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1 queue with customers of n types of impatience
Journal of the Korean Statistical Society 42 (2013) 375–385
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The virtual waiting time of the M /G/1 queue with customers of n types of impatience Jongho Bae ∗ Department of Statistics, Chungnam National University, Gung-dong, Daejeon, South Korea
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Article history: Received 17 July 2012 Accepted 4 January 2013 Available online 24 January 2013 AMS 2000 subject classifications: primary 60K25 secondary 68M20
abstract We consider the M /G/1 queue in which the customers are classified into n + 1 classes by their impatience times. First, we analyze the model with two types of customers; one is the customer with constant impatience time k and the other is the patient customer whose impatience time is ∞. The expected busy period of the server and the limiting distribution of the virtual waiting time process are obtained. Then, the model is generalized to the one in which the impatience time of each customer is anyone in {k1 , k2 , . . . , kn , ∞}. © 2013 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
Keywords: Queue Virtual waiting time Impatient customer Busy period
1. Introduction In this paper, the M /G/1 queue with customers of several types of impatience is considered. The customers who arrive according to the Poisson process of rate ν are classified into n + 1 classes by their impatience times. The impatience time of each customer is anyone in {k1 , k2 , . . . , kn , ∞}. The customer whose impatience time is k is the one who waits no more than k until his/her service is started. We suppose aware customer, in other words, assume that the arriving customer does not enter the system if his/her waiting time would exceed k. The customer with impatience time ∞ means the patient customer who waits until his/her service starts however long the waiting time is. Let the proportion of customers with impatience time n ki be pi , i = 1, 2, . . . , n, and the proportion of patient customers be q = 1 − i=1 pi . We assume that k1 < k2 < · · · < kn , and 0 < pi ≤ 1 and 0 ≤ q < 1, pi + q = 1. The service times of all customers are independent and identically distributed with distribution function G and mean m. There is one server in the system and the service discipline is FIFO. The queue with impatience has been studied by many researchers. Daley (1965) and Bacelli, Boyer, and Hebuterne (1984) studied the waiting time in the GI /G/1 queue with impatient customers. In particular, Bacelli et al. (1984) found the relation between the actual waiting time and the virtual waiting time. In Lillo and Martin (2001), the stability and the ergodic property of the GI /G/1 queue with impatience were studied. Finch (1960) obtained the limiting distribution of the actual waiting time in the GI /M /1 queue with constant impatience and Teghem (1979) studied the stationary distribution of the embedded Markov chain in the GI /M /s queue with impatience. Choi, Kim, and Zhu (2004) obtained many performance measures in the MAP /M /c queue with constant impatience. As for the M /M /s queue with impatient customers, the loss probability under constant impatience was calculated by Barrer (1957). Movaghar (1998) obtained the distributions of the steady state and the waiting time in the general model,
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1226-3192/$ – see front matter © 2013 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2013.01.001
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J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
M (n)/M /s, and Brandt and Brandt (1999) generalized the model of Movaghar (1998) to the model, M (n)/M (n)/s. In Choi, Kim, and Chung (2001) and Brandt and Brandt (2004), both the customer impatience and the priorities among customers were considered. For the M /G/1 queue with impatience, the approximated mean waiting time and the explicit approximation for the distribution of the virtual waiting time are given by de Kok and Tijms (1985). Bacelli et al. (1984) showed the distributional equivalence of the waiting time of the customer and the virtual waiting time in M /G/1 with general impatience and represented the distribution of the virtual waiting time as the series of Laplace–Stieltjes transforms in M /G/1 with exponential impatience. Martin and Artalejo (1995) considered the M /G/1 queue with impatient customers and retrial customers. Bae, Kim, and Lee (2001) obtained the distribution of the virtual waiting time in the M /G/1 queue with constant impatience. Some references are also introduced in Bae and Kim (2010). This study is the extension of Bae et al. (2001). The model of Bae et al. (2001) is the special case of our model — n = 1, p1 = 1, and q = 0. In this paper, the formula for the expected busy period of the server is derived and then the limiting distribution of the virtual waiting time process is obtained by calculating the expected number of downcrossings of the process during a busy period for a given level. Notice that our assumption on G is weaker than that in Bae et al. (2001). This model can be applied to a communication server system, where the customers or job requests wait for limited time durations and they are classified into several types according to their maximum waiting times. There are lots of studies on the queueing system with impatience for application to the communication system. See Jain, Shekhar, and Shukla (2012) and Kim, Dudin, Taramin, and Baek (2013). Let Zt denote the virtual waiting time process. Note that, while ki < Zt ≤ ki+1 , i = 0, 1, 2, . . . , n with k0 = 0 and i kn+1 = ∞, the proportion of entering customers is 1 − j=1 pj . So, ρ0 = ν m can be interpreted as the traffic intensity while Zt ≤ k1 , and ρi = ν m(1 −
i
j =1
pj ) as that while ki < Zt ≤ ki+1 , i = 1, 2, . . . , n. For the stability of the system, it is assumed
that ρn < 1. For x ∈ (a, b], we define the first exit time Tab (x; ρ) = min{t ≥ 0|Zt ̸∈ (a, b]} given that Z0 = x and the traffic x intensity while Zt ∈ (a, b] is ρ . We let Ge (x) = (1/m) 0 (1 − G(s))ds be the equilibrium distribution function of G and denote the Stieltjes-convolution of F and G by (F ∗ G)(x) = of F by F n∗ (x) with F 0∗ (x) = I{x≥0} .
x
0−
G(x − s)dF (s) and the n-fold recursive Stieltjes-convolution
2. The M /G /1 queue with patient and impatient customers We first consider the case of n = 1. The proportion of impatient customers is p1 = 1 − q and their impatience time is k. 2.1. The expected busy period Define P (s, x) = Pr{ZT k (x;ρ ) > k + s}, s ≥ 0. Then, by Bae et al. (2001); Bae, Kim, and Lee (2002), 0 0
(Hρ0 ∗ G˜ ρ0 ,s )(k) ˜ ρ0 ,s )(k − x), Hρ0 (k − x) − (Hρ0 ∗ G (1) Hρ0 (k) ∞ n n∗ ˜ where Hρ (x) = n=0 ρ Ge (x), so called the auxiliary renewal equation of ρ Ge , and Gρ,s (x) = ρ(Ge (x + s) − Ge (s)). Note ˜ ρ0 ,s )(k)/Hρ0 (k). that P (0, x) = 1 − Hρ0 (k − x)/Hρ0 (k) and P (s, k) = (Hρ0 ∗ G Let τ (x) be the busy period when Z0 = x, x ≥ 0. Then, by the Markov property of Zt , we see that P ( s, x) =
τ (x) = T0k (x; ρ0 ) + 1{Z
T k (x;ρ0 ) 0
>k}
{Y + τ (k)} ,
0 ≤ x ≤ k,
where the above equality means ‘‘equal in distribution’’ and random variable Y represents the time duration from the moment where Zt exceeds k to the moment where Zt reaches k thereafter. Since the entering (not arrival) process is the Poisson process of rate qν while Zt > k, Y is equal in distribution to the busy period of the M /G/1 queue in which the input rate is qν , the distribution of the service time is G, and the initial workload is given as ZT k (x;ρ ) − k given that ZT k (x;ρ ) > k. 0
Hence, for 0 ≤ x ≤ k, E [τ (x)] = E [T0k (x; ρ0 )] +
1−
Hρ0 (k − x)
E [Z
T0k (x;ρ0 )
Hρ0 (k)
− k|ZT k (x;ρ0 ) > k] 0
1 − ρ1
0
0
+ E [τ (k)] .
(2)
Lemma 1. For 0 < x ≤ k, E ZT k (x;ρ ) − k| ZT k (x;ρ ) > k = 0 0 0 0
kHρ0 (k − x) − (k − x)Hρ0 (k) Hρ0 (k) − Hρ0 (k − x)
+ (1 − ρ0 )
Hρ0 (k)
k−x 0
0
Hρ0 (u) du − Hρ0 (k − x) Hρ0 (k) − Hρ0 (k − x)
k 0
Hρ0 (u) du
.
J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
377
Proof. Note that
∞
E ZT k (x;ρ ) − k| ZT k (x;ρ ) > k = 0 0 0 0
Pr ZT k (x;ρ ) − k > s| ZT k (x;ρ ) > k ds 0 0 0 0
0
∞
=
Pr ZT k (x;ρ ) > k + s 0 0
= 0
ds.
