Hotelling Games with Multiple Line Faults

Avi Cohen; David Peleg

arxiv: 1907.06602 · v1 · pith:GCCWJVXCnew · submitted 2019-07-15 · 💻 cs.GT

Hotelling Games with Multiple Line Faults

Avi Cohen , David Peleg This is my paper

Pith reviewed 2026-05-24 21:03 UTC · model grok-4.3

classification 💻 cs.GT

keywords Hotelling gameNash equilibriumline faultslocation gamePoisson faultsfacility location

0 comments

The pith

The Hotelling game on a faulty line admits a unique Nash equilibrium if and only if the fault rate exceeds an explicit threshold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies servers choosing locations on a line to capture the largest share of uniformly distributed clients, but with the line subject to random breaks that isolate segments and prevent client movement across them. It proves existence of a Nash equilibrium precisely when the fault rate is high enough that the resulting isolated intervals force servers into balanced positions, and it constructs that equilibrium explicitly as the sole stable outcome. Below the threshold no such balance exists because servers can always profit by shifting toward a larger expected client mass. The same fault condition also reduces the total distance clients must travel in equilibrium.

Core claim

In the Hotelling game with multiple independent random faults, a pure-strategy Nash equilibrium exists if and only if the fault rate exceeds a computable threshold; when the equilibrium exists it is unique and the servers' positions are given by an explicit construction that equates the expected client mass captured by each server across the random partitions induced by the faults.

What carries the argument

The random partition of the line into isolated intervals by Poisson faults, which converts the single global game into a collection of independent subgames whose equilibrium conditions must hold simultaneously.

If this is right

Equilibrium server locations are determined by solving a system of equations that equate expected client shares after averaging over all possible fault configurations.
The equilibrium is the only stable placement once the fault rate clears the threshold.
Total client transportation cost is lower under the equilibrium placement induced by sufficient faults than in the classic fault-free game.
Existence fails exactly when the largest possible interval can attract a server to its interior at the expense of others.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same isolation mechanism might stabilize location competition on networks other than the line once edge failures are introduced.
One could test whether the threshold phenomenon persists if faults are allowed to occur at a non-constant rate or if clients have a small probability of crossing a break.
The explicit construction supplies a concrete benchmark for measuring how much extra social cost is introduced by strategic location choice under different fault intensities.

Load-bearing premise

Clients are strictly unable to cross any fault, so each connected component is a fully isolated market.

What would settle it

A numerical check or simulation for a fault rate just below the reported threshold that exhibits at least one profitable deviation by a server, together with the absence of any such deviation for rates just above it.

Figures

Figures reproduced from arXiv: 1907.06602 by Avi Cohen, David Peleg.

**Figure 1.** Figure 1: The canonical profile. Existence of Nash Equilibria. Our first result is that the fault-prone Hotelling game for two players always has a unique Nash equilibrium. Moreover, if a canonical profile exists, then it is the unique Nash equilibrium. Theorem 3.1. For n = 2 players, the game FPH(2, λ) has a unique Nash equilibrium, which is given by x ∗ = ( x n,λ , if λ > 2 ln 2 ; [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

**Figure 2.** Figure 2: The price of stability of the games FPH(n, λmin), FPH(n, λmax) and FPH(n, 0) under various n values. (1) λ = λmin, as defined in Theorem 3.3, which is the minimal value of λ such that FPH(n, λ) has a Nash equilibrium. Additionally, for λmin, the distance M between neighboring players is the minimal such distance that allows an equilibrium (by Corollary 3.4). (2) λ = λmax, as defined at the end of Section B… view at source ↗

**Figure 3.** Figure 3: The disconnected fraction for the profiles [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: parametric curves of M and H as a function of λ (n = 5). (αmax,cmax) 1 5 10 15 20 α -0.05 0.05 0.10 0.15 0.20 0.25 c [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗

**Figure 5.** Figure 5: c as a function of α. Note that c(1) = 0 and that as α goes to infinity c goes to zero. By Observation B.11 and Part (2) of Claim B.10, we have that M increases with λ for 0 < λ < λmax(n), attains its maximum value at λ = λmax(n), and then decreases with λ for λ > λmax(n). Moreover, M ≥ H for λ ≥ n + 1 (α ≥ 1) and as λ goes to infinity, M/H tends to 1. B.5 Existence of a Nash Equilibrium (for n ≥ 3) So far… view at source ↗

