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A discrete parking problem Results Analysis Outlook On a discrete parking problem Alois Panholzer Institute of Discrete Mathematics and Geometry Vienna University of Technology Alois.Panholzer@tuwien.ac.at International Conference on the

  1. A discrete parking problem Results Analysis Outlook A discrete parking problem: Unsuccessful cars Number of unsuccessful cars: Parking sequence a 1 , . . . , a n ∈ { 1 , . . . , m } n ⇒ k unsuccessful cars (max( n − m , 0) ≤ k ≤ n − 1) Formal description of k = k ( m ; a 1 , . . . , a n ): b i := # ℓ : a ℓ ≥ i ⇒ k = 1 ≤ i ≤ m +1 { b i + i } − m − 1 max k independent of specific order of cars arriving 6 / 37

  2. A discrete parking problem Results Analysis Outlook A discrete parking problem: Unsuccessful cars Number of unsuccessful cars: Parking sequence a 1 , . . . , a n ∈ { 1 , . . . , m } n ⇒ k unsuccessful cars (max( n − m , 0) ≤ k ≤ n − 1) Formal description of k = k ( m ; a 1 , . . . , a n ): b i := # ℓ : a ℓ ≥ i ⇒ k = 1 ≤ i ≤ m +1 { b i + i } − m − 1 max k independent of specific order of cars arriving 6 / 37

  3. A discrete parking problem Results Analysis Outlook A discrete parking problem: Unsuccessful cars Number of unsuccessful cars: Parking sequence a 1 , . . . , a n ∈ { 1 , . . . , m } n ⇒ k unsuccessful cars (max( n − m , 0) ≤ k ≤ n − 1) Formal description of k = k ( m ; a 1 , . . . , a n ): b i := # ℓ : a ℓ ≥ i ⇒ k = 1 ≤ i ≤ m +1 { b i + i } − m − 1 max k independent of specific order of cars arriving 6 / 37

  4. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ = memory addresses) n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  5. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ = memory addresses) n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  6. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  7. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  8. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  9. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  10. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  11. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  12. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  13. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  14. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  15. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  16. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Parking functions: special instance k = 0 ⇒ all cars can be parked Introduced by Konheim and Weiss [1966]: in analysis of linear probing hashing algorithm m places at a round table ( ∼ m = memory addresses) 1 2 m-1 3 n guests arriving sequentially at certain places ( ∼ = data elements) each guest goes clockwise to first empty place 7 / 37

  17. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Topic of active research in combinatorics ⇒ connections to many other objects: labelled trees, major functions, acyclic functions, Pr¨ ufer code, non-crossing partitions, hyperplane arrangements, priority queues, Tutte polynomial of graphs, linear probing hashing algorithm, invesions in trees Generalizations: multiparking functions, G -parking functions, bucket parking functions Authors working on parking functions, amongst others: M. Atkinson, D. Foata, J. Francon, I. Gessel, M. Golin, D. Knuth, G. Kreweras, C. Mallows, J. Pitman, A. Postnikov, J. Riordan, B. Sagan, M. Sch¨ utzenberger, L. Shapiro, R. Stanley, C. Yan, . . . 8 / 37

  18. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Topic of active research in combinatorics ⇒ connections to many other objects: labelled trees, major functions, acyclic functions, Pr¨ ufer code, non-crossing partitions, hyperplane arrangements, priority queues, Tutte polynomial of graphs, linear probing hashing algorithm, invesions in trees Generalizations: multiparking functions, G -parking functions, bucket parking functions Authors working on parking functions, amongst others: M. Atkinson, D. Foata, J. Francon, I. Gessel, M. Golin, D. Knuth, G. Kreweras, C. Mallows, J. Pitman, A. Postnikov, J. Riordan, B. Sagan, M. Sch¨ utzenberger, L. Shapiro, R. Stanley, C. Yan, . . . 8 / 37

  19. A discrete parking problem Results Analysis Outlook A discrete parking problem: Parking functions Topic of active research in combinatorics ⇒ connections to many other objects: labelled trees, major functions, acyclic functions, Pr¨ ufer code, non-crossing partitions, hyperplane arrangements, priority queues, Tutte polynomial of graphs, linear probing hashing algorithm, invesions in trees Generalizations: multiparking functions, G -parking functions, bucket parking functions Authors working on parking functions, amongst others: M. Atkinson, D. Foata, J. Francon, I. Gessel, M. Golin, D. Knuth, G. Kreweras, C. Mallows, J. Pitman, A. Postnikov, J. Riordan, B. Sagan, M. Sch¨ utzenberger, L. Shapiro, R. Stanley, C. Yan, . . . 8 / 37

