- Email: [email protected]
s queue with impatient calls
Performance Evaluation 35 (1999) 1–18
On the M(n)/M(n)/s queue with impatient calls Andreas Brandt a,∗ , Manfred Brandt b,1 a
Wirtschaftswissenschaftliche Fakultät, Humboldt-Universität zu Berlin, Spandauer Str. 1, D-10178 Berlin, Germany b Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustr. 7, D-14195 Berlin, Germany Received 25 June 1997; received in revised form 28 August 1998
Abstract The paper is concerned with the analysis of an s-server queueing system wherein the calls may leave the system due to impatience. The individual maximal waiting times are assumed to be i.i.d. and arbitrarily distributed. The arrival and cumulative service rates may depend on the number n of calls in the system, but the service rate is assumed to be constant for n > s. For this system, denoted by M(n)/M(n)/s + GI , we derive a system of integral equations for the vector of the residual maximal waiting times of the waiting calls and their original maximal waiting times. By solving these equations explicitly we obtain the stability condition and for the steady state of the system, the occupancy distribution and various waiting time distributions. The results are also new for special cases analyzed in earlier papers. As an application of the M(n)/M(n)/s + GI system we give a performance analysis of an automatic call distributor system (ACD system) of finite capacity with outbound and impatient inbound calls; numerical results are given for the case of maximal waiting times as the minimum of constant and exponentially distributed times. ©1999 Elsevier Science B.V. All rights reserved. Keywords: M(n)/M(n)/s queue with impatient calls; State dependent arrivals and services; Finite capacity; Integral equations; Occupancy distribution; Waiting time distribution; ACD system
1. Introduction In this paper we consider an s-server queueing system with an unlimited waiting room with FCFS queueing discipline and where the calls waiting in the queue for service are impatient, cf. Fig. 1. The arrival and service processes are allowed to be state dependent with respect to the number n of calls in the system, but the cumulative service rate is assumed to be constant for n > s. We assume that the sequence of the arrival rates λn is bounded and that λn > 0 for n ≥ 0 or that there exists a positive integer k such that λn > 0 for 0 ≤ n < s + k and λn ≡ 0 for n ≥ s + k. Concerning the cumulative rate µn of finishing service we assume that µ0 = 0 and µn ≡ µ∗ > 0 for n > s. Each call arriving at the system has a maximal waiting time I . If the offered waiting time W o (i.e., the time which a call would have to ∗ 1
Corresponding author. E-mail: [email protected] E-mail: [email protected].
0166-5316/99/$ – see front matter ©1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 9 8 ) 0 0 0 4 2 - X
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
Fig. 1. The M(n)/M(n)/s + GI system with impatient calls and state dependent arrival and service rates, where n denotes the number of calls in the system, W o the offered waiting time and I the maximal waiting time.
wait for accessing a server if it were sufficiently patient) exceeds I , then the call departs from the system after having waited time I . The maximal waiting times are assumed to be i.i.d. with a general distribution C(u) := P (I ≤ u), u ∈ R+ which may be defective, i.e., P (I = ∞) > 0 is not excluded. This system is denoted by M(n)/M(n)/s + GI , where the first M(n) denotes the arrival process, the second M(n) the service process depending on min(n, s + 1) only, i.e., on the number of busy servers and additionally whether there are calls waiting for service. The symbol GI stands for the i.i.d. maximal waiting times. Remark 1.1. Note that if λn > 0 for 0 ≤ n < s + k and λn ≡ 0 for n ≥ s + k, then we have the case of a limited waiting room with k waiting places. In case of λn ≡ λ > 0, µn = min(n, s)µ for n ≥ 0 the model corresponds to a M/M/s + GI system, cf. [2]. The more general case λn ≡ λ > 0 for n ≥ s and µn = min(n, s)µ for n ≥ 0 is treated in [17]. Relations to the results of Baccelli and Hebuterne [2] and Jurkevi˘c [17], which are most relevant papers to ours, and further references are given below. The paper is organized as follows. In Section 2 a system of integral equations for the density of the stationary vector process of the number of calls, residual maximal waiting times and original maximal waiting times of calls waiting for service in the system is derived. By a separation approach corresponding to partial balance, the integral equations are solved explicitly; the densities are of an elementary structure. In Section 3 we derive the distribution of the number of calls in the system (occupancy distribution), the stability condition and performance measures such as the impatience probability, the cumulative arrival rate and the waiting time distributions of the served calls as well as of