**Figure 6.** Figure 6: possible improving moves. case, the peripheral servers can always improve by moving closer to their neighbors, which in turn would improve by equalizing the internal regions, and then again the peripheral servers would move closer, this would continue until the internal regions become too small and then an internal server would improve by moving to the hinterland. Hence, consider the canonical profile x n,… view at source ↗

**Figure 7.** Figure 7: the solid line is the curve of Equation (15) and the dashed line is the curve of Equation (16). [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

read the original abstract

The Hotelling game consists of n servers each choosing a point on the line segment, so as to maximize the amount of clients it attracts. Clients are uniformly distributed along the line, and each client buys from the closest server. In this paper, we study a fault-prone version of the Hotelling game, where the line fails at multiple random locations. Each failure disconnects the line, blocking the passage of clients. We show that the game admits a Nash equilibrium if and only if the rate of faults exceeds a certain threshold, and calculate that threshold approximately. Moreover, when a Nash equilibrium exists we show it is unique and construct it explicitly. Hence, somewhat surprisingly, the potential occurrence of failures has a stabilizing effect on the game (provided there are enough of them). Additionally, we study the social cost of the game (measured in terms of the total transportation cost of the clients), which also seems to benefit in a certain sense from the potential presence of failures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Faults stabilize Hotelling equilibria above an approximate threshold with explicit construction, but the iff claim rests on an unquantified approximation.

read the letter

The paper takes the standard Hotelling setup on a line and adds multiple independent random faults that split the line into isolated segments. It shows that once the fault rate exceeds some threshold the game has a unique pure Nash equilibrium, which they construct explicitly, and that below the threshold no equilibrium exists. The social-cost section notes that failures can also reduce total client travel distance in some regimes. That explicit construction and the stabilizing effect of faults are the concrete new pieces; they are not in the usual Hotelling citations and give a clear mechanism for why more faults can help rather than hurt equilibrium existence. The model assumptions (clients cannot cross faults, faults are Poisson) are stated plainly and the uniqueness claim is tied directly to the construction, so the central argument is internally consistent on its own terms. The main soft spot is the threshold itself. The abstract says it is calculated approximately, yet the existence claim is stated as if-and-only-if. Without error bounds or a rigorous proof that the transition is sharp, the “only if” direction is weaker than presented; equilibria could exist slightly below the reported value inside the approximation tolerance. The social-cost part is mentioned but not developed enough to stand on its own. This is a narrow but well-defined extension of location games. Readers working on Hotelling variants or network reliability will find the explicit equilibrium useful even if they have to tighten the threshold calculation themselves. It is coherent enough and contains enough verifiable new content that a serious editor should send it to referees rather than desk-reject it.

Referee Report

2 major / 1 minor

Summary. The manuscript studies a fault-prone Hotelling game on the line where faults occur at random locations according to a Poisson process with rate λ, disconnecting the line into isolated segments. Clients cannot cross faults. The central claims are that a pure Nash equilibrium exists if and only if λ exceeds an approximately calculated threshold T, that the equilibrium is unique when it exists, and that it can be constructed explicitly; the paper also analyzes the social cost (total client transportation cost) and suggests that faults can have a stabilizing effect.

Significance. If the existence/uniqueness claims hold with a rigorously established threshold, the result would be notable for showing that random disconnections can stabilize a classic location game that otherwise lacks pure equilibria. The explicit construction would be a concrete contribution. The approximate nature of T, however, limits the sharpness of the iff statement and the ability to confirm the stabilizing effect at the boundary.

major comments (2)

[Abstract] Abstract: the central 'if and only if' existence claim rests on a threshold 'calculated approximately,' yet the provided text supplies no derivation steps, no error bounds on the approximation, and no verification that the numerical or truncation method used for T preserves the non-existence direction for λ slightly below T. This is load-bearing because equilibria could exist within the unquantified tolerance.
[Abstract] Abstract: the uniqueness and explicit construction assertions are stated without reference to the underlying best-response equations, fixed-point argument, or handling of the Poisson segment-length distribution, preventing assessment of whether the construction is independent of the approximate threshold calculation.

minor comments (1)

The social-cost section is mentioned only briefly; its dependence on the equilibrium construction and on the fault-rate threshold should be clarified for completeness.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful review and the identification of points where the abstract could better convey the nature of our results. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central 'if and only if' existence claim rests on a threshold 'calculated approximately,' yet the provided text supplies no derivation steps, no error bounds on the approximation, and no verification that the numerical or truncation method used for T preserves the non-existence direction for λ slightly below T. This is load-bearing because equilibria could exist within the unquantified tolerance.

