Hindawi Publishing Corporation
Journal of Applied Mathematics and Stochastic Analysis
Volume 2009, Article ID 308025, 14 pages
doi:10.1155/2009/308025
Research Article
Interloss Time in M/M/1/1 Loss System
Pierpaolo Ferrante
Occupational Medicine Department (DML), National Institute for Occupational Safety and Prevention
(ISPESL), Via Alessandria 220/E, 00198 Rome, Italy
Correspondence should be addressed to Pierpaolo Ferrante, pierpaolo.ferrante@ispesl.it
Received 29 March 2009; Accepted 24 May 2009
Recommended by Ho Lee
We consider the interloss times in the M/M/1/1 Erlang Loss System. Here we present the explicit
form of the probability density function of the time spent between two consecutive losses in the
M/M/1/1 model. This density function solves a Cauchy problem for the second-order differential
equations, which was used to evaluate the corresponding laplace transform. Finally the connection
between the Erlang’s loss rate and the evaluated probability density function is showed.
Copyright q 2009 Pierpaolo Ferrante. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
In this paper, we treat the random variable Ti representing the time spent between the ith
and the i − 1th lost unit or ith interloss time, in the M/M/1/1 loss system.
The M/M/1/1 model is characterized by the Markov property of entering and exiting
processes, by one service channel and by the system capacity to accommodate one customer
at a time for an overview see Medhi 1, page. 77.
Our work has been inspired by the location problem of emergency vehicles
ambulances. Each vehicle can be regarded as an M/M/1/1 or M/Ek /1/1 system,
because its clients cannot wait in the queue.
In Emergency Medical Systems EMSs, the nearest ambulance to the accident place is
called “district unit”, and it assures the best performance to the system. If an emergency
call arrives at EMS while its “district unit” is busy, the nearest ambulance among those
available is dispatched see Larson 2 or Larson 3. The length of interloss times affects
the performance of the system and provides an informative support on efficiency of EMS.
For the exponential M/M/1/1 loss model it was conducted, in Ferrante 4, a detailed
description of the process of losses
{Lt}t>0 ,
1.1
2
Journal of Applied Mathematics and Stochastic Analysis
k
where Lt represent the random number of losses in the time interval 0, t. Let Pm t be the
conditional probability to lose m clients in 0, t with k customers in the system at time t 0:
k
Pm t P {Lt m | k},
k 0, 1,
1.2
the main results found in Ferrante 4 are the explicit values of the conditional probabilities
of no losses in 0, t:
⎡
⎢
1
P0 t e−t/22λμ ⎣cosh
⎡
⎢
0
P0 t e−t/22λμ ⎣cosh
t μ 4λ μ
2
t μ 4λ μ
2
⎤
μ
4λ
μ
t
μ
⎥
⎦,
sinh
2
μ 4λ μ
⎤
t
μ
4λ
μ
2λ μ
⎥
⎦,
sinh
2
μ 4λ μ
1.3
and the iterative procedure to determine the distribution of the total number of losses in
0, t.All is obtained by solving the inhomogeneous differential equations
d k
d k
k
k
k
Pm t − 2λ μ
Pm t − λ2 Pm t λ Pm−1 t λ2 Pm−1 ,
dt
dt
1.4
k
with P−1 t 0 and the initial conditions depending on the k 0, 1 customers in the system
k
at t 0. Furthermore, the generating probability functions Gk s, t of Pm t were evaluated
and their explicit values are the following:
G1 s, t e−t/22λμ−λs
⎡
2
t
λs − μ 4λμ
⎢
× ⎣cosh
2
μ λs
sinh
2
λs − μ 4λμ
t
λs − μ
2
2
⎤
4λμ ⎥
⎦,
G0 s, t e−t/22λμ−λs
⎤
⎡
2
2
t
λs
−
μ
4λμ
λs
−
μ
4λμ
t
2λ μ − λs
⎥
⎢
sinh
× ⎣cosh
⎦.