the calls leaving due to impatience. In Section 4 an application of the results to a performance analysis of an automatic call distributor system (ACD system) with outbound calls and impatient inbound calls is given. In case of maximal waiting times as the minimum of constant and exponentially distributed times some numerical results are presented. The results of the paper can also be applied to more complicated ACD systems directly (e.g. to ACD systems with integrated voice-mail-server) or indirectly for constructing system approximations. A number of papers have dealt with impatience phenomena. It seems that Palm [20] was the first to deal with the impatience problem. Barrer [3,4] analyzed the M/M/1+D system. Brodi [7,8] derived and solved for the M/M/1 + D system the corresponding integro differential equation. The general GI /GI /1 + GI system is treated in [9]. The many-server M/M/s +D system was analyzed by Gnedenko and Kowalenko [13] by giving an explicit solution of the system of integro differential equations for the work load of the s servers, which yields formulae for the performance measures. This method was successfully further applied by Jurkevi˘c [16] to maximal waiting times as the minimum of a constant and an exponentially distributed time and [17] for the general M/M/s + GI system where the arrival intensities may depend on the number of busy servers (cf. also [12, p. 270]). Later, independently Haugen and Skogan [15] and Baccelli and Hebuterne [2] derived results for the M/M/s + GI system, too. Haugen and Skogan [15] started with a result of Wallstrøm [26], cf. also [14], for an s-server system with several Poisson input streams where each call type has an individual constant maximal waiting time and where the calls in the
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system (waiting and served) are exponentially impatient. By an appropriate limiting construction they obtained a formula for the waiting time distribution for the M/M/s + GI system and a generalization of this by allowing departures for the case of a limitation of the total time spent in the system (queue and server). In [2] the Kolmogorov equations are derived and solved for the virtual offered waiting time of a call. By means of this quantity formulae for the relevant performance measures are derived. However, the distribution of the total number of calls in the M/M/s + GI system and in the more general model in [17] was not treated in the references mentioned above. Thus, the results concerning the occupancy distribution given in Section 3 are new – as far as we know – also for the special case of an M/M/s + GI system (treated in [2]) and of the model in [17]. Note that our model includes the case of a finite waiting room, whereas this case is excluded in the models mentioned above. In the literature, there are several other known mechanisms where calls leave the system due to impatience: If the call can calculate its prospective waiting time at its arrival instant, then it leaves immediately if this time exceeds its maximal waiting time. This strategy yields a better utilization of the waiting places because they will not be occupied by calls which later abandon due to impatience. Also impatience may act on the sojourn times (waiting time plus service time). In this case not all work is useful because a call may leave the system due to impatience during its service. For references and other more general models with impatience mechanism we refer to [1,2,18,19,23,25] and the references therein.
2. A system of integral equations and its solution Throughout this section we assume that the queueing system is stable and that the distribution C(u) of the maximal waiting times is non-defective and has a continuous density c(u). If n calls are in the system then l := (n − s)+ calls are waiting in the queue for service. (The notation l := (n − s)+ will be used also in the following.) We number the waiting calls according to their positions in the queue. By the FCFS discipline the first call in the queue will be potentially the next for service. Let N (t) L(t) := (N (t) − s)+ (X1 (t), . . . , XL(t) (t)) (I1 (t), . . . , IL(t) (t))
number of calls in the system at time t number of waiting calls at time t vector of the residual maximal waiting times of waiting calls ordered according to their positions in the queue at time t vector of the original maximal waiting times of the waiting calls ordered according to their positions in the queue at time t stationary distribution of the number of calls in the system
p(n) := P (N (t) = n) P (n; x1 , . . . , xl ; u1 , . . . , ul ) := P (N (t) = n; X1 (t) ≤ x1 , . . . , Xl (t) ≤ xl ; I1 (t) ≤ u1 , . . . , Il (t) ≤ ul ) stationary distribution on N(t) = n, where l := (n − s)+ . Obviously, for fixed n > s the support of P (n; x1 , . . . , xl ; u1 , . . . , ul ) is contained in Ωl := {(x1 , . . . , xl ; u1 , . . . , ul ) ∈ R2l + : u1 − x1 ≥ · · · ≥ ul − xl ≥ 0}.