  20. A discrete parking problem Results Analysis Outlook A discrete parking problem: Enumeration results Enumeration result for parking sequences: Konheim and Weiss [1966] g ( m , n ): number of parking functions for m parking lots and n cars g ( m , n ) = ( m − n + 1)( m + 1) n − 1 Questions for general parking sequences: “Combinatorial question”: What is the number g ( m , n , k ) of parking sequences a 1 , . . . , a n ∈ { 1 , . . . , m } n such that exactly k drivers are unsuccessful? Exact formulæ for g ( m , n , k ) ? 9 / 37

  21. A discrete parking problem Results Analysis Outlook A discrete parking problem: Enumeration results Enumeration result for parking sequences: Konheim and Weiss [1966] g ( m , n ): number of parking functions for m parking lots and n cars g ( m , n ) = ( m − n + 1)( m + 1) n − 1 Questions for general parking sequences: “Combinatorial question”: What is the number g ( m , n , k ) of parking sequences a 1 , . . . , a n ∈ { 1 , . . . , m } n such that exactly k drivers are unsuccessful? Exact formulæ for g ( m , n , k ) ? 9 / 37

  22. A discrete parking problem Results Analysis Outlook A discrete parking problem: Enumeration results “Probabilistic question”: What is the probability that for a randomly chosen parking sequence a 1 , . . . , a n ∈ { 1 , . . . , m } n exactly k drivers are unsuccessful ? r.v. X m , n : counts number of unsuccessful cars for a randomly chosen parking sequence Probability distribution of X m , n ? Limiting distribution results (depending on growth of m , n ) ? 10 / 37

  23. A discrete parking problem Results Analysis Outlook A discrete parking problem: Enumeration results Known results for X m , n : Gonnet and Munro [1984]: X m , n studied in analysis of algorithm “linear probing sort” Exact and asymptotic results for expectation E ( X m , n ) : n n ℓ E ( X m , n ) = 1 � n ≤ m m ℓ , 2 ℓ =2 � π m 8 + 2 m − 1 � 2 � E ( X m , m ) = 3 + O Analysis uses “Poisson model” Transfer of results to “exact filling model” 11 / 37

  24. A discrete parking problem Results Analysis Outlook A discrete parking problem: Enumeration results Known results for X m , n : Gonnet and Munro [1984]: X m , n studied in analysis of algorithm “linear probing sort” Exact and asymptotic results for expectation E ( X m , n ) : n n ℓ E ( X m , n ) = 1 � n ≤ m m ℓ , 2 ℓ =2 � π m 8 + 2 m − 1 � 2 � E ( X m , m ) = 3 + O Analysis uses “Poisson model” Transfer of results to “exact filling model” 11 / 37

  25. A discrete parking problem Results Analysis Outlook Results 12 / 37

  26. A discrete parking problem Results Analysis Outlook Results: Exact enumeration formulæ Exact enumeration results: Cameron, Johannsen, Prellberg and Schweitzer [2007]; Panholzer [2007] Number g ( m , n , k ) of parking sequences for m parking lots and n drivers such that exactly k drivers are unsuccessful ( n ≤ m + k ): n − k � n � � ( m − n + k + ℓ ) ℓ − 1 ( n − k − ℓ ) n − ℓ g ( m , n , k ) = ( m − n + k ) ℓ ℓ =0 n − k − 1 � n � � ( m − n + k + 1 + ℓ ) ℓ − 1 ( n − k − 1 − ℓ ) n − ℓ − ( m − n + k + 1) ℓ ℓ =0 13 / 37

  27. A discrete parking problem Results Analysis Outlook Results: Exact enumeration formulaæ Alternative expression: useful for k small g ( m , n , k ) = ( m − n + k + 1)( m + k + 1) n − 1 k − 1 ✥ ✦ n ❳ ( − 1) ℓ ( m + k − ℓ ) n − ℓ − 2 ( k − ℓ ) ℓ +1 − ( m − n + k + 1) ℓ + 1 ℓ =0 k − 1 ✥ ✦ n ❳ ( − 1) ℓ ( m + k − ℓ ) n − ℓ − 1 ( k − ℓ ) ℓ − ( m − n + k ) ℓ ℓ =0 14 / 37

