Abstract
We analyze the asymptotics of waiting time distributions in the two-class accumulating priority queue with general service times. The accumulating priority queue was suggested by Kleinrock in the 60s—he coined it time-dependent priority—to diversify waiting time objectives of different classes in a paramaterized way. It also avoids the typical starvation problem of regular priority queues. All customers build up priority linearly while waiting in the queue but at a class-dependent rate. At a service opportunity epoch, the customer with highest priority present is served. Stanford and colleagues recently calculated the Laplace–Stieltjes Transform (LST) of the waiting time distributions of the different classes, but only invert these LSTs numerically. In this paper, we analytically calculate the asymptotics of the corresponding distributions from these LSTs. We show that different singularities of the LST can play a role in the asymptotics, depending on the magnitude of service differentiation between both classes.
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Notes
In this paper, we denote the LST corresponding to a density function f(t) by \({\tilde{F}}(s)\).
Not to be confused with class-1 customers.
Other distribution types can be treated as well, but make the (writing down of the) analysis more cumbersome. We refer to [1] for more details.
We follow notation of Stanford et al. [14] as closely as possible.
For exponential service times of class 1, an explicit expression is possible.
\(f(t) \sim g(t), t\rightarrow t_0 \Leftrightarrow \lim _{t\rightarrow t_0}f(t)/g(t)=1\).
Note that our formulas are not entirely identical as in Abate and Whitt [1], since they assumed \(\mu _1=1\).
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The authors wish to thank the two anonymous referees and associate editor for valuable comments that improved the paper.
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Walraevens, J., Van Giel, T., De Vuyst, S. et al. Asymptotics of waiting time distributions in the accumulating priority queue. Queueing Syst 101, 221–244 (2022). https://doi.org/10.1007/s11134-022-09839-7
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DOI: https://doi.org/10.1007/s11134-022-09839-7