## The Annals of Probability

### Greedy walk on the real line

#### Abstract

We consider a self-interacting process described in terms of a single-server system with service stations at each point of the real line. The customer arrivals are given by a Poisson point processes on the space–time half plane. The server adopts a greedy routing mechanism, traveling toward the nearest customer, and ignoring new arrivals while in transit. We study the trajectories of the server and show that its asymptotic position diverges logarithmically in time.

#### Article information

Source
Ann. Probab., Volume 43, Number 3 (2015), 1399-1418.

Dates
First available in Project Euclid: 5 May 2015

https://projecteuclid.org/euclid.aop/1430830285

Digital Object Identifier
doi:10.1214/13-AOP898

Mathematical Reviews number (MathSciNet)
MR3342666

Zentralblatt MATH identifier
1327.60176

#### Citation

Foss, Sergey; Rolla, Leonardo T.; Sidoravicius, Vladas. Greedy walk on the real line. Ann. Probab. 43 (2015), no. 3, 1399--1418. doi:10.1214/13-AOP898. https://projecteuclid.org/euclid.aop/1430830285

#### References

• [1] Altman, E. and Foss, S. (1997). Polling on a space with general arrival and service time distribution. Oper. Res. Lett. 20 187–194.
• [2] Altman, E. and Levy, H. (1994). Queueing in space. Adv. in Appl. Probab. 26 1095–1116.
• [3] Angel, O., Benjamini, I. and Virág, B. (2003). Random walks that avoid their past convex hull. Electron. Commun. Probab. 8 6–16 (electronic).
• [4] Beffara, V., Friedli, S. and Velenik, Y. (2010). Scaling limit of the prudent walk. Electron. Commun. Probab. 15 44–58.
• [5] Benjamini, I. and Berestycki, N. (2010). Random paths with bounded local time. J. Eur. Math. Soc. (JEMS) 12 819–854.
• [6] Benjamini, I. and Wilson, D. B. (2003). Excited random walk. Electron. Commun. Probab. 8 86–92 (electronic).
• [7] Bertsimas, D. J. and Ryzin, G. V. (1991). A stochastic and dynamic vehicle routing problem in the Euclidean plane. Oper. Res. 39 601–615.
• [8] Bordenave, C., Foss, S. and Last, G. (2011). On the greedy walk problem. Queueing Syst. 68 333–338.
• [9] Bousquet-Mélou, M. (2010). Families of prudent self-avoiding walks. J. Combin. Theory Ser. A 117 313–344.
• [10] Carmona, P., Petit, F. and Yor, M. (1998). Beta variables as times spent in $[0,\infty[$ by certain perturbed Brownian motions. J. Lond. Math. Soc. (2) 58 239–256.
• [11] Chaumont, L. and Doney, R. A. (1999). Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion. Probab. Theory Related Fields 113 519–534.
• [12] Coffman, E. G. Jr. and Gilbert, E. N. (1987). Polling and greedy servers on a line. Queueing Systems Theory Appl. 2 115–145.
• [13] Davis, B. (1996). Weak limits of perturbed random walks and the equation $Y_{t}=B_{t}+\alpha\sup\{Y_{s}: s\leq t\}+\beta\inf\{Y_{s}: s\leq t\}$. Ann. Probab. 24 2007–2023.
• [14] Davis, B. (1999). Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113 501–518.
• [15] Foss, S. and Last, G. (1996). Stability of polling systems with exhaustive service policies and state-dependent routing. Ann. Appl. Probab. 6 116–137.
• [16] Foss, S. and Last, G. (1998). On the stability of greedy polling systems with general service policies. Probab. Engrg. Inform. Sci. 12 49–68.
• [17] Kroese, D. P. and Schmidt, V. (1992). A continuous polling system with general service times. Ann. Appl. Probab. 2 906–927.
• [18] Kroese, D. P. and Schmidt, V. (1994). Single-server queues with spatially distributed arrivals. Queueing Systems Theory Appl. 17 317–345.
• [19] Kroese, D. P. and Schmidt, V. (1996). Light-traffic analysis for queues with spatially distributed arrivals. Math. Oper. Res. 21 135–157.
• [20] Kurkova, I. A. (1996). A sequential clearing process. Fundam. Prikl. Mat. 2 619–624.
• [21] Kurkova, I. A. and Menshikov, M. V. (1997). Greedy algorithm, $\mathbf{Z}^{1}$ case. Markov Process. Related Fields 3 243–259.
• [22] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939–995.
• [23] Lawler, G. F., Schramm, O. and Werner, W. (2004). On the scaling limit of planar self-avoiding walk. In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2. Proc. Sympos. Pure Math. 72 339–364. Amer. Math. Soc., Providence, RI.
• [24] Leskelä, L. and Unger, F. (2012). Stability of a spatial polling system with greedy myopic service. Ann. Oper. Res. 198 165–183.
• [25] Litvak, N. and Adan, I. (2001). The travel time in carousel systems under the nearest item heuristic. J. Appl. Probab. 38 45–54.
• [26] Meester, R. and Quant, C. (1999). Stability and weakly convergent approximations of queueing systems on a circle. Available at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.12.7937.
• [27] Merkl, F. and Rolles, S. W. W. (2006). Linearly edge-reinforced random walks. In Dynamics & Stochastics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 48 66–77. IMS, Beachwood, OH.
• [28] Mountford, T. and Tarrès, P. (2008). An asymptotic result for Brownian polymers. Ann. Inst. Henri Poincaré Probab. Stat. 44 29–46.
• [29] Pemantle, R. (2007). A survey of random processes with reinforcement. Probab. Surv. 4 1–79.
• [30] Perman, M. and Werner, W. (1997). Perturbed Brownian motions. Probab. Theory Related Fields 108 357–383.
• [31] Raimond, O. and Schapira, B. (2011). Excited Brownian motions. ALEA Lat. Am. J. Probab. Math. Stat. 8 19–41.
• [32] Robert, P. (2010). The evolution of a spatial stochastic network. Stochastic Process. Appl. 120 1342–1363.
• [33] Rojas-Nandayapa, L., Foss, S. and Kroese, D. P. (2011). Stability and performance of greedy server systems. A review and open problems. Queueing Syst. 68 221–227.
• [34] Rolla, L. T. and Sidoravicius, V. Stability of the greedy algorithm on the circle. Available at arXiv:1112.2389.
• [35] Schaßberger, R. (1995). Stability of polling networks with state-dependent server routing. Probab. Engrg. Inform. Sci. 9 539–550.
• [36] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
• [37] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239–244.
• [38] Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 1435–1467.
• [39] Tóth, B. (1995). The “true” self-avoiding walk with bond repulsion on $\mathbf{Z}$: Limit theorems. Ann. Probab. 23 1523–1556.
• [40] Tóth, B. (1999). Self-interacting random motions—A survey. In Random Walks (Budapest, 1998). Bolyai Soc. Math. Stud. 9 349–384. János Bolyai Math. Soc., Budapest.
• [41] Tóth, B. and Werner, W. (1998). The true self-repelling motion. Probab. Theory Related Fields 111 375–452.
• [42] Zerner, M. P. W. (2005). On the speed of a planar random walk avoiding its past convex hull. Ann. Inst. Henri Poincaré Probab. Stat. 41 887–900.