P (s, x) ds, we first show that for x > 0,
∞
P ( s, x) P (0, x)
0
∞
ds
∞
Pr ZT k (x;ρ ) > k 0 0
0
In order to calculate
(Hρ0 ∗ G˜ ρ0 ,s )(x) ds =
∞
0
0
=
ρ0
˜ ρ0 ,s (u) ds Hρ0 (x − u) dG
0 x
m
x
Hρ0 (x − u)
∞
1 − G(u + s) ds du
0
0
= x − (1 − ρ0 )
x
Hρ0 (u) du, 0
where the last equality is justified in Kim, Bae, and Lee (2001), Appendix C. ∞ Returning to 0 P (s, x) ds, ∞
P (s, x) ds =
Hρ0 (k − x) Hρ0 (k)
0
∞
∞
(Hρ0 ∗ G˜ ρ0 ,s )(k) ds −
0
(Hρ0 ∗ G˜ ρ0 ,s )(k − x) ds 0
k−x Hρ0 (k − x) k Hρ (k − x) + (1 − ρ0 ) Hρ0 (u) du − Hρ0 (u) du . = x−k 1− 0 Hρ0 (k) Hρ0 (k) 0 0 Then, some algebras complete the proof.
Lemma 2. For 0 < x ≤ k, E
T0k
H ρ ( k − x) (x; ρ0 ) = 0 Hρ0 (k)
k
Hρ0 (u) du −
k−x
0
Hρ0 (u) du. 0
Proof. Using the Optional Stopping Theorem of the martingale, we have E[
T0k
x − kP (0, x) −
(x; ρ0 )] =
∞ 0
P (s, x) ds
1 − ρ0
(See Bae et al. (2001).) By using the equality on Putting x = k in Eq. (2) gives E [τ (k)] =
1 1 − ρ1
.
∞ 0
P (s, x) ds, we get the lemma.
k (ρ0 − ρ1 ) Hρ0 (u)du + k , 0
and substituting the above equation into Eq. (2) yields E [τ (x)] =
x + (ρ0 − ρ1 )
k k−x
Hρ0 (u)du
1 − ρ1
,
0 < x ≤ k,
(3)
and E [τ (k + s)] =
s 1 − ρ1
+ E [τ (k)] =
k + s + (ρ0 − ρ1 )
k
1 − ρ1
0
Hρ0 (u)du
.
Theorem 1. The formula for the expected busy period of the queue is given by E [τ ] =
m(ρ0 − ρ1 )
ρ0 (1 − ρ1 )
Hρ0 (k) +
mρ1
ρ0 (1 − ρ1 )
.
(4)
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J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
Proof. It can be seen that
E τ =
k
E τ (x) dG(x) +
= = = = =
x−k 1 − ρ1
k
0
=
∞
+ E τ (k) dG(x)
(ρ0 − ρ1 ) ∞ k Hρ0 (u) du dG(x) 1 − ρ1 0 1 − ρ1 0 k−x k 0 k (ρ0 − ρ1 ) m (ρ0 − ρ1 ) Hρ0 (u) dG(x) du + + Hρ0 (u) du 1 − G(k) 1 − ρ1 1 − ρ1 1 − ρ1 k−u 0 0 m (ρ0 − ρ1 ) k Hρ0 (u) 1 − G(k − u) du + 1 − ρ1 1 − ρ1 0 m m(ρ0 − ρ1 ) k Hρ0 (k − u) dGe (u) + 1 − ρ1 1 − ρ1 0 m m(ρ0 − ρ1 ) 1 + Hρ0 (k) − 1 1 − ρ1 1 − ρ1 ρ0 m(ρ0 − ρ1 ) mρ1 Hρ (k) + . ρ0 (1 − ρ1 ) 0 ρ0 (1 − ρ1 ) 1
(ρ0 − ρ1 ) 1 − ρ1 k k
∞
x dG(x) +
k
k
Hρ0 (u) du dG(x) +
This formula can be rewritten as E [τ ] =
m
1−
1 − ρ0
(ρ0 − ρ1 )(1 − ρ0 ) ρ0 (1 − ρ1 )
1 1 − ρ0
− Hρ0 (k)
,
which is less than the expected busy period of an ordinary M /G/1 queue, m/(1 − ρ0 ), noting that Hρ0 (k) increases to 1/(1 −ρ0 ) as k goes to ∞. The impatience of customers causes the reduction of the expected busy period and the proportion of the amount of the reduction to the expected busy period without impatience is
(ρ0 − ρ1 )(1 − ρ0 ) ρ0 (1 − ρ1 )
1 1 − ρ0
− Hρ0 (k) .
2.2. The expected number of downcrossings Let D(z ) be the number of {Zt } downcrossings of level z during a busy period, and D(z , x) be that of level z during a busy period given that Z0 = x, (x > 0). Then, for 0 < z ≤ k, by the same argument as that of Bae et al. (2001), E [D(z , x)] =
Hρ0 (z ) − Hρ0 (z − x) Hρ0 (z )
if 0 < x ≤ z , if x > z ,
and hence, E [D(z )] =
z
Hρ0 (z ) − Hρ0 (z − x) dG(x) +
0
∞
Hρ0 (z )dG(x) z
= Hρ0 (z ) −
z
Hρ0 (z − x)dG(x) 0
= (m/ρ0 ) Hρ′ 0 (z ),
0 < z ≤ k,
where we use identity (G ∗ Hρ0 )(x) = Hρ0 (x) − For z > k, if k < x ≤ z,
(5)
(m/ρ0 ) Hρ′ 0 (x)
whose proof is given in Appendix.
E [D(z , x)] = Pr{ZT z (x;ρ1 ) > z } (1 + E [D(z , z )]) + Pr{ZT z (x;ρ1 ) = k}E [D(z , k)]. k
k
That is, E [D(z , x)] =
1−
Hρ1 (z − x)
Hρ1 (z − k)
(1 + E [D(z , z )]) +
Hρ1 (z − x) Hρ1 (z − k)
E [D(z , k)].
By putting x = z in Eq. (6), we have E [D(z , z )] =
1−
1 Hρ1 (z − k)
(1 + E [D(z , z )]) +
1 Hρ1 (z − k)
E [D(z , k)].
(6)
J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
379
Therefore, E [D(z , z )] = E [D(z , k)] + Hρ1 (z − k) − 1. Substituting E [D(z , z )] in the above equation into Eq. (6) yields
E [D(z , x)] =
1−
Hρ1 (z − x)
Hρ1 (z − k)
E [D(z , k)] + Hρ1 (z − k) +
Hρ1 (z − x) Hρ1 (z − k)
E [D(z , k)].
Hence, E [D(z , x)] = E [D(z , k)] + Hρ1 (z − k) − Hρ1 (z − x).
(7)
For z > k, if x > z, E [D(z , x)] = 1 + E [D(z , z )] = E [D(z , k)] + Hρ1 (z − k).
(8)
For z > k, if 0 < x ≤ k, we first denote the distribution function of ZT k (x;ρ ) by (y; x, ρ0 ) or simply F (y; x) in this 0 0 section. Note that T0k (x; ρ0 ) has a mass of Hρ0 (k − x)/Hρ0 (k) at 0 and continuous distribution in (k, ∞). Namely, F0k
0 Hρ0 (k − x)/Hρ0 (k) 1 − P (y − k; x)
F (y; x) =
if y < 0, if 0 ≤ y ≤ k, if y ≥ k,
where the formula of P (s; x) is in Eq. (1). Now, we derive E [D(z , x)] for z > k, 0 < x ≤ k by conditioning on T0k (x; ρ0 ) as follows:
z
E [D(z , x)] =
Hρ1 (z − k) − Hρ1 (z − y) + E [D(z , k)] dy F (y; x) +
∞
k
Hρ1 (z − k) + E [D(z , k)] dy F (y; x)
z
= Hρ1 (z − k) + E [D(z , k)]
Hρ0 (k − x)
z
1− − Hρ1 (z − y) dy F (y; x) Hρ0 (k) k Hρ (z )Hρ0 (k − x) Hρ (k − x) − (F ∗ Hρ1 )(z ; x) + 1 , = Hρ1 (z − k) + E [D(z , k)] 1 − 0 Hρ0 (k) Hρ0 (k) z where (F ∗ Hρ )(z ; x) = 0− Hρ (z − y) dy F (y; x). Letting x = k gives us Hρ ( z ) 1 1 E [D(z , k)] = Hρ1 (z − k) 1 − − (F ∗ Hρ1 )(z ; k) + 1 , Hρ0 (k) Hρ0 (k) Hρ0 (k) that is
E [D(z , k)] = Hρ1 (z − k) Hρ0 (k) − 1 − Hρ0 (k)(F ∗ Hρ1 )(z ; k) + Hρ1 (z ), and it follows that
E [D(z , x)] = Hρ0 (k) − Hρ0 (k − x)
Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k) + Hρ1 (z ) − (F ∗ Hρ1 )(z ; x).