(2.1)
In view of the assumptions on C(u) the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) :=
∂ 2l P (n; x1 , . . . , xl ; u1 , . . . , ul ) ∂x1 · · · ∂xl ∂u1 · · · ∂ul
(2.2)
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
is continuous on Ωl ; from a representation of this density given at the end of this section its continuity on Ωl follows, too. In case of n ≤ s we have the balance equations (λn + µn )p(n) = 1{n > 0}λn−1 p(n − 1) + µn+1 p(n + 1), n = 0, 1, . . . , s − 1, Z Z p(s + 1; 0; u) du + µ∗ p(s + 1; x; u) dx du. (λs + µs )p(s) = λs−1 p(s − 1) + R+
R2+
(2.3) (2.4)
In view of µ0 = 0 by summing the first n + 1 equations of (2.3) we obtain λn p(n) = µn+1 p(n + 1),
n = 0, 1, . . . , s − 1.
Therefore, (2.4) is equivalent to Z Z p(s + 1; 0; u) du + µ∗ λs p(s) = R+
R2+
(2.5)
p(s + 1; x; u) dx du.
(2.6)
In case of n > s and (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl , which in view of (2.1) especially implies 0 ≤ xl ≤ ul , we have the balance conditions p(n; x1 , . . . , xl ; u1 , . . . , ul ) = p(n; x1 + h, . . . , xl + h; u1 , . . . , ul )(1 − hλn − hµ∗ ) l+1 Z X p(n + 1; x1 , . . . , xi−1 , 0, xi , . . . , xl ; u1 , . . . , ui−1 , u, ui , . . . , ul ) du +h i=1
Z
+hµ∗
R+
R2+
p(n + 1; x, x1 , . . . , xl ; u, u1 , . . . , ul ) dx du + o(h),
h > 0, xl < ul ,
p(n; x1 , . . . , xl−1 , ul ; u1 , . . . , ul ) = λn−1 p(n − 1; x1 , . . . , xl−1 ; u1 , . . . , ul−1 )c(ul ).
(2.7) (2.8)
In the following from (2.7) and (2.8) we shall derive an equivalent system of integral equations for the density p(n; x1 , . . . , xl ; u1 , . . . , ul ). Let n > s and (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl with xl < ul be fixed. For t ∈ [0, ul − xl ] we define Z ∞ p(n; x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul ) ϕ(t) := p(n; x1 + t, . . . , xl + t; u1 , . . . , ul )e−µ∗ t + λn ×e−µ∗ ξ dξ −
l+1 Z X i=1
t
∞Z t
R+
p(n + 1; x1 + ξ, . . . , xi−1 + ξ, 0, xi + ξ, . . . , xl + ξ ;
u1 , . . . , ui−1 , u, ui , . . . , ul )e−µ∗ ξ dudξ − µ∗
Z t
∞Z
R2+
p(n + 1; x, x1 + ξ, . . . , xl + ξ ;
u, u1 , . . . , ul )e−µ∗ ξ dx du dξ. By continuity of the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) on Ωl the function ϕ(t) is also continuous, and for t ∈ [0, ul − xl ), h ∈ (0, ul − xl − t] by some algebra and applying (2.7) at the point (x1 + t, . . . , xl + t; u1 , . . . , ul ) we find ϕ(t + h) − ϕ(t) = o(h), t ∈ [0, ul − xl ), h ∈ (0, ul − xl − t]. By continuity of ϕ(t)
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
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therefore we conclude that ϕ(t) is constant, especially it follows the relation ϕ(0) = ϕ(ul − xl ). In view of the definition of ϕ(t), since the support of the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) is contained in Ωl and because of the boundary condition (2.8) thus we obtain the following system of integral equations for n > s and (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl : Z p(n; x1 , . . . , xl ; u1 , . . . , ul ) + λn
R+
p(n; x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul )e−µ∗ ξ dξ
= λn−1 p(n − 1; x1 + ul − xl , . . . , xl−1 + ul − xl ; u1 , . . . , ul−1 )c(ul )e−µ∗ (ul −xl ) l+1 Z X + p(n + 1; x1 + ξ, . . . , xi−1 + ξ, 0, xi + ξ, . . . , xl + ξ ; u1 , . . . , ui−1 , u, ui , . . . , ul ) i=1
×e
−µ∗ ξ
R2+
Z
du dξ + µ∗
R3+
p(n + 1; x, x1 + ξ, . . . , xl + ξ ; u, u1 , . . . , ul )e−µ∗ ξ dx du dξ.
