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Mathematical modelof train delaysAssumptions and notations The accepted notations:nis the total number of trains,s0is the minimum safe distance between trains,v0is the average speed of trains,X j  s0is the distance from train j to train  j  1 , whereX j , 2  j  n, are random variables,s01T j  X j , t0  , T  m  is the departure time of the train m, i.e.v0v0T mm  T j   m  1 t0 , m  2, 3, ... , n.1j 22The first scenarioScheduled departure times of trains 1, 2 and 3 from station AFig. 1 is the random duration of unscheduled stop of train 1, k is the time interval between the arrivals of trainswith numbers  k  1 and k at the destination station B.It is required to determine distribution function Wk  t     k  t of the random variable  k , k  2, 3, ...

, n.3The second scenarioLet the train 1 was delayed at the station A at time t = 0 for a random time If   T2 ,then trains 2, 3, etc. departat scheduled points in timeT  2 , T 3 , etc.If   T2 ,then the train 2 will be delayed and start 2tT.moving at the moment0Trains 3, … , n depart by the same rule.Now  k is the time interval between the departures of trains (k – 1) and k4An example to thesecond scenario Letn  5,Tk  2, k  2,5,t0  1,T    3  k  1 ,kk  1,5.The timeintervals_ kfor some_Fig. 25Main results (Theorem 1)Theorem 11. If   T2 , then  2  T2  t0   ,  k  Tk  t0 , 3  k  n.2. Let k be a fixed integer , 2  k  n.kIf    T j , then  2  ...   k  t0 .j 2 k 13. If  T j     T j , then  k 1  I  k  1  n    T j  t0    ,j 2j 2 j 2 m  I  m  n Tm  t0  , m  k  2, ...

, n,kk 11 ifwhere I  x  A   0 ifx  A,x  A.6Main results (Theorem 2)Theorem 2Let n  2. The following formulas hold :W2  t   I  t  t0     t  t0  T2  ,2 k 1kk 1 Wk  t   I  t  t0    Tk  t  t0 ,    T j      t  t0   T j ,    T j   ,  3j 2j 2j 2  3  k  n,1 ifwhere I  x  A   0 ifx  A,x  A.7Main results (Corollary 1)Corollary 1Let all T j , 2  j  n, be arbitrary constants,g  x  is the density function of a random variable  .Then, the following formula holds :Wk  t   I  t0  t  Tk  t0   kg  x  dx  I  t  Tk  t0  ,Tttj 0j 22  k  n,in particular , W2  t   I  t  t0  T2 t  t0g  x  dx.8Main results (Corollary 2)Corollary 2Let all T j , 2  j  n, be constants, which are equal to the same number T ,g  x  is the density function of a random variable  .Then, the following formula holds :Wk  t   I  t0  t  T  t0   k 1T t  t0g  x  dx  I  t  T  t0  , 42  k  n,in particular , W2  t   I  t  t0  T  t  t0g  x  dx. 59An example within theframework of Corollary 2Let all T j , 2  j  n, be constants, which are equal to T  0.

Ifg  x   I  x  0  e x ,with   0, then for k  2Wk  t   I  0  t  t0  T  e    k 1T t  t0  I  t  t0  T  .6 Moreover, k  t0  T D k 12e1e   k  2 T   k  2 T1  e  , T 2 1  e  T   e    k  2T 1  e  T 2  2Te  T  .7 8 10The distribution functions of  2 and 3Fig. 3Wk  t   I  t  t0  T k Parameters:1  0.26,mint0  4 min,T  7 min .tE 2  7.77702,D 2  5.68009,D 2  2.38329,E 3  10.4778,D 3  2.33068,D 3  1.52666.11Main results (Corollary 3)Corollary 3Let Tk , 2  k  n, be independent identically distributed random variableswith the density function   x  .Let  has a density function g  x  and be independent of Tk .

Then,W2  t   I  t  t0  Wk  t   I  t  t0 t  t0z  t  t0g  x  dx   z  dz ,9   z  dz    t  t0gxdxzdzz u t t0 k  2  u  du,10 where   k  2 denotes  k  2  - fold convolution of  .12An example within theframework of Corollary 3Let all Tk , 2  k  n, have the normal distribution with the mean T andthe variance  2 , which is sufficiently small, at least 4   2  T . Ifg  x   I  x  0  e x1,   x e 2 x T 222   x, T , x2 ,   x     t , 0,1 dt,a   k  2  T , b 2   k  2   2 ,   a  b 2 , then for k  2  t  t  T0Wk  t   I  t  t0        2   t  t0     exp   k  1  T 2   t  t0  T   2      0  t  t0  T   2  u  2  1          1       u,  , b  du    b 0  t  t0  T  u  t  t0  T  2  u,a,bdu11,   2 t  t T T  t  t0   t  t0  T   2   20in particular, W2  t   I  t  t0   e1     .  13The distribution functions of  2 and 3 ,when   t  is the normal densityFig.

4Parameters:  0.26,t0  4,T  7,  0.5.t14The distribution functions of  2with different variancesParameters:  0.26,t0  4,T  7,  0.5,  5.tFig. 515The behavior of W2  t ,   , E 2 and D 2as   0Table 1Table 216Main results(Lemma 1 and its corollaries)Lemma 1For each fixed integer m, 1  m  n  1, we havem 1  N  m       T j  ,j 2where N is the number of secondary delays.12Corollary 1’Corollary 2’If T j  T is a constant value, thenIf T2 , ... , Tn are independent identically  N  m    g  x  dx.mTdistributed random variables withthe density function  , then  N  m  ug  x  dx  m  u  du.17The condition for ensuringminimum number of delaysIt is needed to define the minimum interval between departures of trains Tdep ,wherein the number of secondary stops mknock does not exceed mwith probability   1  p,t0 s0 Tdep  Т  t0 .v0If T j  const and g  t   I  t  t0   e  t , then an additional departureinterval T must satisfy the following condition:11ln ,13m pwhere p is maximum probability that at least m of secondary delays occur.T18The condition for ensuringminimum number of delays11T  m, p,   lnm pParameters:  0.26,p  0.1,p  0.05.mknockTTdepm=24.58.5m=1913Fig.

6m19.

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