After substituting the above equation into Eqs. (7) and (8), we get the following summary: for z > k
H ( k ) − H ( k − x ) H ( z − k ) − ( F ∗ H )( z ; k ) ρ ρ ρ ρ 0 0 1 1 + Hρ1 (z ) − (F ∗ Hρ1 )(z ; x), E [D(z , x)] = H ( k ) H ( z − k ) − ( F ∗ H )( z ; k ) + Hρ1 (z ) − Hρ1 (z − x), ρ ρ ρ 0 1 1 Hρ0 (k) Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k) + Hρ1 (z ),
if 0 < x ≤ k, if k < x ≤ z , if x > z .
Therefore, we have that for z > k, E [D(z )] =
∞
E [D(z , x)] dG(x) 0
k = Hρ0 (k) Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k) + Hρ1 (z ) − (F ∗ Hρ1 )(z ; x) dG(x) 0
− Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k)
k
Hρ0 (k − x) dG(x) − 0
z
Hρ1 (z − x) dG(x) k
k (F ∗ Hρ1 )(z ; x) dG(x) = (m/ρ0 )Hρ′ 0 (k) Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k) + Hρ1 (z ) − 0
z
Hρ1 (z − x) dG(x).
− k
(9)
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J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
Now, the limiting probability density function f (z ) of the virtual waiting time process is calculated by Brill and Posner (1977) as follows: f (z ) =
E [D(z )] E [τ ] + m/ρ0
,
z > 0,
and the limiting distribution has a mass 1 − Eqs. (5) and (9).
∞ 0
f (z )dz at 0. The formula for E [τ ] is given in Theorem 1, and E [D(z )] in
2.3. The M /G/1 queue with impatient customers only We present the result in the special case where all customers are impatient, that is q = 0. The following are consistent with Kim et al. (2001) and Bae et al. (2001). First, we have
k Hρ0 (u) du x + ρ 0 k−k x E [τ (x)] = x + ρ 0 Hρ0 (u) du
if 0 < x ≤ k, if x > k,
0
and E [τ ] = mHρ0 (k). For 0 < z ≤ k, we get E [D(z )] = (m/ρ0 ) Hρ′ 0 (z ). Note that when q = 0, Hρ1 (x) ≡ 1,
(F ∗ Hρ1 )(z ; x) = 1 − P (z − k, x).
˜ and Hρ0 gives us Therefore, for z > k, some calculus on G, G, ˜ ρ0 ,z −k )′ (k). E [D(z )] = E [D(k + s)] = (m/ρ0 ) (Hρ0 ∗ G Hence, the limiting probability density function of the virtual waiting time process is given by
f (z ) =
Hρ′ 0 (z )
1 + ρ0 Hρ0 (k) ′ ˜ (Hρ0 ∗ Gρ0 ,z −k ) (k)
1 + ρ0 Hρ0 (k)
if 0 < z ≤ k, if z > k,
and the mass at z = 0 is ∞
f (z ) dz =
1− 0
1 1 + ρ0 Hρ0 (k)
,
which is equal to the expected idle period divided by the sum of the expected idle period and busy period. 3. The M /G /1 queue with patient and n types of impatient customers 3.1. The expected busy period and the number of downcrossings We let E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] be the expected busy period of the queue given that Z0 = x, and also let E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )] be the expected number of {Zt } downcrossings of level z during a busy period given that Z0 = x. In the previous section, we have obtained E [τ (x; k1 ; ρ0 , ρ1 )] and E [D(z , x; k1 ; ρ0 , ρ1 )]. In this section, it is provided the recursive method of deriving E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] and E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )]. Assume that E [τ (x; k1 , . . . , kj ; ρ0 , . . . , ρj )] – the expected busy period of the queue with patient and j types of impatient customers – is known for j = n − 1. When j = n, for x > k1 the busy period of the queue with {Zt } starting at x is equal in distribution to the sum of the time for reaching k1 of Zt and the busy period starting at k1 . Hence, E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )], abbreviated by E [τ (x)], can be expressed as E [τ (x)] = E [τ (x − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] + E [τ (k1 )].
(10)
J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
For 0 < x ≤ k1 , by conditioning on Z k
E [τ (x)] = E [T0 1 (x; ρ0 )] +
k
T0 1 (x;ρ0 )
381
, E [τ (x)] is represented as
∞
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ) k1
+ E [τ (k1 )] Pr{ZT k1 (x;ρ ) > k1 } 0 0 ∞ k E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ) = k1
Hρ0 (k1 − x)
+ Notice that Z
Hρ0 (k1 )
k
T0 1 (x;ρ0 )
E [τ (k1 )] =
Hρ0 (u)du −
k 1 −x
Hρ0 (u)du + E [τ (k1 )] 1 −
0
0
Hρ0 (k1 − x)
Hρ0 (k1 )
.
has a mass Hρ0 (k1 − x)/Hρ0 (k1 ) at 0. Putting x = k1 gives us k1
k1
Hρ0 (u)du + Hρ0 (k1 )
∞
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ). k1
0
Using this, we get that for 0 < x ≤ k1 E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] k1
=
Hρ0 (u)du −
k1 −x
0
Hρ0 (u)du +
∞
0
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ) k1
+ (Hρ0 (k1 ) − Hρ0 (k1 − x))
∞
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ), k1
and from Eq. (10), for x > k1 E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] = E [τ (x − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] +
k1
Hρ0 (u)du + Hρ0 (k1 )
0
∞
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ).
×
(11)
k1
Therefore, E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] can be obtained inductively. E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )], simply E [D(z , x)], is also derived similarly to E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )]. First, observe that for 0 < z ≤ kn and x > 0 E [D(z , x)] = E [D(z , x; k1 , . . . , kn−1 ; ρ0 , . . . , ρn−1 )]. When z > kn and x > k1 , E [D(z , x)] = E [D(z − k1 , x − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] + E [D(z , k1 )],
(12)
and when z > kn and 0 < x ≤ k1 , E [D(z , x)] =
∞
k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ) k1
+ E [D(z , k1 )] 1 −
Hρ0 (k1 − x) Hρ0 (k1 )
.
To obtain E [D(z , k1 )], we let x = k1 in the above equation, then E [D(z , k1 )] = Hρ0 (k1 )
∞ k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ). k1
Therefore, it follows that for z > kn and 0 < x ≤ k1 E [D(z , x)] = (Hρ0 (k1 ) − Hρ0 (k1 − x))
∞ k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ) k1
∞
+
k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ),
(13)
k1
and from Eq. (12), for z > kn and x > k1 E [D(z , x)] = E [D(z − k1 , x − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )]
+ Hρ0 (k1 )
∞ k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ). k1
Hence, E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )] is derived by using the induction hypothesis.
(14)
382
J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
Finally, if we denote the busy period of the queue by τ (k1 , . . . , kn ; ρ0 , . . . , ρn ) and the number of {Zt } downcrossings of level z during a busy period by D(z ; k1 , . . . , kn ; ρ0 , . . . , ρn ), the limiting distribution of the virtual waiting time process {Zt } has the density function f (z ) given by f (z ) =
∞
E [D(z ; k1 , . . . , kn ; ρ0 , . . . , ρn )] E [τ (k1 , . . . , kn ; ρ0 , . . . , ρn )] + m/ρ0
and has a mass 1 −
∞ 0
= ∞ 0 0
E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )] dG(x)
E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] dG(x) + m/ρ0
,
z > 0,
f (z )dz at 0.
4. The M /M /1 queue with patient and n types of impatient customers In this section, we provide the explicit formulas of the expected busy period and the limiting distribution when the service time of a customer is exponentially distributed. Let G(x) = 1 − e−x/m and let θi = (1 − ρi )/m. Then it can be shown that Hρi (x) =
1 1 − ρi
−
ρi 1 − ρi
e−θi x ,
i = 0 , 1 , 2 , . . . , n.