2
2
2
λs − μ 4λμ
1.5
The aim of this work is to identify the type of the process of interloss times
Ti i∈Z
1.6
Journal of Applied Mathematics and Stochastic Analysis
3
for the M/M/1/1 loss model and to find the differential equation which governs it, in order
to determine the probability density functions
fTi t d P Ti < t ,
dt
1.7
with i 1, 2, . . . and the related properties.
Our results show unexpected connections among very different branches of probability such as random motion on hyperbolic space and queueing systems. In effect, the
probabilities appearing below have a structure quite similar to the Hyperbolic distances of
moving particles envisaged in Cammarota and Orsingher 5.
In Section 2, we establish that {Ti }i∈Z is a renewal process for the Mλ/Mμ/1/1
loss system and that the density functions 1.7 solve the second-order linear homogeneous
differential equations
d
d2
fT t − λ2 fTi t.
fTi t − 2λ μ
2
dt i
dt
1.8
Let νt be the number of customers in the system at the moment t, and let tli be the
moment of the ith loss with tl0 0. The initial conditions for 1.8 depend on νtli−1 , and the
renewal process {Ti }i∈Z has the following property:
fT1 t fTi t,
if ν0 1,
fTi t,
fT1 t /
if ν0 0
1.9
for i > 1.
In Section 2, we also present the derivation of 1.8 and its solution conditionally by
ν0.
Let fT1 t; i be the conditional density function of the 1th interloss time with i 0,1
customers in the system at time t 0:
fT1 t; i d P T1 < t | ν0 i .
dt
1.10
For the Mλ/Mμ/1/1 model, the explicit values obtained for 1.10 are the
following:
⎡
⎢
fT1 t; 1 λe−t/22λμ ⎣cosh
2 −t/22λμ
2λ e
fT1 t; 0 sinh
μ 4λ μ
t μ 4λ μ
2
t μ 4λ μ
2
⎤
t
μ
4λ
μ
μ
⎥
− ⎦,
sinh
2
μ 4λ μ
.
1.11
4
Journal of Applied Mathematics and Stochastic Analysis
In Section 3, we compute the laplace transforms of 1.10:
FT∗1 s; i ∞
0
e−st fT1 t; idt
1.12
for i 0, 1, using 1.8.
The explicit values obtained for 1.12 are the following:
FT∗1 s; 1 FT∗1 s; 0
λλ s
λ s2 sμ
λ2
λ s2 sμ
,
1.13
.
i
Finally, let Θ1 be the conditional means of the 1th interloss time
i
Θ1 E T1 | ν0 i ,
1.14
with i 0,1; it has been checked that their values are
1
Θ1 1
,
r
0
Θ1 1 1
,
r λ
1.15
where r is the Erlang loss rate, and λ−1 is the interarrival mean time.
2. First Interloss Time
In the Mλ/Mμ/1/1 model, let νt be the number of customers in the system at the
moment t, let τk be the kth interarrival time, let tk be the moment when the kth client enters
the system, let Sk be the service time of the kth served customer, let Ti be the ith interloss
time. Furthermore, let li t be the arrival order of ith loss happened in t, starting from the
i − 1th loss, and let tli be the moment when the ith loss happen, with tl0 0 and tli t t.
If we consider that the system is busy at time t 0, the event “The 1th interloss time ist”
is represented by Figure 1.
The random variable T1 can be expressed as follows:
T1 l
1 t
τk ,
2.1
k1
where l1 t represents the arrival order of the 1th loss happened in tl1 t t, starting from zero.
1
Now, let Pn t be the conditional probability that the arrival order of 1th loss
happened in t with ν0 1 is equal to n :
1
Pn t P {l1 t n | ν0 1},
2.2
Journal of Applied Mathematics and Stochastic Analysis
τ1
0 S0
τ2
τ3
t1 S1
5
τl1 t
t3 · · · tl1 t−1
t2 S2
Sl1 t−1 tl1 t t
Figure 1: The 1th interloss time is t with P {ν0 1} 1.
and it can be expressed as follows
1
Pn t P S0 < τ1 , . . . , Sn−2 < τn−1 , Sn−1 > t − tn−1
2.3
and can be computed conditionally by the n − 1 moments when the served customers have
arrived at the system
1
Pn t n − 1!
tn−1
t
dt1 · · ·
t
0
dtn−1 e−μt−tn−1 tn−2
n−1 1 − e−μti −ti−1 2.4
i1
n − 1! 0
Fn−1,1 t,
tn−1
where
0
Fn−1,1 t t
dt1 · · ·
t
0
dtn−1 e−μt−tn−1 n−1 tn−2
1 − e−μti −ti−1 .
