# Queueing theoretical analysis of first passage processes in foreign by utg65734

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```									  Modeling of internet
Queueing theoretical approaches
Jun-ichi Inoue (1) and Naoya Sazuka (2)
(1) Hokkaido University          (2) Sony Corporation

To appear in Quantitative Finance (2007) as
“Queueing theoretical analysis of foreign currency
exchange rates”

International Conference on Modeling and Simulation 07
Kolkata, India 4th December 2007
Fluctuations in intervals of events
ISI in a single BUND future Sony bank
neuron          (“Bond” in  rate
German word)
Average time ～3 [ms]           ～10 [s]          ～ 20 [min]
interval
PDF of       Gamma             Mittag-Leffler       ?
duration
duration

time

Price change, Neuronal spikes, etc..
Aim of this study
Evaluate time fluctuations in intervals between events:
duration t

w                             time
Price change
Observation time

How long does a trader wait until the next price changes ?
Time between observing the rate and the next price change

We evaluate the waiting time by queueing theoretical approach
Data: Sony Bank rate
Sony bank USD/JPY exchange rate:
・Rate for individual customers of the Sony bank (http://moneykit.net/)
・Tradable on the web 24 hours a day
・The rate depends on the market rate, not customers’ order
Sony bank rate
124
123
122
121
120
119
118
ticks
500    1000 1500 2000 2500 3000
FPP in financial markets
Sony bank rate
124    http://moneykit.net/
123                                                How do we estimate
122
121
120
P(t )
119                                                    FPT distribution ?
118
ticks
500   1000 1500 2000 2500 3000
Sony bank rate

Sony bank rate
Market rate                 ε = 0.1 [yen]
ε = 0.1 [yen]
is regarded as
a first passage
process
t        First passage time
Sony Bank rate as a FPP
The duration gets longer than that of market rate

X
2ε

market rate

sony bank rate

first passage time

t
FPT pdf of the Sony Bank rate

P (t ) =
W
mt m−1
a          ( )
exp −   tm
a   Weibull distribution

Sazuka (2005)

Weibull paper analysis

m = 0.59, a = 50.855
Lorentz curve and Gini index
Gini index is originally a quantity to measure earning inequality
・ It takes a value between 0 and 1
・ It gives 1 for perfect inequality and 0 for perfect equality
・ Japan (0.314), USA (0.357), Mexico (0.480), Denmark (0.225)        r
r
X (r ) = ∫ P(t )dt , Y (r ) = ∞∫0 tP ( t ) dt
0
Income distribution ∫0 tP (t ) dt
We regard P(t) as FPT pdf
Y = f ( X ) : Lorentz curve

For a Weibull distribution:

Y = Q ( m + 1, − log(1 − X ) ) , Q( z , x) = ∫ t z −1e − t dt
x
1
0
Gini index for a Weibull pdf
Gini index is originally a quantity to measure earning inequality
・ It takes a value between 0 and 1
・ It gives 1 for perfect inequality and 0 for perfect equality

Sazuka and Inoue (2007)
Gini index for Weibull FPT pdf

Sony bank rate
P (t ) =
W
mt m−1
a         ( )
exp −   tm
a
Empirical data evaluation
N
G=       1
∑ ( 2i − N − 1)x
G = 1 − ( 12 )
1                 N 2μ                   i
m                    i =1

0.693079
Theoretical result
Poisson arrival process                                       31,000 points from
0.694618
September 2002 to
May 2004
0.59
Queueing theoretical analysis
Sony bank USD/JPY exchange rates
Waiting time

Time
FPT

How long does the customer wait until the next price change ?
The waiting time should depend on the login time.

The renewal-reward theorem gives

( )                      ∞
, E (⋅⋅⋅) = ∫ ( ⋅⋅⋅)P(t )dt
E t2
w=            2 E(t )                    0
FPT distribution
Results for the Weibull FPT pdf
P (t ) =                 ( )
We assume :
mt m−1
W             a      exp −   tm
a

From empirical data analysis
m = 0.59, a = 50.855
Γ( m )
2
w=a
1
m
w = 42.236 [min]
Γ( m )
1
t2
wsampling =    2t
49 [min]

cf. For Poisson processes

P(t ) = λ e − λt
w= E(t) =20 [m w pling /2
in] sam
W versus E(t)
E (t ) = w gives
mΓ(2 / m) = {Γ(1/ m)}
2

w > E (t )

“Inspection paradox”                     w < E (t )
Summary
• The queueing theoretical approach provides us the average
waiting time until the next price change after the customer
checks the price
• The average waiting time obtained by the renewal reward
theorem assuming Weibull FPT distribution is good
agreement with empirical evidence
• FPT of Sony bank rate is not exponentially distributed, the
jump process is non-Markovian
• The deviation from the average waiting time or more general
formulation is possible. The result is available as the IEEE
proceedings of fundation of computer intelligence 2007
• Explanation of the gap between theory and empirical data is
explained: Sazuka, Inoue and Scalas (working paper)

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