We begin from the case n = 1. From Eqs. (3), (4) and Theorem 1, we have
E [τ (x; k1 ; ρ0 , ρ1 )] =
x
1 − ρ0 k1
mρ0 (ρ0 − ρ1 )
−
e−θ0 (k1 −x) − e−θ0 k1
(1 − ρ0 ) (1 − ρ1 ) x − k1 mρ0 (ρ0 − ρ1 ) + − 1 − e−θ0 k1 1 − ρ0 1 − ρ1 (1 − ρ0 )2 (1 − ρ1 ) 2
if 0 < x ≤ k1 , (15) if x > k1 ,
and E [τ (k1 ; ρ0 , ρ1 )] = m
1 1 − ρ0
−
ρ0 − ρ1 e−θ0 k1 . (1 − ρ0 )(1 − ρ1 )
When 0 < z ≤ k1 , the expected number of downcrossings is given by
ρ −θ (z −x) 0 e 0 − e−θ0 z 1 − ρ0 E [D(z , x; k1 ; ρ0 , ρ1 )] = 1 ρ0 −θ0 z − e 1 − ρ0 1 − ρ0
if 0 < x ≤ z , (16)
if x > z ,
and E [D(z ; k1 ; ρ0 , ρ1 )] = e−θ0 z . When z > k1 ,
ρ0 (eθ0 x − 1) e−θ0 k1 −θ1 (z −k1 ) , 1 − ρ 0 ρ0 ρ1 (eθ0 k1 − 1) e−θ0 k1 −θ1 (z −k1 ) + (eθ1 (x−k1 ) − 1) e−θ1 (z −k1 ) , E [D(z , x; k1 ; ρ0 , ρ1 )] = 1 − ρ0 1 − ρ1 ρ0 1 ρ1 (eθ0 k1 − 1) e−θ0 k1 −θ1 (z −k1 ) + − e−θ1 (z −k1 ) , 1 − ρ0 1 − ρ1 1 − ρ1
if 0 < x ≤ k1 , if k1 < x ≤ z , (17) if x > z ,
and E [D(z ; k1 ; ρ0 , ρ1 )] = e−θ0 k1 −θ1 (z −k1 ) . Consider the case n = 2. After obtaining E [τ (x − k1 ; k2 − k1 ; ρ1 , ρ2 )] from Eq. (15) and calculating E [τ (x; k1 , k2 ; ρ0 , ρ1 , ρ2 )] by using Eq. (11), we get E [τ (k1 , k2 ; ρ0 , ρ1 , ρ2 )] = m
1 1 − ρ0
−
ρ0 − ρ1 ρ1 − ρ2 e−θ0 k1 − e−θ0 k1 −θ1 (k2 −k1 ) , (1 − ρ0 )(1 − ρ1 ) (1 − ρ1 )(1 − ρ2 )
and the expected number of downcrossings is given by E [D(z ; k1 , k2 ; ρ0 , ρ1 , ρ2 )] =
−θ z e 0
e−θ0 k1 −θ1 (z −k1 ) −θ0 k1 −θ1 (k2 −k1 )−θ2 (z −k2 ) e
if 0 < z ≤ k1 , if k1 < z ≤ k2 , if z > k2 ,
where the last formula of the above is derived by using Eqs. (13), (14), (16), and (17).
J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
383
Table 1 Expected busy periods. n=0
n=1
n=2
n=3
10.0000
3.9678
3.0740
2.8417
In general, the formulas for the expected busy period and the expected number of downcrossings are given as follows:
E [τ (k1 , k2 , . . . , kn ; ρ0 , ρ1 , . . . , ρn )] = m
1 1 − ρ0
−
n i=1
ρi−1 − ρi − ij=1 θj−1 (kj −kj−1 ) e , (1 − ρi−1 )(1 − ρi )
and E [D(z ; k1 , k2 , . . . , kn ; ρ0 , ρ1 , . . . , ρn )] = e
−θi (z −ki )−
i
j=1 θj−1 (kj −kj−1 )
for ki < z ≤ ki+1 , i = 0, 1, 2, . . . , n with k0 = 0 and kn+1 = ∞, whose proofs can be done by mathematical induction on n. Hence, the limiting distribution of the virtual waiting time process has the density function −θl (z −kl )−
e
f (z ) =
m
1 ρ0 (1−ρ0 )
−
l
θj−1 (kj −kj−1 )
j=1
n i=1
ρi−1 −ρi − e (1−ρi−1 )(1−ρi )
i
j=1 θj−1 (kj −kj−1 )
for kl < z ≤ kl+1 , l = 0, 1, 2, . . . , n, and has a mass 1/ρ0
∞
f (z ) dz =
1− 0
1
ρ0 (1−ρ0 )
−
n i=1
ρi−1 −ρi − e (1−ρi−1 )(1−ρi )
i
j=1 θj−1 (kj −kj−1 )
at z = 0. The mass at z = 0 is equal to the expected idle period divided by the sum of the expected idle period and busy period. 5. Illustrations In this section, some illustrations are given to show the effects of the impatience of customers on the busy period and the limiting distribution of the virtual waiting time process. We consider some M /M /1 queues with several cases of impatience as follows:
• • • •
n n n n
= 0; ρ0 = 0.9 (ordinary M /M /1 queue) = 1; k1 = 1; ρ0 = 0.9, ρ1 = 0.7 = 2; k1 = 1, k2 = 2; ρ0 = 0.9, ρ1 = 0.7, ρ2 = 0.5 = 3; k1 = 1, k2 = 2, k3 = 3; ρ0 = 0.9, ρ1 = 0.7, ρ2 = 0.5, ρ3 = 0.3.
It is assumed that the expected service time m of a customer is 1. Note that θi = (1 − ρi )/m = 1 − ρi , i = 0, 1, 2, . . . , n. We can observe in Table 1 that the impatience of customers reduces the expected busy period and in Figs. 1 and 2 that the more impatient the customers are, the shorter the virtual waiting times are stochastically. 6. Conclusion We consider the virtual waiting time process of the M /G/1 queue with patient customers and impatient customers, in which the customers are classified into n + 1 classes by their impatience times. It has been obtained the limiting distribution of the virtual waiting time process by deriving the formulas for the expected busy period and the expected number of process downcrossings of any level z. We have studied the case of n = 1 first and generalized the result to the case of n > 1 by an inductive argument. The research will be extended to the case of n = ∞, and to the case that the impatience time is any random variable. Acknowledgments We would like to thank the anonymous referees for valuable comments, which result in a great improvement of the paper. This study was financially supported by research fund of Chungnam National University in 2010.
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J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
Fig. 1. The limiting cdf’s of the virtual waiting time processes.
Fig. 2. The limiting pdf’s of the virtual waiting time processes.
Appendix When ρ > 0, we derive the formula for G ∗ Hρ (x) as follows:
x
Hρ (x − u) dG(u) = 0
x
G(x − u) dHρ (u)
0−
= Hρ (x) −
x
1 − G(x − u) dHρ (u)
0−
= Hρ (x) − m
x
G′e (x − u) dHρ (u)
0−
= Hρ (x) − m
d dx
x
Ge (x − u) dHρ (u) noting that Ge (0) = 0, 0−
′ = Hρ (x) − m Hρ (x) − 1 /ρ = Hρ (x) − (m/ρ)Hρ′ (x). References Bacelli, F., Boyer, P., & Hebuterne, G. (1984). Single-server queues with impatient customers. Advances in Applied Probability, 16, 887–905. Bae, J., & Kim, S. (2010). The stationary workload of the M /G/1 queue with impatient customers. Queueing Systems, 64, 253–265.
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385
Bae, J., Kim, S., & Lee, E. Y. (2001). The virtual waiting time of the M /G/1 queue with impatient customers. Queueing Systems, 38, 485–494. Bae, J., Kim, S., & Lee, E. Y. (2002). A PλM -policy for an M /G/1 queueing system. Applied Mathematical Modeling, 26, 929–939. Barrer, D. Y. (1957). Queuing with impatient customers and ordered service. Operations Research, 5, 650–656. Brandt, A., & Brandt, M. (1999). On the M (n)/M (n)/s queue with impatient calls. Performance Evaluation, 35, 1–18. Brandt, A., & Brandt, M. (2004). On the two-class M /M /1 system under preemptive resume and impatience of the prioritized customers. Queueing Systems, 47, 147–168. Brill, P. H., & Posner, M. J. M. (1977). Level crossings in point process applied to queues: single-server case. Operations Research, 25, 662–674. Choi, B. D., Kim, B., & Chung, J. (2001). M /M /1 queue with impatient customers of higher priority. Queueing Systems, 38, 49–66. Choi, B. D., Kim, B., & Zhu, D. (2004). MAP /M /c queue with constant impatience time. Mathematics of Operations Research, 29, 309–325. Daley, D. J. (1965). General customer impatience in the queue GI /G/1. Journal of Applied Probability, 2, 186–205. de Kok, A. G., & Tijms, H. C. (1985). A queueing system with impatient customers. Journal of Applied Probability, 22, 688–696. Finch, P. D. (1960). Deterministic customer impatience in the queueing system GI /M /1. Biometrika, 47, 45–52. Jain, M., Shekhar, C., & Shukla, S. (2012). Queueing analysis of a multi-component machining system having unreliable heterogeneous servers and impatient customers. American Journal of Operational Research, 2, 16–26. Kim, S., Bae, J., & Lee, E. Y. (2001). Busy periods of poisson arrival queues with loss. Queueing Systems, 39, 201–212. Kim, C., Dudin, S., Taramin, O., & Baek, J. (2013). Queueing system MAP |PH |N |N + R with impatient heterogeneous customers as a model of call center. Applied Mathematical Modeling, 37, 958–976. Lillo, R. E., & Martin, M. (2001). Stability in queues with impatience customers. Stochastic Models, 17, 375–389. Martin, M., & Artalejo, J. R. (1995). Analysis of an M /G/1 queue with two types of impatient units. Advances in Applied Probability, 27, 840–861. Movaghar, A. (1998). On queueing with customer impatience until the beginning of service. Queueing Systems, 29, 337–350. Teghem, J., Jr. (1979). Use of discrete transforms for the study of a GI /M /S queue with impatienct cusotmer phenomena. Zeitschrift fur Operations Research, Serie A-B, 23, 95–106.