Authors: The threshold T is obtained by numerically solving the fixed-point equation that equates the best-response locations across Poisson-generated segments (detailed in Section 4). We agree the abstract omits these steps and the error analysis. In revision we will add a paragraph summarizing the truncation procedure, supplying explicit error bounds derived from the tail of the Poisson process, and verifying non-existence for λ < T − ε by exhibiting a profitable deviation that grows with the number of segments. revision_made = yes revision: yes
Referee: [Abstract] Abstract: the uniqueness and explicit construction assertions are stated without reference to the underlying best-response equations, fixed-point argument, or handling of the Poisson segment-length distribution, preventing assessment of whether the construction is independent of the approximate threshold calculation.

Authors: Uniqueness follows from a contraction-mapping argument on the best-response operator (Theorem 3), and the explicit locations are obtained by solving the resulting system of integral equations that incorporate the exponential segment-length distribution; both are independent of the numerical value of T. The abstract, being a summary, does not cite these elements. We will revise it to include a one-sentence pointer to the fixed-point argument and the segment-length handling. revision_made = partial revision: partial

standing simulated objections not resolved

The threshold T is defined only approximately via numerical solution of a transcendental equation; an exact closed-form expression is not available, so the iff statement necessarily remains approximate.

Circularity Check

0 steps flagged

No circularity detected; claims rest on model derivation without reduction to inputs by construction

full rationale

The abstract states existence of NE iff fault rate exceeds a threshold (calculated approximately), with uniqueness and explicit construction when it exists. No equations, self-citations, or derivation steps are visible in the provided text. The approximation concerns numerical evaluation of the threshold value but does not indicate any fitted parameter renamed as prediction, self-definitional loop, or load-bearing self-citation. The model assumptions (independent faults, isolated components) are stated as primitives, not derived from the claimed result. Without quoted reductions showing Eq. X equivalent to input Y by construction, the derivation is treated as self-contained. This is the expected outcome for a theoretical game-theory paper whose central claims are not shown to collapse into their own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; full model assumptions, parameter definitions, and any invented entities are not visible. The central claim rests on an unstated probabilistic fault model and isolation of components, but no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5687 in / 1125 out tokens · 18797 ms · 2026-05-24T21:03:39.040580+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

[1]

Ashtiani

M. Ashtiani. Competitive location: a state-of-art review. Int. J. Industr. Eng. Comput., 7:1–18, 2016

work page 2016
[2]

C. Avin, A. Cohen, Z. Lotker and D. Peleg. Fault-tolerant Hotelling games. In 8th Int. (EAI) Conf. on Game Theory for Networks (GameNets), 2018

work page 2018
[3]

K. E. Boulding. Economic Analysis; Vol. 1, Microeconomics. 1966

work page 1966
[4]

Chechik and D

S. Chechik and D. Peleg. Robust fault tolerant uncapacitated facility location. TCS, 543:9–23, 2014

work page 2014
[5]

Chechik and D

S. Chechik and D. Peleg. The fault-tolerant capacitated k-center problem. TCS, 566:12–25, 2015

work page 2015
[6]

Stability in Competition

C. d’Aspremont, J. J. Gabszewicz, and J.-F. Thisse. On Hotelling’s “Stability in Competition”. Econometrica: J. Econometric Soc., 1145–1150, 1979

work page 1979
[7]

De Palma, V

A. De Palma, V . Ginsburgh, and J.-F. Thisse. On existence of location equilibria in the 3-ﬁrm Hotelling problem. J. Industrial Economics, 245–252, 1987

work page 1987
[8]

B. C. Eaton and R. G. Lipsey. The principle of minimum differentiation reconsidered: Some new developments in the theory of spatial competition. Rev. Economic Studies, 42:27–49, 1975

work page 1975
[9]

H. A. Eiselt. Equilibria in competitive location models. In Found. of Loc. Analysis , 139–162. Springer, 2011

work page 2011
[10]

H. A. Eiselt, G. Laporte, and J.-F. Thisse. Competitive location models: A framework and bibliogra- phy. Transportat. Sci., 27:44–54, 1993

work page 1993
[11]