  28. A discrete parking problem Results Analysis Outlook Results: Exact enumeration formulaæ Alternative expression: useful for k small g ( m , n , k ) = ( m − n + k + 1)( m + k + 1) n − 1 k − 1 ✥ ✦ n ❳ ( − 1) ℓ ( m + k − ℓ ) n − ℓ − 2 ( k − ℓ ) ℓ +1 − ( m − n + k + 1) ℓ + 1 ℓ =0 k − 1 ✥ ✦ n ❳ ( − 1) ℓ ( m + k − ℓ ) n − ℓ − 1 ( k − ℓ ) ℓ − ( m − n + k ) ℓ ℓ =0 Examples for small numbers k of unsuccessful cars: g ( m , n , 0) = ( m − n + 1)( m + 1) n − 1 g ( m , n , 1) = ( m − n + 2)( m + 2) n − 1 + ( n 2 − n − m 2 − 2 m − 1)( m + 1) n − 2 g ( m , n , 2) = ( m − n + 3)( m + 3) n − 1 + (2 n 2 − mn − m 2 − 4 n − 4 m − 4)( m + 2) n − 2 + 1 2 n ( − n 2 − mn + 2 m 2 + 2 n − 5 m + 1)( m + 1) n − 3 14 / 37

  29. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Exact probability distribution of X m , n : P { X m , n = k } = g ( m , n , k ) m n Limiting distribution results for X m , n : Panholzer [2007] Depending on growth of m , n ⇒ nine different phases m (parking lots) ≥ n (cars) m (parking lots) < n (cars) ∆ := n − m ≪ √ n n ≪ m ∆ ∼ ρ √ n , ρ > 0 n ∼ ρ m , 0 < ρ < 1 √ m ≪ ∆ := m − n ≪ m √ n ≪ ∆ ≪ n ∆ ∼ ρ √ m , ρ > 0 n ∼ ρ m , ρ > 1 ∆ ≪ √ m m ≪ n 15 / 37

  30. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Exact probability distribution of X m , n : P { X m , n = k } = g ( m , n , k ) m n Limiting distribution results for X m , n : Panholzer [2007] Depending on growth of m , n ⇒ nine different phases m (parking lots) ≥ n (cars) m (parking lots) < n (cars) ∆ := n − m ≪ √ n n ≪ m ∆ ∼ ρ √ n , ρ > 0 n ∼ ρ m , 0 < ρ < 1 √ m ≪ ∆ := m − n ≪ m √ n ≪ ∆ ≪ n ∆ ∼ ρ √ m , ρ > 0 n ∼ ρ m , ρ > 1 ∆ ≪ √ m m ≪ n 15 / 37

  31. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): ( d ) n ≪ m : − − → X X m , n P { X = 0 } = 1 degenerate limit law 16 / 37

  32. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): ( d ) n ∼ ρ m , 0 < ρ < 1 : − − → X ρ X m , n k ( − 1) k − ℓ ( ℓ + 1) k − ℓ � ( k − ℓ )! ρ k − ℓ e ( ℓ +1) ρ P { X ρ ≤ k } = (1 − ρ ) ℓ =0 discrete limit law 16 / 37

  33. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): √ m ≪ ∆ := m − n ≪ m : ( d ) ( d ) ∆ − − → X = EXP(2) m X m , n survival function: P { X ≥ x } = e − 2 x , x ≥ 0 asymptotically exponential distributed 16 / 37

  34. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): ∆ := m − n ∼ ρ √ m , ρ > 0 : ( d ) ( d ) 1 − − → X rho = LINEXP(2 , ρ ) √ m X m , n survival function: P { X ρ ≥ x } = e − 2 x ( x + ρ ) , x ≥ 0 asymptotically linear-exponential distributed 16 / 37

  35. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): 0 ≤ ∆ := m − n ≪ √ m : ( d ) ( d ) 1 − − → X = RAYLEIGH(2) √ m X m , n survival function: P { X ≥ x } = e − 2 x 2 , x ≥ 0 asymptotically Rayleigh distributed 16 / 37

  36. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): 0 ≤ ∆ := n − m ≪ √ n : ( d ) ( d ) X m , n + m − n − − → X = RAYLEIGH(2) √ n survival function: P { X ≥ x } = e − 2 x 2 , x ≥ 0 asymptotically Rayleigh distributed 16 / 37