Contents lists available at SciVerse ScienceDirect
Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss
The virtual waiting time of the M /G/1 queue with customers of n types of impatience Jongho Bae ∗ Department of Statistics, Chungnam National University, Gung-dong, Daejeon, South Korea
article
info
Article history: Received 17 July 2012 Accepted 4 January 2013 Available online 24 January 2013 AMS 2000 subject classifications: primary 60K25 secondary 68M20
abstract We consider the M /G/1 queue in which the customers are classified into n + 1 classes by their impatience times. First, we analyze the model with two types of customers; one is the customer with constant impatience time k and the other is the patient customer whose impatience time is ∞. The expected busy period of the server and the limiting distribution of the virtual waiting time process are obtained. Then, the model is generalized to the one in which the impatience time of each customer is anyone in {k1 , k2 , . . . , kn , ∞}. © 2013 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
Keywords: Queue Virtual waiting time Impatient customer Busy period
1. Introduction In this paper, the M /G/1 queue with customers of several types of impatience is considered. The customers who arrive according to the Poisson process of rate ν are classified into n + 1 classes by their impatience times. The impatience time of each customer is anyone in {k1 , k2 , . . . , kn , ∞}. The customer whose impatience time is k is the one who waits no more than k until his/her service is started. We suppose aware customer, in other words, assume that the arriving customer does not enter the system if his/her waiting time would exceed k. The customer with impatience time ∞ means the patient customer who waits until his/her service starts however long the waiting time is. Let the proportion of customers with impatience time n ki be pi , i = 1, 2, . . . , n, and the proportion of patient customers be q = 1 − i=1 pi . We assume that k1 < k2 < · · · < kn , and 0 < pi ≤ 1 and 0 ≤ q < 1, pi + q = 1. The service times of all customers are independent and identically distributed with distribution function G and mean m. There is one server in the system and the service discipline is FIFO. The queue with impatience has been studied by many researchers. Daley (1965) and Bacelli, Boyer, and Hebuterne (1984) studied the waiting time in the GI /G/1 queue with impatient customers. In particular, Bacelli et al. (1984) found the relation between the actual waiting time and the virtual waiting time. In Lillo and Martin (2001), the stability and the ergodic property of the GI /G/1 queue with impatience were studied. Finch (1960) obtained the limiting distribution of the actual waiting time in the GI /M /1 queue with constant impatience and Teghem (1979) studied the stationary distribution of the embedded Markov chain in the GI /M /s queue with impatience. Choi, Kim, and Zhu (2004) obtained many performance measures in the MAP /M /c queue with constant impatience. As for the M /M /s queue with impatient customers, the loss probability under constant impatience was calculated by Barrer (1957). Movaghar (1998) obtained the distributions of the steady state and the waiting time in the general model,
∗
Tel.: +82 42 821 5432. E-mail address: [email protected].
1226-3192/$ – see front matter © 2013 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2013.01.001
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J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
M (n)/M /s, and Brandt and Brandt (1999) generalized the model of Movaghar (1998) to the model, M (n)/M (n)/s. In Choi, Kim, and Chung (2001) and Brandt and Brandt (2004), both the customer impatience and the priorities among customers were considered. For the M /G/1 queue with impatience, the approximated mean waiting time and the explicit approximation for the distribution of the virtual waiting time are given by de Kok and Tijms (1985). Bacelli et al. (1984) showed the distributional equivalence of the waiting time of the customer and the virtual waiting time in M /G/1 with general impatience and represented the distribution of the virtual waiting time as the series of Laplace–Stieltjes transforms in M /G/1 with exponential impatience. Martin and Artalejo (1995) considered the M /G/1 queue with impatient customers and retrial customers. Bae, Kim, and Lee (2001) obtained the distribution of the virtual waiting time in the M /G/1 queue with constant impatience. Some references are also introduced in Bae and Kim (2010). This study is the extension of Bae et al. (2001). The model of Bae et al. (2001) is the special case of our model — n = 1, p1 = 1, and q = 0. In this paper, the formula for the expected busy period of the server is derived and then the limiting distribution of the virtual waiting time process is obtained by calculating the expected number of downcrossings of the process during a busy period for a given level. Notice that our assumption on G is weaker than that in Bae et al. (2001). This model can be applied to a communication server system, where the customers or job requests wait for limited time durations and they are classified into several types according to their maximum waiting times. There are lots of studies on the queueing system with impatience for application to the communication system. See Jain, Shekhar, and Shukla (2012) and Kim, Dudin, Taramin, and Baek (2013). Let Zt denote the virtual waiting time process. Note that, while ki < Zt ≤ ki+1 , i = 0, 1, 2, . . . , n with k0 = 0 and i kn+1 = ∞, the proportion of entering customers is 1 − j=1 pj . So, ρ0 = ν m can be interpreted as the traffic intensity while Zt ≤ k1 , and ρi = ν m(1 −
i
j =1
pj ) as that while ki < Zt ≤ ki+1 , i = 1, 2, . . . , n. For the stability of the system, it is assumed
that ρn < 1. For x ∈ (a, b], we define the first exit time Tab (x; ρ) = min{t ≥ 0|Zt ̸∈ (a, b]} given that Z0 = x and the traffic x intensity while Zt ∈ (a, b] is ρ . We let Ge (x) = (1/m) 0 (1 − G(s))ds be the equilibrium distribution function of G and denote the Stieltjes-convolution of F and G by (F ∗ G)(x) = of F by F n∗ (x) with F 0∗ (x) = I{x≥0} .
x
0−
G(x − s)dF (s) and the n-fold recursive Stieltjes-convolution
2. The M /G /1 queue with patient and impatient customers We first consider the case of n = 1. The proportion of impatient customers is p1 = 1 − q and their impatience time is k. 2.1. The expected busy period Define P (s, x) = Pr{ZT k (x;ρ ) > k + s}, s ≥ 0. Then, by Bae et al. (2001); Bae, Kim, and Lee (2002), 0 0
(Hρ0 ∗ G˜ ρ0 ,s )(k) ˜ ρ0 ,s )(k − x), Hρ0 (k − x) − (Hρ0 ∗ G (1) Hρ0 (k) ∞ n n∗ ˜ where Hρ (x) = n=0 ρ Ge (x), so called the auxiliary renewal equation of ρ Ge , and Gρ,s (x) = ρ(Ge (x + s) − Ge (s)). Note ˜ ρ0 ,s )(k)/Hρ0 (k). that P (0, x) = 1 − Hρ0 (k − x)/Hρ0 (k) and P (s, k) = (Hρ0 ∗ G Let τ (x) be the busy period when Z0 = x, x ≥ 0. Then, by the Markov property of Zt , we see that P ( s, x) =
τ (x) = T0k (x; ρ0 ) + 1{Z
T k (x;ρ0 ) 0
>k}
{Y + τ (k)} ,
0 ≤ x ≤ k,
where the above equality means ‘‘equal in distribution’’ and random variable Y represents the time duration from the moment where Zt exceeds k to the moment where Zt reaches k thereafter. Since the entering (not arrival) process is the Poisson process of rate qν while Zt > k, Y is equal in distribution to the busy period of the M /G/1 queue in which the input rate is qν , the distribution of the service time is G, and the initial workload is given as ZT k (x;ρ ) − k given that ZT k (x;ρ ) > k. 0
Hence, for 0 ≤ x ≤ k, E [τ (x)] = E [T0k (x; ρ0 )] +
1−
Hρ0 (k − x)
E [Z
T0k (x;ρ0 )
Hρ0 (k)
− k|ZT k (x;ρ0 ) > k] 0
1 − ρ1
0
0
+ E [τ (k)] .