2.5
i1
Lemma 2.1. The functions
s
Fn,1 t t
dt1 · · ·
t
s
dtn e−μt−tn tn−1
n 1 − e−μti −ti−1 2.6
i1
with t0 s do not depend on t but on the time interval s, t:
s
0
Fn,1 t Fn,1 t − s.
2.7
Proof. We proceed by showing that 2.7 is true for n 1:
s
F1,1 t t
dt1 e−μt−t1 1 − e−μt1 −s
s
1 − e−μt−s
− t − se−μt−s
μ
0
F1,1 t − s.
2.8
6
Journal of Applied Mathematics and Stochastic Analysis
Then, we suppose that it is true for n − 1, and we obtain that
s
Fn,1 t t
s
t
s
t1 dt1 1 − e−μt1 −s Fn−1,1
t
2.9
s
0
dt1 G1 t1 Fn−1,1 t
− t1 ,
where
s
G1 t1 t1
dxμe−μt1 −x .
2.10
s
Finally, by the Markov property of the exponential distribution, the 2.7 appears
s
Fn,1 t t−s
0
0
0
dt1 G1 t1 Fn−1,1 t − s − t1 .
2.11
The conditional density function
fT1 t; 1 d P T1 < t | ν0 1
dt
2.12
can be evaluated as mean of convolution of l1 t exponential probability density functions,
and thus we have that
fT1 t; 1 ∞
1
fnk1 τk tPn t e−λt
∞
0
λn1 Fn,1 t.
2.13
n0
n1
0
At first, we state the following result concerning the evaluation of the integrals Fn,1 t, n ≥ 1.
Lemma 2.2. The functions
0
Fn,1 t
t
dt1 · · ·
0
t
tn−1
dtn e−μt−tn n 1 − e−μti −ti−1 2.14
i1
satisfy the difference-differential equations
d2 0
d 0
0
Fn,1 t −μ Fn,1 t μFn−1,1 t,
2
dt
dt
where t0 0, t > 0, n ≥ 1.
2.15
Journal of Applied Mathematics and Stochastic Analysis
7
Proof. We first note that
t
d 0
d
F t dt n,1
dt
t
dt1 · · ·
t
0
dtn e−μt−tn n tn−1
dt1 · · ·
t
0
1 − e−μti −ti−1 i1
n−1 1 − e−μti −ti−1 dtn−1 1 − e−μt−tn−1 tn−2
−μ
t
dt1 · · ·
t
0
2.16
i1
dtn e−μt−tn tn−1
n 1 − e−μti −ti−1 ,
i1
and therefore
d2 0
F t μ
dt2 n,1
t
dt1 · · ·
t
0
tn−2
t
t
−μ
dt1 · · ·
0
μ2
n−1 1 − e−μti −ti−1 i1
n−1 1 − e−μti −ti−1 dtn−1 1 − e−μt−tn−1 tn−2
t
dt1 · · ·
0
−μ
dtn−1 e−μt−tn−1 t
i1
dtn e−μt−tn tn−1
n 1 − e−μti −ti−1 2.17
i1
d 0
0
F t μFn−1,1 t.
dt n,1
In view of Lemma 2.2 we can prove also the following.