K. Eliaz. Fault tolerant implementation. Rev. Economic Studies, 69:589–610, 2002

work page 2002
[12]

Feige and M

U. Feige and M. Tennenholtz. Mechanism design with uncertain inputs: (to err is human, to forgive divine). In Proc. 43rd ACM Symp. on Theory of computing, 549–558, 2011

work page 2011
[13]

Feldmann, M

R. Feldmann, M. Mavronicolas and B. Monien. Nash equilibria for V oronoi games on transitive graphs. In Int. Workshop on Internet and Network Economics, 280–291. Springer, 2009

work page 2009
[14]

Fournier

G. Fournier. General distribution of consumers in pure Hotelling games. arXiv:1602.04851, 2016. 14

work page arXiv 2016
[15]

Fournier and M

G. Fournier and M. Scarsini. Hotelling games on networks: existence and efﬁciency of equilibria. 2016

work page 2016
[16]

Edward N. Gilbert. Random plane networks. J. Soc. for Industr. and Applied Math., 9(4):533–543, 1961

work page 1961
[17]

Gradwohl and O

R. Gradwohl and O. Reingold. Fault tolerance in large games. Games & Econ. Beha., 86:438–457, 2014

work page 2014
[18]

Hotelling

H. Hotelling. Stability in competition. Economic J., 39:41–57, 1929

work page 1929
[19]

E. Kalai. Large robust games. Econometrica, 72:1631–1665, 2004

work page 2004
[20]

Khuller, R

S. Khuller, R. Pless and Y . J. Sussmann. Fault tolerant k-center problems. TCS, 242:237–245, 2000

work page 2000
[21]

Queuing systems, volume i: Theory, 1975

Leonard Kleinrock. Queuing systems, volume i: Theory, 1975

work page 1975
[22]

A. P. Lerner and H. W. Singer. Some notes on duopoly and spatial competition. J. Polit. Econ., 45(2):145–186, 1937

work page 1937
[23]

Mavronicolas, B

M. Mavronicolas, B. Monien, V . G. Papadopoulou and F. Schoppmann. V oronoi games on cycle graphs. In Int. Symp. on Math. Foundations of Computer Sci., 503–514. Springer, 2008

work page 2008
[24]

Continuum percolation, volume 119

Ronald Meester and Rahul Roy. Continuum percolation, volume 119. Cambridge University Press, 1996

work page 1996
[25]

R. Meir, M. Tennenholtz, Y . Bachrach and P. Key. Congestion games with agent failures. In Proc. AAAI, vol. 12, 1401–1407, 2012

work page 2012
[26]

Mitzenmacher and E

M. Mitzenmacher and E. Upfal. Probability and computing: Randomized algorithms and probabilis- tic analysis. Cambridge Univ. press, 2005

work page 2005
[27]

P ´alv¨olgyi

D. P ´alv¨olgyi. Hotelling on graphs. In URL http://media. coauthors. net/konferencia/conferences/5/palvolgyi. pdf. Mimeo, 2011

work page 2011
[28]

H. J. M. Peters, M. J. W. Schr ¨oder and A. Joseph Vermeulen. Waiting in the queue on Hotelling’s main street. 2015

work page 2015
[29]

L. V . Snyder, Z. Atan, P. Peng, Y . Rong, A. J. Schmitt and B. Sinsoysal. OR/MS models for supply chain disruptions: A review. IIE Trans., 48:89–109, 2016

work page 2016
[30]

Sviridenko

M. Sviridenko. An improved approximation algorithm for the metric uncapacitated facility location problem. In Int. Conf. Integer Prog. & Combin. Opt. (IPCO), 240–257. Springer, 2002

work page 2002
[31]

Swamy and D

C. Swamy and D. B. Shmoys. Fault-tolerant facility location. ACM Tr. Algorithms (TALG), 4:51, 2008

work page 2008
[32]

Wang and Y

X. Wang and Y . Ouyang. A continuum approximation approach to competitive facility location design under facility disruption risks. Transportat. Res. Part B: Methodological, 50:90–103, 2013

work page 2013
[33]

Zhang, L

Y . Zhang, L. V . Snyder, T. K. Ralphs and Z. Xue. The competitive facility location problem under disruption risks. Transportat. Res. Part E: Logistics and Transportation Review, 93:453–473, 2016. 15 Appendix A The Poisson Process In this section we present a few well known results that are necessary for our proofs. Namely, we discuss the Poisson process...