(2)
Lemma 1. For 0 < x ≤ k, E ZT k (x;ρ ) − k| ZT k (x;ρ ) > k = 0 0 0 0
kHρ0 (k − x) − (k − x)Hρ0 (k) Hρ0 (k) − Hρ0 (k − x)
+ (1 − ρ0 )
Hρ0 (k)
k−x 0
0
Hρ0 (u) du − Hρ0 (k − x) Hρ0 (k) − Hρ0 (k − x)
k 0
Hρ0 (u) du
.
J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
377
Proof. Note that
∞
E ZT k (x;ρ ) − k| ZT k (x;ρ ) > k = 0 0 0 0
Pr ZT k (x;ρ ) − k > s| ZT k (x;ρ ) > k ds 0 0 0 0
0
∞
=
Pr ZT k (x;ρ ) > k + s 0 0
= 0
ds.
P (s, x) ds, we first show that for x > 0,
∞
P ( s, x) P (0, x)
0
∞
ds
∞
Pr ZT k (x;ρ ) > k 0 0
0
In order to calculate
(Hρ0 ∗ G˜ ρ0 ,s )(x) ds =
∞
0
0
=
ρ0
˜ ρ0 ,s (u) ds Hρ0 (x − u) dG
0 x
m
x
Hρ0 (x − u)
∞
1 − G(u + s) ds du
0
0
= x − (1 − ρ0 )
x
Hρ0 (u) du, 0
where the last equality is justified in Kim, Bae, and Lee (2001), Appendix C. ∞ Returning to 0 P (s, x) ds, ∞
P (s, x) ds =
Hρ0 (k − x) Hρ0 (k)
0
∞
∞
(Hρ0 ∗ G˜ ρ0 ,s )(k) ds −
0
(Hρ0 ∗ G˜ ρ0 ,s )(k − x) ds 0
k−x Hρ0 (k − x) k Hρ (k − x) + (1 − ρ0 ) Hρ0 (u) du − Hρ0 (u) du . = x−k 1− 0 Hρ0 (k) Hρ0 (k) 0 0 Then, some algebras complete the proof.
Lemma 2. For 0 < x ≤ k, E
T0k
H ρ ( k − x) (x; ρ0 ) = 0 Hρ0 (k)
k
Hρ0 (u) du −
k−x
0
Hρ0 (u) du. 0
Proof. Using the Optional Stopping Theorem of the martingale, we have E[
T0k
x − kP (0, x) −
(x; ρ0 )] =
∞ 0
P (s, x) ds
1 − ρ0
(See Bae et al. (2001).) By using the equality on Putting x = k in Eq. (2) gives E [τ (k)] =
1 1 − ρ1
.
∞ 0
P (s, x) ds, we get the lemma.
k (ρ0 − ρ1 ) Hρ0 (u)du + k , 0
and substituting the above equation into Eq. (2) yields E [τ (x)] =
x + (ρ0 − ρ1 )
k k−x
Hρ0 (u)du
1 − ρ1
,
0 < x ≤ k,
(3)
and E [τ (k + s)] =
s 1 − ρ1
+ E [τ (k)] =
k + s + (ρ0 − ρ1 )
k
1 − ρ1
0
Hρ0 (u)du
.
Theorem 1. The formula for the expected busy period of the queue is given by E [τ ] =
m(ρ0 − ρ1 )
ρ0 (1 − ρ1 )
Hρ0 (k) +
mρ1
ρ0 (1 − ρ1 )
.
(4)
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J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
Proof. It can be seen that
E τ =
k
E τ (x) dG(x) +
= = = = =
x−k 1 − ρ1
k
0
=
∞
+ E τ (k) dG(x)
(ρ0 − ρ1 ) ∞ k Hρ0 (u) du dG(x) 1 − ρ1 0 1 − ρ1 0 k−x k 0 k (ρ0 − ρ1 ) m (ρ0 − ρ1 ) Hρ0 (u) dG(x) du + + Hρ0 (u) du 1 − G(k) 1 − ρ1 1 − ρ1 1 − ρ1 k−u 0 0 m (ρ0 − ρ1 ) k Hρ0 (u) 1 − G(k − u) du + 1 − ρ1 1 − ρ1 0 m m(ρ0 − ρ1 ) k Hρ0 (k − u) dGe (u) + 1 − ρ1 1 − ρ1 0 m m(ρ0 − ρ1 ) 1 + Hρ0 (k) − 1 1 − ρ1 1 − ρ1 ρ0 m(ρ0 − ρ1 ) mρ1 Hρ (k) + . ρ0 (1 − ρ1 ) 0 ρ0 (1 − ρ1 ) 1
(ρ0 − ρ1 ) 1 − ρ1 k k
∞
x dG(x) +
k
k
Hρ0 (u) du dG(x) +
This formula can be rewritten as E [τ ] =
m
1−
1 − ρ0
(ρ0 − ρ1 )(1 − ρ0 ) ρ0 (1 − ρ1 )
1 1 − ρ0
− Hρ0 (k)
,
which is less than the expected busy period of an ordinary M /G/1 queue, m/(1 − ρ0 ), noting that Hρ0 (k) increases to 1/(1 −ρ0 ) as k goes to ∞. The impatience of customers causes the reduction of the expected busy period and the proportion of the amount of the reduction to the expected busy period without impatience is
(ρ0 − ρ1 )(1 − ρ0 ) ρ0 (1 − ρ1 )
1 1 − ρ0
− Hρ0 (k) .
2.2. The expected number of downcrossings Let D(z ) be the number of {Zt } downcrossings of level z during a busy period, and D(z , x) be that of level z during a busy period given that Z0 = x, (x > 0). Then, for 0 < z ≤ k, by the same argument as that of Bae et al. (2001), E [D(z , x)] =
Hρ0 (z ) − Hρ0 (z − x) Hρ0 (z )
if 0 < x ≤ z , if x > z ,
and hence, E [D(z )] =
z
Hρ0 (z ) − Hρ0 (z − x) dG(x) +
0
∞
Hρ0 (z )dG(x) z
= Hρ0 (z ) −
z
Hρ0 (z − x)dG(x) 0
= (m/ρ0 ) Hρ′ 0 (z ),
0 < z ≤ k,
where we use identity (G ∗ Hρ0 )(x) = Hρ0 (x) − For z > k, if k < x ≤ z,
(5)
(m/ρ0 ) Hρ′ 0 (x)
whose proof is given in Appendix.
E [D(z , x)] = Pr{ZT z (x;ρ1 ) > z } (1 + E [D(z , z )]) + Pr{ZT z (x;ρ1 ) = k}E [D(z , k)]. k
k
That is, E [D(z , x)] =
1−
Hρ1 (z − x)
Hρ1 (z − k)
(1 + E [D(z , z )]) +
Hρ1 (z − x) Hρ1 (z − k)
E [D(z , k)].
By putting x = z in Eq. (6), we have E [D(z , z )] =
1−
1 Hρ1 (z − k)
(1 + E [D(z , z )]) +
1 Hρ1 (z − k)
E [D(z , k)].
(6)
J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
379
Therefore, E [D(z , z )] = E [D(z , k)] + Hρ1 (z − k) − 1. Substituting E [D(z , z )] in the above equation into Eq. (6) yields
E [D(z , x)] =
1−
Hρ1 (z − x)
Hρ1 (z − k)
E [D(z , k)] + Hρ1 (z − k) +
Hρ1 (z − x) Hρ1 (z − k)
E [D(z , k)].
Hence, E [D(z , x)] = E [D(z , k)] + Hρ1 (z − k) − Hρ1 (z − x).
(7)
For z > k, if x > z, E [D(z , x)] = 1 + E [D(z , z )] = E [D(z , k)] + Hρ1 (z − k).