  37. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): ∆ := n − m ∼ ρ √ n , ρ > 0 : ( d ) ( d ) X m , n + m − n − − → X ρ = LINEXP(2 , ρ ) √ n survival function: P { X ≥ x } = e − 2 x ( x + ρ ) , x ≥ 0 asymptotically linear-exponential distributed 16 / 37

  38. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): √ n ≪ ∆ := n − m ≪ n : ( d ) ( d ) ∆ n ( X m , n + m − n ) − − → X = EXP(2) survival function: P { X ≥ x } = e − 2 x , x ≥ 0 asymptotically exponential distributed 16 / 37

  39. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): ( d ) n ∼ ρ m , ρ > 1 : X m , n + m − n − − → X ρ ∞ ( ℓ + k ) ℓ − 1 P { X ρ ≥ k } = ke − ρ k � ( ρ e − ρ ) ℓ , k ≥ 1 ℓ ! ℓ =0 discrete limit law 16 / 37

  40. A discrete parking problem Results Analysis Outlook Results: Limiting distributions Weak convergence of X m , n ( m parking lots, n cars): ( d ) n ≪ m : X m , n + m − n − − → X P { X = 0 } = 1 degenerate limit law 16 / 37

  41. A discrete parking problem Results Analysis Outlook Analysis 17 / 37

  42. A discrete parking problem Results Analysis Outlook Analysis: Outline Outline of proof Exact enumeration results: Recursive description of parameter Generating functions approach Limiting distribution results: Asymptotic evaluation of distribution function Asymptotic evaluation of positive integer moments (Method of moments) 18 / 37

  43. A discrete parking problem Results Analysis Outlook Analysis: Outline Outline of proof Exact enumeration results: Recursive description of parameter Generating functions approach Limiting distribution results: Asymptotic evaluation of distribution function Asymptotic evaluation of positive integer moments (Method of moments) 18 / 37

  44. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Exact enumeration results Quantity of interest: g ( m , n , k ): number of sequences ∈ { 1 , . . . , m } n , such that exactly k cars are unsuccessful Recursive description of g ( m , n , k ) : Auxiliary quantities: f ( n ) = ( n + 1) n − 1 : number of parking functions ∈ { 1 , . . . , n } n s ( m , k ): number of sequences ∈ { 1 , . . . , m } m + k , such that all parking lots are occupied ⇔ exactly k cars are unsuccessful 19 / 37

  45. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Exact enumeration results Quantity of interest: g ( m , n , k ): number of sequences ∈ { 1 , . . . , m } n , such that exactly k cars are unsuccessful Recursive description of g ( m , n , k ) : Auxiliary quantities: f ( n ) = ( n + 1) n − 1 : number of parking functions ∈ { 1 , . . . , n } n s ( m , k ): number of sequences ∈ { 1 , . . . , m } m + k , such that all parking lots are occupied ⇔ exactly k cars are unsuccessful 19 / 37

  46. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Case n < m + k : decomposition after first empty lot j : j m 1 � n m � � g ( m , n , k ) = f ( j − 1) g ( m − j , n − j + 1 , k ) j − 1 j =1 20 / 37

  47. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Case n < m + k : decomposition after first empty lot j : j m 1 � n m � � g ( m , n , k ) = f ( j − 1) g ( m − j , n − j + 1 , k ) j − 1 j =1 Case n = m + k : all parking lots are occupied: m 1 g ( m , n , k ) = s ( m , k ) 20 / 37

  48. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Introducing suitable generating functions: g ( m , n , k ) z n � � � n ! u m v k G ( z , u , v ) := m ≥ 0 n ≥ 0 k ≥ 0 u m v k � � S ( u , v ) := s ( m , k ) ( m + k )! m ≥ 0 k ≥ 0 n n − 1 z n z n � � T ( z ) := n ! = f ( n − 1) ( n − 1)! n ≥ 1 n ≥ 1 T ( z ): satisfies functional equation T ( z ) = ze T ( z ) Equation for generating functions: G ( z , u , v ) = S ( zu , zv ) 1 − T ( zu ) z 21 / 37

  49. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Introducing suitable generating functions: g ( m , n , k ) z n � � � n ! u m v k G ( z , u , v ) := m ≥ 0 n ≥ 0 k ≥ 0 u m v k � � S ( u , v ) := s ( m , k ) ( m + k )! m ≥ 0 k ≥ 0 n n − 1 z n z n � � T ( z ) := n ! = f ( n − 1) ( n − 1)! n ≥ 1 n ≥ 1 T ( z ): satisfies functional equation T ( z ) = ze T ( z ) Equation for generating functions: G ( z , u , v ) = S ( zu , zv ) 1 − T ( zu ) z 21 / 37