Theorem 2.3. The function fT1 t; 1 satisfies the second-order linear homogeneous differential
equation
d
d2
fT t; 1 − λ2 fT1 t; 1,
fT1 t; 1 − 2λ μ
2
dt 1
dt
2.18
with the initial conditions
fT1 0; 1 λ,
d
fT1 t; 1|t0 −λ λ μ .
dt
2.19
The explicit value of fT1 t; 1 is
⎡ ⎤
√
√
μ
4λ
μ
−
μ
μ
4λ
μ
μ
⎢
−t/2 μ4λμ ⎥
t/2 μ4λμ
fT1 t; 1 λe−t/22λμ ⎣ ⎦.
e
e
2 μ 4λ μ
2 μ 4λ μ
2.20
8
Journal of Applied Mathematics and Stochastic Analysis
Proof. From 2.13, it follows that
∞
d
d 0
fT1 t; 1 −λfT1 t; 1 e−λt λn1 Fn,1 t,
dt
dt
n0
2.21
0
and thus, in view of 2.17 and by letting F−1,1 t 0 , we have that
∞
∞
2
d2
d
0
−λt
n1 d 0
−λt
n1 d
f
F
f
λ
λ
F t
1
−λ
1
−
λe
e
t;
t;
t
T
T
1
1
n,1
2 n,1
dt
dt
dt2
dt
n0
n0
d
d
fT1 t; 1 λfT1 t; 1
−2λ fT1 t; 1 − λ2 fT1 t; 1 λμfT1 t; 1 − μ
dt
dt
d
fT t; 1 − λ2 fT1 t; 1.
− 2λ μ
dt 1
2.22
While the first condition is straightforward to verify, the second one needs some explanations:
if we write
fT1 Δt; 1 − fT1 0; 1
d
fT t; 1|t0 lim
,
Δt → 0
dt 1
Δt
2.23
0
fT1 Δt; 1 e−λΔt λF0,1 Δt λe−λμΔt λ 1 − λ μ Δt oΔt,
2.24
and observe that
by substituting 2.24 in 2.23 the second condition emerges.
The general solution to 2.22 has the form
√
√
e−t/22λμ Aet/2 μ4λμ Be−t/2 μ4λμ .
2.25
By imposing the initial conditions 2.19 to 2.18 we obtain 2.20.
Remark 2.4. By 2.7 derive that the functions fTi t; 1 do not depend on t, but on the time
interval tli−1 , t, in fact if tli−1 s, we have that
fTi t; 1, s e−λt−s
∞
n0
for i 1, 2, . . . .
s
λn1 Fn,1 t fTi t − s; 1,
2.26
Journal of Applied Mathematics and Stochastic Analysis
τ1
τ2
0
t1 S1
τ3
t2 S2
9
τl1 t
t3 · · · tl1 t−1
Sl1 t−1 tl1 t t
Figure 2: The 1th interloss time is t with P {ν0 0} 1.
Furthermore, by the Markov properties of the M/M/1/1 system, the random
variables T1 , T2 , . . . are independent, and {Ti }i∈Z is a renewal process with
fT1 t fTi t
if ν0 1,
fTi t
fT1 t /
if ν0 0.
2.27
Remark 2.5. If λ → 0 we get fT1 t; 1 0, because the 1th interloss time is greater than t,
∨t > 0, when nobody enters.
If μ → 0, we have that fT1 t; 1 λe−λt because without exits and with the system
busy at t 0, the 1th interloss time has the same distribution of the interarrival time.
Remark 2.6. The probability density function fT1 t; 1 can be expressed by the following
hyperbolic functions:
⎡
⎢
fT1 t; 1 λe−t/22λμ ⎣cosh
t μ 4λ μ
2
⎤
t
μ
4λ
μ
μ
⎥
− ⎦.
sinh
2
μ 4λ μ
2.28
Now, if we assume that the system is free at the starting point, the event “The 1th
interloss time ist” is represented by Figure 2.
0
Let Pn t be the conditional probability that the nth entered customer is the 1th lost
at time t, when ν0 0,
0
Pn t P {l1 t n | ν0 0}
P S1 < τ2 , . . . , Sn−2 < τn−1 , Sn−1 > t − tn−1 ,
2.29
it can be computed conditionally by the n − 1 moments when the served customers have
arrived at the system
0
Pn t
n − 1!
tn−1
t
dt1 · · ·
0
t
dtn−1 e−μt−tn−1 tn−2
n−1 n − 1!
