omega _0^4+\omega _0^2 (\delta (\delta \mu (\Delta _0 \mu +1)\right. \\&\left. -1)+\Delta _0)+\mu (-\delta \mu +\Delta _0 \mu +1)\right) \big )\Big ) \\&/\Big (3 A^2 \theta ^2 \left( \delta ^2 \omega _0^2+1\right) \left( \delta ^2 \Delta _0 \omega _0^4+\omega _0^2 (\delta (\delta \mu (\Delta _0 \mu +1)\right. \\&\left. -1)+\Delta _0)+\mu (-\delta \mu +\Delta _0 \mu +1)\right) \\&+16 \Delta _1 \omega _0^2 \left( \delta ^2 \mu ^2-1\right) \Big ). \end{aligned}$$

To reproduce our numerical results from Sects. 5.35.4, set \(\epsilon = 1\) and \(\Delta _1 = \frac{1}{\epsilon }(\Delta - \Delta _0)\), with \(\Delta _0\) given by Eq. (3.52). Note that in the equations above there is no presence of \(\Delta _2\), because we have set \(\Delta _2 = 0\). There is no equation that determines \(\Delta _2\) and \(\Delta _1\) uniquely, and the only restriction is that \(\Delta = \Delta _0 + \epsilon \Delta _1 + \epsilon ^2 \Delta _2\). Prior to choosing \(\Delta _2 \) to be 0, we experimented numerically with different combinations of \(\Delta _1\) and \(\Delta _2\) and determined that the pair \(\Delta _1 = \frac{1}{\epsilon }(\Delta - \Delta _0)\) and \(\Delta _2 = 0\) results in nearly the most accurate approximation.

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Novitzky, S., Pender, J., Rand, R.H. et al. Limiting the oscillations in queues with delayed information through a novel type of delay announcement. Queueing Syst (2020). https://doi.org/10.1007/s11134-020-09657-9

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Keywords

  • Neutral delay-differential equation
  • Hopf bifurcation
  • Perturbations method
  • Operations research
  • Queueing theory
  • Fluid limits
  • Delay announcement
  • Velocity

Mathematics Subject Classification

  • 34K40
  • 34K18
  • 41A10
  • 37G15
  • 34K27