(8)
For z > k, if 0 < x ≤ k, we first denote the distribution function of ZT k (x;ρ ) by (y; x, ρ0 ) or simply F (y; x) in this 0 0 section. Note that T0k (x; ρ0 ) has a mass of Hρ0 (k − x)/Hρ0 (k) at 0 and continuous distribution in (k, ∞). Namely, F0k
0 Hρ0 (k − x)/Hρ0 (k) 1 − P (y − k; x)
F (y; x) =
if y < 0, if 0 ≤ y ≤ k, if y ≥ k,
where the formula of P (s; x) is in Eq. (1). Now, we derive E [D(z , x)] for z > k, 0 < x ≤ k by conditioning on T0k (x; ρ0 ) as follows:
z
E [D(z , x)] =
Hρ1 (z − k) − Hρ1 (z − y) + E [D(z , k)] dy F (y; x) +
∞
k
Hρ1 (z − k) + E [D(z , k)] dy F (y; x)
z
= Hρ1 (z − k) + E [D(z , k)]
Hρ0 (k − x)
z
1− − Hρ1 (z − y) dy F (y; x) Hρ0 (k) k Hρ (z )Hρ0 (k − x) Hρ (k − x) − (F ∗ Hρ1 )(z ; x) + 1 , = Hρ1 (z − k) + E [D(z , k)] 1 − 0 Hρ0 (k) Hρ0 (k) z where (F ∗ Hρ )(z ; x) = 0− Hρ (z − y) dy F (y; x). Letting x = k gives us Hρ ( z ) 1 1 E [D(z , k)] = Hρ1 (z − k) 1 − − (F ∗ Hρ1 )(z ; k) + 1 , Hρ0 (k) Hρ0 (k) Hρ0 (k) that is
E [D(z , k)] = Hρ1 (z − k) Hρ0 (k) − 1 − Hρ0 (k)(F ∗ Hρ1 )(z ; k) + Hρ1 (z ), and it follows that
E [D(z , x)] = Hρ0 (k) − Hρ0 (k − x)
Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k) + Hρ1 (z ) − (F ∗ Hρ1 )(z ; x).
After substituting the above equation into Eqs. (7) and (8), we get the following summary: for z > k
H ( k ) − H ( k − x ) H ( z − k ) − ( F ∗ H )( z ; k ) ρ ρ ρ ρ 0 0 1 1 + Hρ1 (z ) − (F ∗ Hρ1 )(z ; x), E [D(z , x)] = H ( k ) H ( z − k ) − ( F ∗ H )( z ; k ) + Hρ1 (z ) − Hρ1 (z − x), ρ ρ ρ 0 1 1 Hρ0 (k) Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k) + Hρ1 (z ),
if 0 < x ≤ k, if k < x ≤ z , if x > z .
Therefore, we have that for z > k, E [D(z )] =
∞
E [D(z , x)] dG(x) 0
k = Hρ0 (k) Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k) + Hρ1 (z ) − (F ∗ Hρ1 )(z ; x) dG(x) 0
− Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k)
k
Hρ0 (k − x) dG(x) − 0
z
Hρ1 (z − x) dG(x) k
k (F ∗ Hρ1 )(z ; x) dG(x) = (m/ρ0 )Hρ′ 0 (k) Hρ1 (z − k) − (F ∗ Hρ1 )(z ; k) + Hρ1 (z ) − 0
z
Hρ1 (z − x) dG(x).
− k
(9)
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Now, the limiting probability density function f (z ) of the virtual waiting time process is calculated by Brill and Posner (1977) as follows: f (z ) =
E [D(z )] E [τ ] + m/ρ0
,
z > 0,
and the limiting distribution has a mass 1 − Eqs. (5) and (9).
∞ 0
f (z )dz at 0. The formula for E [τ ] is given in Theorem 1, and E [D(z )] in
2.3. The M /G/1 queue with impatient customers only We present the result in the special case where all customers are impatient, that is q = 0. The following are consistent with Kim et al. (2001) and Bae et al. (2001). First, we have
k Hρ0 (u) du x + ρ 0 k−k x E [τ (x)] = x + ρ 0 Hρ0 (u) du
if 0 < x ≤ k, if x > k,
0
and E [τ ] = mHρ0 (k). For 0 < z ≤ k, we get E [D(z )] = (m/ρ0 ) Hρ′ 0 (z ). Note that when q = 0, Hρ1 (x) ≡ 1,
(F ∗ Hρ1 )(z ; x) = 1 − P (z − k, x).
˜ and Hρ0 gives us Therefore, for z > k, some calculus on G, G, ˜ ρ0 ,z −k )′ (k). E [D(z )] = E [D(k + s)] = (m/ρ0 ) (Hρ0 ∗ G Hence, the limiting probability density function of the virtual waiting time process is given by
f (z ) =
Hρ′ 0 (z )
1 + ρ0 Hρ0 (k) ′ ˜ (Hρ0 ∗ Gρ0 ,z −k ) (k)
1 + ρ0 Hρ0 (k)
if 0 < z ≤ k, if z > k,
and the mass at z = 0 is ∞
f (z ) dz =
1− 0
1 1 + ρ0 Hρ0 (k)
,
which is equal to the expected idle period divided by the sum of the expected idle period and busy period. 3. The M /G /1 queue with patient and n types of impatient customers 3.1. The expected busy period and the number of downcrossings We let E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] be the expected busy period of the queue given that Z0 = x, and also let E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )] be the expected number of {Zt } downcrossings of level z during a busy period given that Z0 = x. In the previous section, we have obtained E [τ (x; k1 ; ρ0 , ρ1 )] and E [D(z , x; k1 ; ρ0 , ρ1 )]. In this section, it is provided the recursive method of deriving E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] and E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )]. Assume that E [τ (x; k1 , . . . , kj ; ρ0 , . . . , ρj )] – the expected busy period of the queue with patient and j types of impatient customers – is known for j = n − 1. When j = n, for x > k1 the busy period of the queue with {Zt } starting at x is equal in distribution to the sum of the time for reaching k1 of Zt and the busy period starting at k1 . Hence, E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )], abbreviated by E [τ (x)], can be expressed as E [τ (x)] = E [τ (x − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] + E [τ (k1 )].
(10)
J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
For 0 < x ≤ k1 , by conditioning on Z k
E [τ (x)] = E [T0 1 (x; ρ0 )] +
k
T0 1 (x;ρ0 )
381
, E [τ (x)] is represented as
∞
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ) k1
+ E [τ (k1 )] Pr{ZT k1 (x;ρ ) > k1 } 0 0 ∞ k E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ) = k1
Hρ0 (k1 − x)
+ Notice that Z
Hρ0 (k1 )
k
T0 1 (x;ρ0 )
E [τ (k1 )] =
Hρ0 (u)du −
k 1 −x
Hρ0 (u)du + E [τ (k1 )] 1 −
0
0
Hρ0 (k1 − x)
Hρ0 (k1 )
.
has a mass Hρ0 (k1 − x)/Hρ0 (k1 ) at 0. Putting x = k1 gives us k1
k1
Hρ0 (u)du + Hρ0 (k1 )
∞
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ). k1
0
Using this, we get that for 0 < x ≤ k1 E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] k1
=
Hρ0 (u)du −
k1 −x
0
Hρ0 (u)du +
∞
0
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ) k1
+ (Hρ0 (k1 ) − Hρ0 (k1 − x))
∞
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ), k1
and from Eq. (10), for x > k1 E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] = E [τ (x − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] +
k1
Hρ0 (u)du + Hρ0 (k1 )
0
∞
k
E [τ (y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ).
×
(11)
k1
Therefore, E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] can be obtained inductively. E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )], simply E [D(z , x)], is also derived similarly to E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )]. First, observe that for 0 < z ≤ kn and x > 0 E [D(z , x)] = E [D(z , x; k1 , . . . , kn−1 ; ρ0 , . . . , ρn−1 )]. When z > kn and x > k1 , E [D(z , x)] = E [D(z − k1 , x − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] + E [D(z , k1 )],
(12)
and when z > kn and 0 < x ≤ k1 , E [D(z , x)] =
∞
k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ) k1
+ E [D(z , k1 )] 1 −
Hρ0 (k1 − x) Hρ0 (k1 )
.
To obtain E [D(z , k1 )], we let x = k1 in the above equation, then E [D(z , k1 )] = Hρ0 (k1 )
∞ k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ). k1
Therefore, it follows that for z > kn and 0 < x ≤ k1 E [D(z , x)] = (Hρ0 (k1 ) − Hρ0 (k1 − x))
∞ k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ) k1
∞
+
k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; x, ρ0 ),
(13)
k1
and from Eq. (12), for z > kn and x > k1 E [D(z , x)] = E [D(z − k1 , x − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )]
+ Hρ0 (k1 )
∞ k
E [D(z − k1 , y − k1 ; k2 − k1 , . . . , kn − k1 ; ρ1 , . . . , ρn )] dy F0 1 (y; k1 , ρ0 ). k1
Hence, E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )] is derived by using the induction hypothesis.