0
1 − e−μti −ti−1 Fn−1,0 t,
n−1
t
i2
2.30
where
0
Fn−1,0 t t
0
dt1 · · ·
t
tn−2
dtn−1 e−μt−tn−1 n−1 i2
1 − e−μti −ti−1 .
2.31
10
Journal of Applied Mathematics and Stochastic Analysis
The function fT1 t; 0 can be computed as mean of convolution of l1 t exponential
probability density functions. So we have that
fT1 t; 0 ∞
∞
0
0
fnk1 τk tPn t e−λt λn Fn−1,0 t.
n2
2.32
n2
Lemma 2.7. The functions
0
Fn,0 t
t
dt1 · · ·
t
0
dtn e−μt−tn tn−1
n 1 − e−μti −ti−1 2.33
i2
satisfy the difference-differential equations
d2 0
d 0
0
Fn,0 t −μ Fn,0 t μFn−1,0 t,
2
dt
dt
2.34
0
where t > 0, n ≥ 1, F0,0 t 0.
Proof. See proof of Lemma 2.2.
In view of Lemma 2.7 we can prove also the following.
Theorem 2.8. The function fT1 t; 0 satisfies the second-order linear homogeneous differential
equation
d
d2
fT t; 0 − λ2 fT1 t; 0,
fT1 t; 0 − 2λ μ
2
dt 1
dt
2.35
with the initial conditions
fT1 0; 0 0,
d
fT t; 0|t0 λ2 .
dt 1
2.36
The explicit value of fT1 t; 0 is
√
√
λ2
−t/22λμ t/2 μ4λμ
e
− e−t/2 μ4λμ .
fT1 t; 0 e
μ 4λ μ
0
2.37
0
Proof. By substituting in 2.21 and 2.22 Fn,1 t with Fn,0 t, 2.35 emerges. While the first
condition is straightforward to verify, the second one needs some explanations: if we write
fT1 Δt; 0 − fT1 0; 0
d
fT1 t; 0|t0 lim
,
Δt → 0
dt
Δt
2.38
Journal of Applied Mathematics and Stochastic Analysis
11
and observe that
fT1 Δt; 0 e−λΔt λ2 e−μΔt
Δt
eμt1 dt1 λ2 Δt oΔt,
2.39
0
by substituting 2.39 in 2.38, the second condition emerges.
By imposing the initial conditions 2.36 to 2.35, we obtain 2.37.
Remark 2.9. If λ → 0, we get fT1 t; 0 0 because the 1th interloss time is greater than t,
∨t > 0, when nobody enters.
If μ → 0, we have that fT1 t; 0 λ2 te−λt because without exits and with the system
free at t 0, the 1th interloss time is equal to the sum of 1th and 2th interarrival times.
Remark 2.10. The function fT1 t; 0 can be expressed by the following hyperbolic function:
2 −t/22λμ
2λ e
fT1 t; 0 sinh
μ 4λ μ
t μ 4λ μ
2
.
2.40
Remark 2.11. The function fT1 t; 0 can also be computed as the convolution of the
exponential density with rate λ and fT1 t; 1:
fT1 t; 0 t
0
λe−λt1 fT1 t − t1 ; 1dt1 .
2.41
In fact, if we observe that
0
Fn,0 t t
t
dt1
0
dt2 · · ·
t
t1
dtn e−μt−tn tn−1
n t
t1 1 − e−μti −ti−1 dt1 Fn−1,1
t,
i2
2.42
0
by 2.7, 2.32, and 2.42 we obtain that
fT1 t; 0 e
−λt
∞
n2
n
t
λ
0
0
dt1 Fn−2,1 t − t1 fτ1 ∗ fT1 t; 1.
2.43
3. Interloss Mean Time
In this section we compute the laplace transforms of the density functions 2.20 and 2.37,
and we evaluate the averages of the 1th interloss time, conditionally by ν0.