work page 2016
[34]

The process has independent and stationary increments. That is, for any t, s > 0, the distribution of Q(t + s)− Q(s) is identical to the distribution of Q(t), and for any two disjoint intervals[t1, t2] and [t3, t4], the distribution of Q(t2)− Q(t1) is independent of the distribution of Q(t4)− Q(t3)

work page
[35]

That is, the probability of a single event in a short interval tends to λt

limt→0 Pr(Q(t) = 1)/t = λ. That is, the probability of a single event in a short interval tends to λt

work page
[36]

That is, the probability of more than one event in a short interval tends to zero

limt→0 Pr(Q(t)≥ 2) = 0 . That is, the probability of more than one event in a short interval tends to zero. Theorem A.1. Let{Q(t)| t≥ 0} be a Poisson process with parameter λ. For any t, s≥ 0 and any integer k≥ 0, Pr(Q(t + s)− Q(s) = k) = e−λt (λt)n n! . Theorem A.2. Given that Q(t) = k, the k arrival times have the same distribution as that ofn independe...

work page

[1] [1]

Ashtiani

M. Ashtiani. Competitive location: a state-of-art review. Int. J. Industr. Eng. Comput., 7:1–18, 2016

work page 2016

[2] [2]

C. Avin, A. Cohen, Z. Lotker and D. Peleg. Fault-tolerant Hotelling games. In 8th Int. (EAI) Conf. on Game Theory for Networks (GameNets), 2018

work page 2018

[3] [3]

K. E. Boulding. Economic Analysis; Vol. 1, Microeconomics. 1966

work page 1966

[4] [4]

Chechik and D

S. Chechik and D. Peleg. Robust fault tolerant uncapacitated facility location. TCS, 543:9–23, 2014

work page 2014

[5] [5]

Chechik and D

S. Chechik and D. Peleg. The fault-tolerant capacitated k-center problem. TCS, 566:12–25, 2015

work page 2015

[6] [6]

Stability in Competition

C. d’Aspremont, J. J. Gabszewicz, and J.-F. Thisse. On Hotelling’s “Stability in Competition”. Econometrica: J. Econometric Soc., 1145–1150, 1979

work page 1979

[7] [7]

De Palma, V

A. De Palma, V . Ginsburgh, and J.-F. Thisse. On existence of location equilibria in the 3-ﬁrm Hotelling problem. J. Industrial Economics, 245–252, 1987

work page 1987

[8] [8]

B. C. Eaton and R. G. Lipsey. The principle of minimum differentiation reconsidered: Some new developments in the theory of spatial competition. Rev. Economic Studies, 42:27–49, 1975

work page 1975

[9] [9]

H. A. Eiselt. Equilibria in competitive location models. In Found. of Loc. Analysis , 139–162. Springer, 2011

work page 2011

[10] [10]

H. A. Eiselt, G. Laporte, and J.-F. Thisse. Competitive location models: A framework and bibliogra- phy. Transportat. Sci., 27:44–54, 1993

work page 1993

[11] [11]

K. Eliaz. Fault tolerant implementation. Rev. Economic Studies, 69:589–610, 2002

work page 2002

[12] [12]

Feige and M

U. Feige and M. Tennenholtz. Mechanism design with uncertain inputs: (to err is human, to forgive divine). In Proc. 43rd ACM Symp. on Theory of computing, 549–558, 2011

work page 2011

[13] [13]

Feldmann, M

R. Feldmann, M. Mavronicolas and B. Monien. Nash equilibria for V oronoi games on transitive graphs. In Int. Workshop on Internet and Network Economics, 280–291. Springer, 2009

work page 2009

[14] [14]

Fournier

G. Fournier. General distribution of consumers in pure Hotelling games. arXiv:1602.04851, 2016. 14

work page arXiv 2016

[15] [15]

Fournier and M

G. Fournier and M. Scarsini. Hotelling games on networks: existence and efﬁciency of equilibria. 2016

work page 2016

[16] [16]

Edward N. Gilbert. Random plane networks. J. Soc. for Industr. and Applied Math., 9(4):533–543, 1961

work page 1961

[17] [17]

Gradwohl and O

R. Gradwohl and O. Reingold. Fault tolerance in large games. Games & Econ. Beha., 86:438–457, 2014

work page 2014

[18] [18]