(14)
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Finally, if we denote the busy period of the queue by τ (k1 , . . . , kn ; ρ0 , . . . , ρn ) and the number of {Zt } downcrossings of level z during a busy period by D(z ; k1 , . . . , kn ; ρ0 , . . . , ρn ), the limiting distribution of the virtual waiting time process {Zt } has the density function f (z ) given by f (z ) =
∞
E [D(z ; k1 , . . . , kn ; ρ0 , . . . , ρn )] E [τ (k1 , . . . , kn ; ρ0 , . . . , ρn )] + m/ρ0
and has a mass 1 −
∞ 0
= ∞ 0 0
E [D(z , x; k1 , . . . , kn ; ρ0 , . . . , ρn )] dG(x)
E [τ (x; k1 , . . . , kn ; ρ0 , . . . , ρn )] dG(x) + m/ρ0
,
z > 0,
f (z )dz at 0.
4. The M /M /1 queue with patient and n types of impatient customers In this section, we provide the explicit formulas of the expected busy period and the limiting distribution when the service time of a customer is exponentially distributed. Let G(x) = 1 − e−x/m and let θi = (1 − ρi )/m. Then it can be shown that Hρi (x) =
1 1 − ρi
−
ρi 1 − ρi
e−θi x ,
i = 0 , 1 , 2 , . . . , n.
We begin from the case n = 1. From Eqs. (3), (4) and Theorem 1, we have
E [τ (x; k1 ; ρ0 , ρ1 )] =
x
1 − ρ0 k1
mρ0 (ρ0 − ρ1 )
−
e−θ0 (k1 −x) − e−θ0 k1
(1 − ρ0 ) (1 − ρ1 ) x − k1 mρ0 (ρ0 − ρ1 ) + − 1 − e−θ0 k1 1 − ρ0 1 − ρ1 (1 − ρ0 )2 (1 − ρ1 ) 2
if 0 < x ≤ k1 , (15) if x > k1 ,
and E [τ (k1 ; ρ0 , ρ1 )] = m
1 1 − ρ0
−
ρ0 − ρ1 e−θ0 k1 . (1 − ρ0 )(1 − ρ1 )
When 0 < z ≤ k1 , the expected number of downcrossings is given by
ρ −θ (z −x) 0 e 0 − e−θ0 z 1 − ρ0 E [D(z , x; k1 ; ρ0 , ρ1 )] = 1 ρ0 −θ0 z − e 1 − ρ0 1 − ρ0
if 0 < x ≤ z , (16)
if x > z ,
and E [D(z ; k1 ; ρ0 , ρ1 )] = e−θ0 z . When z > k1 ,
ρ0 (eθ0 x − 1) e−θ0 k1 −θ1 (z −k1 ) , 1 − ρ 0 ρ0 ρ1 (eθ0 k1 − 1) e−θ0 k1 −θ1 (z −k1 ) + (eθ1 (x−k1 ) − 1) e−θ1 (z −k1 ) , E [D(z , x; k1 ; ρ0 , ρ1 )] = 1 − ρ0 1 − ρ1 ρ0 1 ρ1 (eθ0 k1 − 1) e−θ0 k1 −θ1 (z −k1 ) + − e−θ1 (z −k1 ) , 1 − ρ0 1 − ρ1 1 − ρ1
if 0 < x ≤ k1 , if k1 < x ≤ z , (17) if x > z ,
and E [D(z ; k1 ; ρ0 , ρ1 )] = e−θ0 k1 −θ1 (z −k1 ) . Consider the case n = 2. After obtaining E [τ (x − k1 ; k2 − k1 ; ρ1 , ρ2 )] from Eq. (15) and calculating E [τ (x; k1 , k2 ; ρ0 , ρ1 , ρ2 )] by using Eq. (11), we get E [τ (k1 , k2 ; ρ0 , ρ1 , ρ2 )] = m
1 1 − ρ0
−
ρ0 − ρ1 ρ1 − ρ2 e−θ0 k1 − e−θ0 k1 −θ1 (k2 −k1 ) , (1 − ρ0 )(1 − ρ1 ) (1 − ρ1 )(1 − ρ2 )
and the expected number of downcrossings is given by E [D(z ; k1 , k2 ; ρ0 , ρ1 , ρ2 )] =
−θ z e 0
e−θ0 k1 −θ1 (z −k1 ) −θ0 k1 −θ1 (k2 −k1 )−θ2 (z −k2 ) e
if 0 < z ≤ k1 , if k1 < z ≤ k2 , if z > k2 ,
where the last formula of the above is derived by using Eqs. (13), (14), (16), and (17).
J. Bae / Journal of the Korean Statistical Society 42 (2013) 375–385
383
Table 1 Expected busy periods. n=0
n=1
n=2
n=3
10.0000
3.9678
3.0740
2.8417
In general, the formulas for the expected busy period and the expected number of downcrossings are given as follows:
E [τ (k1 , k2 , . . . , kn ; ρ0 , ρ1 , . . . , ρn )] = m
1 1 − ρ0
−
n i=1
ρi−1 − ρi − ij=1 θj−1 (kj −kj−1 ) e , (1 − ρi−1 )(1 − ρi )
and E [D(z ; k1 , k2 , . . . , kn ; ρ0 , ρ1 , . . . , ρn )] = e
−θi (z −ki )−
i
j=1 θj−1 (kj −kj−1 )
for ki < z ≤ ki+1 , i = 0, 1, 2, . . . , n with k0 = 0 and kn+1 = ∞, whose proofs can be done by mathematical induction on n. Hence, the limiting distribution of the virtual waiting time process has the density function −θl (z −kl )−
e
f (z ) =
m
1 ρ0 (1−ρ0 )
−
l
θj−1 (kj −kj−1 )
j=1
n i=1
ρi−1 −ρi − e (1−ρi−1 )(1−ρi )
i
j=1 θj−1 (kj −kj−1 )
for kl < z ≤ kl+1 , l = 0, 1, 2, . . . , n, and has a mass 1/ρ0
∞
f (z ) dz =
1− 0
1
ρ0 (1−ρ0 )
−
n i=1
ρi−1 −ρi − e (1−ρi−1 )(1−ρi )
i
j=1 θj−1 (kj −kj−1 )
at z = 0. The mass at z = 0 is equal to the expected idle period divided by the sum of the expected idle period and busy period. 5. Illustrations In this section, some illustrations are given to show the effects of the impatience of customers on the busy period and the limiting distribution of the virtual waiting time process. We consider some M /M /1 queues with several cases of impatience as follows:
• • • •
n n n n
= 0; ρ0 = 0.9 (ordinary M /M /1 queue) = 1; k1 = 1; ρ0 = 0.9, ρ1 = 0.7 = 2; k1 = 1, k2 = 2; ρ0 = 0.9, ρ1 = 0.7, ρ2 = 0.5 = 3; k1 = 1, k2 = 2, k3 = 3; ρ0 = 0.9, ρ1 = 0.7, ρ2 = 0.5, ρ3 = 0.3.
It is assumed that the expected service time m of a customer is 1. Note that θi = (1 − ρi )/m = 1 − ρi , i = 0, 1, 2, . . . , n. We can observe in Table 1 that the impatience of customers reduces the expected busy period and in Figs. 1 and 2 that the more impatient the customers are, the shorter the virtual waiting times are stochastically. 6. Conclusion We consider the virtual waiting time process of the M /G/1 queue with patient customers and impatient customers, in which the customers are classified into n + 1 classes by their impatience times. It has been obtained the limiting distribution of the virtual waiting time process by deriving the formulas for the expected busy period and the expected number of process downcrossings of any level z. We have studied the case of n = 1 first and generalized the result to the case of n > 1 by an inductive argument. The research will be extended to the case of n = ∞, and to the case that the impatience time is any random variable. Acknowledgments We would like to thank the anonymous referees for valuable comments, which result in a great improvement of the paper. This study was financially supported by research fund of Chungnam National University in 2010.
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Fig. 1. The limiting cdf’s of the virtual waiting time processes.
Fig. 2. The limiting pdf’s of the virtual waiting time processes.
Appendix When ρ > 0, we derive the formula for G ∗ Hρ (x) as follows:
x
Hρ (x − u) dG(u) = 0
x
G(x − u) dHρ (u)
0−
= Hρ (x) −
x
1 − G(x − u) dHρ (u)
0−
= Hρ (x) − m
x
G′e (x − u) dHρ (u)
0−
= Hρ (x) − m
d dx
x
Ge (x − u) dHρ (u) noting that Ge (0) = 0, 0−
′ = Hρ (x) − m Hρ (x) − 1 /ρ = Hρ (x) − (m/ρ)Hρ′ (x). References Bacelli, F., Boyer, P., & Hebuterne, G. (1984). Single-server queues with impatient customers. Advances in Applied Probability, 16, 887–905. Bae, J., & Kim, S. (2010). The stationary workload of the M /G/1 queue with impatient customers. Queueing Systems, 64, 253–265.
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