Theorem 3.1. The laplace transform of fT1 t; 1
FT∗1 s; 1
∞
0
e−st fT1 t; 1dt
3.1
12
Journal of Applied Mathematics and Stochastic Analysis
satisfies the equation
s2 FT∗1 s; 1 − sλ λ λ μ − 2λ μ sFT∗1 s; 1 λ 2λ μ − λ2 FT∗1 s; 1,
3.2
and its explicit value is
λλ s
FT∗1 s; 1 λ s2 sμ
.
3.3
Proof. By 2.18, 2.19, and the property
∞
0
dte−st
d
fT t; 1 sFT∗1 s; 1 − fT1 0; 1,
dt 1
3.4
the results 3.2 and 3.3 emerge.
Remark 3.2. The 1th conditional interloss mean time
1
Θ1 E T1 | ν0 1
3.5
can be found by evaluating the derivative of 3.3 with respect to s in s 0, as follows:
−
λμ
d ∗
FT1 s; 1|s0 .
ds
λ2
3.6
In the M/M/1/1 model, the interloss mean time 3.5 is equal to the inverse of the Erlang
loss rate.
Theorem 3.3. The laplace transform of fT1 t; 0
FT∗1 s; 0
∞
0
e−st fT1 t; 0dt
3.7
satisfies the equation
s2 FT∗1 s; 0 − λ2 − 2λ μ sFT∗1 s; 0 − λ2 FT∗1 s; 0,
3.8
and its explicit value is
FT∗1 s; 0 λ2
λ s2 sμ
.
Proof. By 2.35, 2.36, and 3.4, the results 3.8 and 3.9 emerge.
3.9
Journal of Applied Mathematics and Stochastic Analysis
13
Remark 3.4. The conditional interloss mean time
0
Θ1 E T1 | ν0 0
3.10
can be found by evaluating the derivative of 3.9 with respect to s in s 0, as follows:
−
d ∗
1
1
F s; 0|s0 Θ1 .
ds T1
λ
3.11
Remark 3.5. By 2.41, the result 3.9 can be obtained using 3.3, as follows:
FT∗1 s; 0 λ
F ∗ s; 1.
λ s T1
3.12
Now, let Tn be the time spent between t 0 and the nth loss, the conditional density functions
fTn t; i d
P {Tn < t; i},
dt
3.13
with i 0, 1, can be evaluated by convolutions as follows: if i 0, we have that
fTn t; 0 fτ1 ∗ fT1 t; 1 ∗ · · · ∗ fTn t; 1,
3.14
while if i 1, we have that
fTn t; 1 fT1 t; 1 ∗ · · · ∗ fTn t; 1.
3.15
By the independence between interloss times, the laplace transforms of 3.14 and 3.15 can
be expressed as power of 3.3 as follows:
i if i 0,
FT∗n s; 0
Fτ∗0 n Tk s; 1
k1
n
λλ s
λ
,
λ s λ s2 μs
3.16
ii if i 1,
FT∗n s; 1
∗
F
n
k1
Tk
s; 1 λλ s
λ s2 μs
n
.
3.17
Remark 3.6. The conditional averages
i
Θn ETn | ν0 i,
3.18
14
Journal of Applied Mathematics and Stochastic Analysis
with i 0, 1, can be permuted by evaluating the derivatives of 3.16 and 3.17 with respect
to s in s 0, thus obtaining
i if i 0, we have that
−
1
d ∗
1
F s; 0|s0 nΘ1 ,
ds Tn
λ
3.19
d ∗
1
F s; 1|s0 nΘ1 .
ds Tn
3.20
ii if i 1, we have that
−
Acknowledgments
The author wants to acknowledge his PhD tutor Professor Enzo Orsingher for the theoretical
support and Professor Francesca Guerriero and Dr. Patrizia Scano for their contributions.
References
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3 R. C. Larson, “Approximating the performance of urban emergency service systems,” Operations
Research, vol. 23, no. 5, pp. 845–868, 1975.
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