  50. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Evaluating at v = 1 : m n z n 1 n ! u m = G ( z , u , 1) = S ( zu , z ) � � 1 − ue z = 1 − T ( zu ) m ≥ 0 n ≥ 0 z ⇒ S ( zu , z ) = 1 − T ( zu ) z 1 − ue z Substituting z ← zv , u ← u v : v , zv ) = 1 − T ( zu ) S ( zu , zv ) = S ( zv · u zv 1 − u v e zv Exact expression for generating function: 1 − T ( zu ) zv G ( z , u , v ) = 1 − T ( zu ) � � � 1 − u v e zv � · z 22 / 37

  51. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Evaluating at v = 1 : m n z n 1 n ! u m = G ( z , u , 1) = S ( zu , z ) � � 1 − ue z = 1 − T ( zu ) m ≥ 0 n ≥ 0 z ⇒ S ( zu , z ) = 1 − T ( zu ) z 1 − ue z Substituting z ← zv , u ← u v : v , zv ) = 1 − T ( zu ) S ( zu , zv ) = S ( zv · u zv 1 − u v e zv Exact expression for generating function: 1 − T ( zu ) zv G ( z , u , v ) = 1 − T ( zu ) � � � 1 − u v e zv � · z 22 / 37

  52. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Evaluating at v = 1 : m n z n 1 n ! u m = G ( z , u , 1) = S ( zu , z ) � � 1 − ue z = 1 − T ( zu ) m ≥ 0 n ≥ 0 z ⇒ S ( zu , z ) = 1 − T ( zu ) z 1 − ue z Substituting z ← zv , u ← u v : v , zv ) = 1 − T ( zu ) S ( zu , zv ) = S ( zv · u zv 1 − u v e zv Exact expression for generating function: 1 − T ( zu ) zv G ( z , u , v ) = 1 − T ( zu ) � � � 1 − u v e zv � · z 22 / 37

  53. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Extracting coefficients ⇒ exact formula: n − k � n � � ( m − n + k + ℓ ) ℓ − 1 ( n − k − ℓ ) n − ℓ g ( m , n , k ) = ( m − n + k ) ℓ ℓ =0 n − k − 1 � n � � ( m − n + k + 1 + ℓ ) ℓ − 1 ( n − k − 1 − ℓ ) n − ℓ − ( m − n + k + 1) ℓ ℓ =0 Exact distribution of X m , n : P { X m , n = k } = g ( m , n , k ) m n 23 / 37

  54. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Extracting coefficients ⇒ exact formula: n − k � n � � ( m − n + k + ℓ ) ℓ − 1 ( n − k − ℓ ) n − ℓ g ( m , n , k ) = ( m − n + k ) ℓ ℓ =0 n − k − 1 � n � � ( m − n + k + 1 + ℓ ) ℓ − 1 ( n − k − 1 − ℓ ) n − ℓ − ( m − n + k + 1) ℓ ℓ =0 Exact distribution of X m , n : P { X m , n = k } = g ( m , n , k ) m n 23 / 37

  55. A discrete parking problem Results Analysis Outlook Analysis: Exact enumeration results Abel’s generalization of the binomial theorem: n ( x + y ) n = � x ( x − ℓ z ) ℓ − 1 ( y + ℓ z ) n − ℓ ℓ =0 ⇒ alternative expression for g ( m , n , k ) : g ( m , n , k ) = ( m − n + k + 1)( m + k + 1) n − 1 k − 1 ✥ ✦ n ❳ ( − 1) ℓ ( m + k − ℓ ) n − ℓ − 2 ( k − ℓ ) ℓ +1 − ( m − n + k + 1) ℓ + 1 ℓ =0 k − 1 ✥ ✦ n ❳ ( − 1) ℓ ( m + k − ℓ ) n − ℓ − 1 ( k − ℓ ) ℓ − ( m − n + k ) ℓ ℓ =0 24 / 37

  56. A discrete parking problem Results Analysis Outlook Analysis: Limiting distribution results (I) Limiting distribution results for X m , n (I) Special instance: m (parking lots) = n (cars) complex-analytic techniques Generating function of diagonal: m m P { X m , m = k } u m � � m ! v k F ( u , v ) = m ≥ 0 k ≥ 0 Computed via contour integral: � G ( t , u t − T ( u ) � � t , v ) dt 1 1 � v F ( u , v ) = dt = � � � t − u v e tv � 2 π i t 2 π i t − T ( u ) · 25 / 37

