Understanding interarrival and interdeparture time statistics from interactions in queuing systems

We discuss the statistics of long queues, in which the interdeparture time statistics is dominated by spatial interactions among the elements in a queue rather than the arrival or exit processes. Based on a Fokker–Planck approach, it is possible to calculate the stationary distance distribution among the elements in a queue as a function of their interaction potential. The results relate to the ones known from Random Matrix Theory. Together with the velocity distribution, one can determine the time-gap distribution as well. This yields an analytical approach to the interdeparture and interarrival time distributions of queuing systems with spatially interacting elements. While these distributions are usually determined from empirical data or from theoretical assumptions about the arrival or exit process, we offer here an alternative interpretation of interdeparture time distributions as an effect of interactions in a queue. This is relevant for the understanding of traffic and production systems and for the optimization of the statistical behavior of some queuing systems.

[1]  Albert-László Barabási,et al.  The origin of bursts and heavy tails in human dynamics , 2005, Nature.

[2]  D. Helbing,et al.  Stability analysis and stabilization strategies for linear supply chains , 2003, cond-mat/0304476.

[3]  Dirk Helbing,et al.  SUPPLY AND PRODUCTION NETWORKS: FROM THE BULLWHIP EFFECT TO BUSINESS CYCLES , 2004 .

[4]  D. Helbing,et al.  DERIVATION, PROPERTIES, AND SIMULATION OF A GAS-KINETIC-BASED, NONLOCAL TRAFFIC MODEL , 1999, cond-mat/9901240.

[5]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

[6]  Dieter Armbruster.  Dynamical Systems and Production Systems , 2005 .

[7]  Hd Dieter Armbruster,et al.  Control and synchronization in switched arrival systems , 2003 .

[8]  Christian A. Ringhofer,et al.  A Continuum Model for a Re-entrant Factory , 2006, Oper. Res..

[9]  Kunihiko Kaneko,et al.  Networks of Interacting Machines – Production Organization in Complex Industrial Systems and Biological Cells , 2005 .

[10]  Nakayama,et al.  Dynamical model of traffic congestion and numerical simulation. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  A.A.J. Lefeber,et al.  Nonlinear models for control of manufacturing systems , 2003 .

[12]  Carlos F. Daganzo,et al.  A theory of supply chains , 2003 .

[13]  Ulrich Witt,et al.  Myopic Behavior and Cycles in Aggregate Output. A Note on the Role of Correlated Quantity Adjustments / Myopisches Verhalten und der Konjunkturzyklus. Bemerkungen zur Rolle korrelierter Mengenanpassungen , 2002 .

[14]  Dirk Helbing,et al.  Jam-Avoiding Adaptive Cruise Control (ACC) and its Impact on Traffic Dynamics , 2005 .

[15]  D. Helbing,et al.  Physics, stability, and dynamics of supply networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  D. Helbing,et al.  Phase diagram of tra c states in the presence of inhomogeneities , 1998, cond-mat/9809324.

[17]  Dirk Helbing,et al.  Analytical investigation of oscillations in intersecting flows of pedestrian and vehicle traffic. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Helbing,et al.  Congested traffic states in empirical observations and microscopic simulations , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  Dirk Helbing,et al.  Fokker-Planck equation approach to vehicle statistics , 2003 .

[20]  Carl M. Harris,et al.  Fundamentals of queueing theory , 1975 .

[21]  Dirk Helbing,et al.  Determination of Interaction Potentials in Freeway Traffic From Steady-State Statistics , 2003, cond-mat/0301484.

[22]  A. Barabasi,et al.  Human dynamics: Darwin and Einstein correspondence patterns , 2005, Nature.

[23]  Dirk Helbing,et al.  Network-induced oscillatory behavior in material flow networks and irregular business cycles. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Christian A. Ringhofer,et al.  A Mesoscopic Approach to the Simulation of Semiconductor Supply Chains , 2003, Simul..

[25]  Dirk Helbing,et al.  Understanding widely scattered traffic flows, the capacity drop, and platoons as effects of variance-driven time gaps. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Günter Radons,et al.  Nonlinear Dynamics of Production Systems , 2004 .