Generalized priority-queue network dynamics: Impact of team and hierarchy

Won-kuk Cho, Byungjoon Min, K.-I. Goh, and I.-M. Kim

Phys. Rev. E 81, 066109 – Published 15 June 2010

Abstract

We study the effect of team and hierarchy on the waiting-time dynamics of priority-queue networks. To this end, we introduce generalized priority-queue network models incorporating interaction rules based on team-execution and hierarchy in decision making, respectively. It is numerically found that the waiting-time distribution exhibits a power law for long waiting times in both cases, yet with different exponents depending on the team size and the position of queue nodes in the hierarchy, respectively. The observed power-law behaviors have in many cases a corresponding single or pairwise-interacting queue dynamics, suggesting that the pairwise interaction may constitute a major dynamic consequence in the priority-queue networks. It is also found that the reciprocity of influence is a relevant factor for the priority-queue network dynamics.

Received 8 September 2009

DOI:https://doi.org/10.1103/PhysRevE.81.066109

Authors & Affiliations

Won-kuk Cho, Byungjoon Min, K.-I. Goh^*, and I.-M. Kim

Department of Physics, Korea University, Seoul 136-713, Korea

^*kgoh@korea.ac.kr

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Vol. 81, Iss. 6 — June 2010

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Images

Figure 1
Schematic illustrations of interaction rules of the models. (a) $(N, r)$ -model with $N = 4$ queues and team-size $r = 3$ . Each queue has $(\binom{3}{2}) = 3$ $I$ -tasks (denoted by different shades and labeled with team members) and one $O$ -task (white). If an $I$ -task, say $I {1, 2, 4}$ (checkerboard), is chosen as highest priority task by a team member, all the queues in the team, queues 1, 2, and 4, execute the task. (b) $H$ -model with $N = 4$ . Each queue has one $I$ -task and one $O$ -task. Arranged by the hierarchy value $h$ from top to bottom, when a node, say the $h = 2$ node, executes its $I$ -task as highest priority task, other nodes in the lower hierarchy, $h = 3$ and $h = 4$ nodes, follow to execute the $I$ -task (indicated as dark).Reuse & Permissions
Figure 2
(Color online) Waiting-time distribution $P (τ)$ of $(N, r)$ -model. [(a)–(f)] $P (τ)$ of (a)–(c) $I$ -tasks and (d)–(f) $O$ -tasks for various team sizes $r$ with $N = 4$ [(a),(d)], 5 [(b),(e)], and 6 [(c),(f)]. Slope of the straight lines in (a)–(c) is $- 2$ ; those in (d)–(f) are $- 1.3$ (upper) and $- 2$ (lower), respectively, drawn for the eyes. [(g),(h)] Maximum waiting time $τ_{m}$ vs the simulation time $T$ for (g) $I$ -tasks and (h) $O$ -tasks for $N = 6$ . Slopes of the straight lines in (g) are 1 and 0.8 from top to bottom, and that in (h) is 1, drawn for the eyes. These slope values are to be compared with $β_{theor} = min {1 / (α - 1), 1}$ , expected from measured $α = 2.0 (1)$ and 2.3(1), respectively, for (g) and $α = 1.3 (1)$ for (h). Symbols in (g) and (h) follow those of (c) and (f).Reuse & Permissions
Figure 3
(Color online) $P (τ)$ for various $n_{O}$ for (a) $I$ -tasks and (b) $O$ -tasks in the (4,3)-model, showing that the power laws are robust with the number of $O$ -tasks, $n_{O}$ . Same symbols are used for (a) and (b).Reuse & Permissions
Figure 4
(Color online) $P (τ)$ of the hierarchy-based $H$ -model. [(a)–(c)] $P (τ)$ of $I$ -tasks for various hierarchy index $h$ with $N = 2$ (a), 4 (b), and 8 (c). Slope of the straight lines is $- 1$ , drawn for the eyes. [(d)–(f)] $P (τ)$ of $O$ -tasks for various hierarchy index $h$ with $N = 2$ (d), 4 (e), and 8 (f). Slopes of the straight lines in each panel are $- 2$ (upper) and $- 1$ (lower), respectively, drawn for the eyes. (g) Statistical weight of the distribution tail $A (T)$ vs simulation time $T$ for the $H$ -model with $N = 8$ , showing the inverse relation between them, $A (T) \sim 1 / T$ . (h) Exponential cutoff of $P (τ)$ $τ_{c}$ vs hierarchy index $h$ for the $H$ -model with $N = 8$ , obtained by fitting the distribution tails in (c) with Eq. (2), shows the approximate exponential decrease of $τ_{c}$ with $h$ .Reuse & Permissions

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