Hotelling

H. Hotelling. Stability in competition. Economic J., 39:41–57, 1929

work page 1929

[19] [19]

E. Kalai. Large robust games. Econometrica, 72:1631–1665, 2004

work page 2004

[20] [20]

Khuller, R

S. Khuller, R. Pless and Y . J. Sussmann. Fault tolerant k-center problems. TCS, 242:237–245, 2000

work page 2000

[21] [21]

Queuing systems, volume i: Theory, 1975

Leonard Kleinrock. Queuing systems, volume i: Theory, 1975

work page 1975

[22] [22]

A. P. Lerner and H. W. Singer. Some notes on duopoly and spatial competition. J. Polit. Econ., 45(2):145–186, 1937

work page 1937

[23] [23]

Mavronicolas, B

M. Mavronicolas, B. Monien, V . G. Papadopoulou and F. Schoppmann. V oronoi games on cycle graphs. In Int. Symp. on Math. Foundations of Computer Sci., 503–514. Springer, 2008

work page 2008

[24] [24]

Continuum percolation, volume 119

Ronald Meester and Rahul Roy. Continuum percolation, volume 119. Cambridge University Press, 1996

work page 1996

[25] [25]

R. Meir, M. Tennenholtz, Y . Bachrach and P. Key. Congestion games with agent failures. In Proc. AAAI, vol. 12, 1401–1407, 2012

work page 2012

[26] [26]

Mitzenmacher and E

M. Mitzenmacher and E. Upfal. Probability and computing: Randomized algorithms and probabilis- tic analysis. Cambridge Univ. press, 2005

work page 2005

[27] [27]

P ´alv¨olgyi

D. P ´alv¨olgyi. Hotelling on graphs. In URL http://media. coauthors. net/konferencia/conferences/5/palvolgyi. pdf. Mimeo, 2011

work page 2011

[28] [28]

H. J. M. Peters, M. J. W. Schr ¨oder and A. Joseph Vermeulen. Waiting in the queue on Hotelling’s main street. 2015

work page 2015

[29] [29]

L. V . Snyder, Z. Atan, P. Peng, Y . Rong, A. J. Schmitt and B. Sinsoysal. OR/MS models for supply chain disruptions: A review. IIE Trans., 48:89–109, 2016

work page 2016

[30] [30]

Sviridenko

M. Sviridenko. An improved approximation algorithm for the metric uncapacitated facility location problem. In Int. Conf. Integer Prog. & Combin. Opt. (IPCO), 240–257. Springer, 2002

work page 2002

[31] [31]

Swamy and D

C. Swamy and D. B. Shmoys. Fault-tolerant facility location. ACM Tr. Algorithms (TALG), 4:51, 2008

work page 2008

[32] [32]

Wang and Y

X. Wang and Y . Ouyang. A continuum approximation approach to competitive facility location design under facility disruption risks. Transportat. Res. Part B: Methodological, 50:90–103, 2013

work page 2013

[33] [33]

Zhang, L

Y . Zhang, L. V . Snyder, T. K. Ralphs and Z. Xue. The competitive facility location problem under disruption risks. Transportat. Res. Part E: Logistics and Transportation Review, 93:453–473, 2016. 15 Appendix A The Poisson Process In this section we present a few well known results that are necessary for our proofs. Namely, we discuss the Poisson process...

work page 2016

[34] [34]

The process has independent and stationary increments. That is, for any t, s > 0, the distribution of Q(t + s)− Q(s) is identical to the distribution of Q(t), and for any two disjoint intervals[t1, t2] and [t3, t4], the distribution of Q(t2)− Q(t1) is independent of the distribution of Q(t4)− Q(t3)

work page

[35] [35]

That is, the probability of a single event in a short interval tends to λt

limt→0 Pr(Q(t) = 1)/t = λ. That is, the probability of a single event in a short interval tends to λt

work page

[36] [36]

That is, the probability of more than one event in a short interval tends to zero

limt→0 Pr(Q(t)≥ 2) = 0 . That is, the probability of more than one event in a short interval tends to zero. Theorem A.1. Let{Q(t)| t≥ 0} be a Poisson process with parameter λ. For any t, s≥ 0 and any integer k≥ 0, Pr(Q(t + s)− Q(s) = k) = e−λt (λt)n n! . Theorem A.2. Given that Q(t) = k, the k arrival times have the same distribution as that ofn independe...

work page