  57. A discrete parking problem Results Analysis Outlook Analysis: Limiting distribution results (I) Limiting distribution results for X m , n (I) Special instance: m (parking lots) = n (cars) complex-analytic techniques Generating function of diagonal: m m P { X m , m = k } u m � � m ! v k F ( u , v ) = m ≥ 0 k ≥ 0 Computed via contour integral: � G ( t , u t − T ( u ) � � t , v ) dt 1 1 � v F ( u , v ) = dt = � � � t − u v e tv � 2 π i t 2 π i t − T ( u ) · 25 / 37

  58. A discrete parking problem Results Analysis Outlook Analysis: Limiting distribution results (I) Limiting distribution results for X m , n (I) Special instance: m (parking lots) = n (cars) complex-analytic techniques Generating function of diagonal: m m P { X m , m = k } u m � � m ! v k F ( u , v ) = m ≥ 0 k ≥ 0 Computed via contour integral: � G ( t , u t − T ( u ) � � t , v ) dt 1 1 � v F ( u , v ) = dt = � � � t − u v e tv � 2 π i t 2 π i t − T ( u ) · 25 / 37

  59. A discrete parking problem Results Analysis Outlook Analysis: Limiting distribution results (I) Explicit formula: simple pole at t = T ( u ) computing residue ( v − 1) T ( u ) F ( u , v ) = vT ( u ) − ue T ( u ) v Method of moments: m m [ u m ] ∂ r � m , m ) = m ! E ( X r � ∂ v r F ( u , v ) � � v =1 Studying derivatives of F ( u , v ) evaluated at v = 1 : local expansion around dominant singularity u = 1 e Singularity analysis, Flajolet and Odlyzko [1990] 26 / 37

  60. A discrete parking problem Results Analysis Outlook Analysis: Limiting distribution results (I) Explicit formula: simple pole at t = T ( u ) computing residue ( v − 1) T ( u ) F ( u , v ) = vT ( u ) − ue T ( u ) v Method of moments: m m [ u m ] ∂ r � m , m ) = m ! E ( X r � ∂ v r F ( u , v ) � � v =1 Studying derivatives of F ( u , v ) evaluated at v = 1 : local expansion around dominant singularity u = 1 e Singularity analysis, Flajolet and Odlyzko [1990] 26 / 37

  61. A discrete parking problem Results Analysis Outlook Analysis: Limiting distribution results (I) Explicit formula: simple pole at t = T ( u ) computing residue ( v − 1) T ( u ) F ( u , v ) = vT ( u ) − ue T ( u ) v Method of moments: m m [ u m ] ∂ r � m , m ) = m ! E ( X r � ∂ v r F ( u , v ) � � v =1 Studying derivatives of F ( u , v ) evaluated at v = 1 : local expansion around dominant singularity u = 1 e Singularity analysis, Flajolet and Odlyzko [1990] 26 / 37

  62. A discrete parking problem Results Analysis Outlook Analysis: Limiting distribution results (I) r -th moments converge to moments of Rayleigh r.v.: �� X m , m � r � r � → 2 − r 2 Γ � √ m 2 + 1 E Theorem of Fr´ echet and Shohat: X m , m ( d ) √ m − − → RAYLEIGH(2) 27 / 37

  63. A discrete parking problem Results Analysis Outlook Analysis: Limiting distribution results (I) r -th moments converge to moments of Rayleigh r.v.: �� X m , m � r � r � → 2 − r 2 Γ � √ m 2 + 1 E Theorem of Fr´ echet and Shohat: X m , m ( d ) √ m − − → RAYLEIGH(2) 27 / 37

  64. A discrete parking problem Results Analysis Outlook Analysis: Limiting distribution results (II) Limiting distribution results for X m , n (II) Instance: m (parking lots) > n (cars) Extension of previous approach Generating function for ∆ := m − n : u m v k � � m m − ∆ P { X m , m − ∆ = k } F ∆ ( u , v ) = ( m − ∆)! m ≥ ∆ k ≥ 0 Computed via contour integral: � G ( t , u t − T ( u ) t ∆ dt t , v ) t ∆ � � 1 1 � v F ∆ ( u , v ) = dt = � � � t − u v e tv � 2 π i t 2 π i t − T ( u ) · 28 / 37

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