(2.9)
On the other hand, from the system of integral equations (2.9) the balance conditions (2.7) and (2.8) may be derived. In the following we shall solve the system of equations (2.5),(2.6) and (2.9). From (2.5) we find p(n) = g
n−1 Y
! λi
i=0
s Y
! µi ,
n ≤ s,
(2.10)
i=n+1
where g > 0 is a normalizing factor. Remark 2.1. In case of µn > 0 for n = 1, 2, . . . , s the representation p(n) = g ∗
n−1 Y
λi , µ i+1 i=0
g∗ = g
s Y
µi ,
i=1
is possible for n ≤ s, too, being more closely to the birth death process notations. In view of the representation (2.10) of p(n) for n ≤ s, in case of n ≥ s we choose the substitution p(n; x1 , . . . , xl ; u1 , . . . , ul ) = g
n−1 Y
! λi q(n; x1 , . . . , xl ; u1 , . . . , ul ).
(2.11)
i=0
This substitution is verified by the fact that obviously we have p(n; x1 , . . . , xl ; u1 , . . . , ul ) ≡ 0 in case of λ0 · · · λn−1 = 0. From (2.10) and (2.11) it follows q(s) = 1. Hence, Eq. (2.6) now reads Z
Z 1=
R+
q(s + 1; 0; u) du + µ∗
R2+
q(s + 1; x; u) dx du,
and for n > s, (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl the system of integral equations (2.9) reads
(2.12)
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
Z
q(n; x1 , . . . , xl ; u1 , . . . , ul ) + λn
R+
q(n; x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul )e−µ∗ ξ dξ
= q(n − 1; x1 + ul − xl , . . . , xl−1 + ul − xl ; u1 , . . . , ul−1 )c(ul )e−µ∗ (ul −xl ) l+1 Z X +λn q(n + 1; x1 + ξ, . . . , xi−1 + ξ, 0, xi + ξ, . . . , xl + ξ ; u1 , . . . , ui−1 , u, ui , . . . , ul ) R2+
i=1
×e
−µ∗ ξ
Z
du dξ + λn µ∗
R3+
q(n + 1; x, x1 + ξ, . . . , xl + ξ ; u, u1 , . . . , ul )e−µ∗ ξ dx du dξ. (2.13)
In case of a finite waiting room with k waiting places, i.e., λn > 0 for 0 ≤ n < s + k and λn ≡ 0 for n ≥ s + k, Eqs. (2.13) are relevant only for s < n ≤ s + k. The structure of (2.12) and (2.13) suggests to conjecture that q(n; x1 , . . . , xl ; u1 , . . . , ul ) is independent on λn . Assuming this independence, then for n > s, (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl from (2.13) it follows q(n; x1 , . . . , xl ; u1 , . . . , ul ) = q(n − 1; x1 + ul − xl , . . . , xl−1 + ul − xl ; u1 , . . . , ul−1 )c(ul )e−µ∗ (ul −xl ) , Z R+
(2.14)
q(n; x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul )e−µ∗ ξ dξ
=
l+1 Z X 2 i=1 R+
×e
−µ∗ ξ
q(n + 1; x1 + ξ, . . . , xi−1 + ξ, 0, xi + ξ, . . . , xl + ξ ; u1 , . . . , ui−1 , u, ui , . . . , ul ) Z du dξ + µ∗
R3+
q(n + 1; x, x1 + ξ, . . . , xl + ξ ; u, u1 , . . . , ul )e−µ∗ ξ dx du dξ. (2.15)
In view of q(s) = 1 and of the definition (2.1) of Ωl from (2.14) by induction over n > s we get ! l Y c(ui ) e−µ∗ (u1 −x1 ) , n > s. q(n; x1 , . . . , xl ; u1 , . . . , ul ) = 1{(x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl } i=1
(2.16)
Since the distribution C(u) of the maximal waiting times is non-defective by integration it follows that the function q(s + 1; x1 ; u1 ) defined by (2.16) satisfies (2.12). Moreover, using the convention xl+1 = ul+1 = 0 for the functions q(n; x1 , . . . , xl ; u1 , . . . , ul ) defined by (2.16) for n > s, (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl we obtain l+1 Z X q(n + 1; x1 , . . . , xi−1 , 0, xi , . . . , xl ; u1 , . . . , ui−1 , u, ui , . . . , ul ) du i=1
R+
Z
+µ∗ ×
R2+
l Y i=1
q(n + 1; x, x1 , . . . , xl ; u, u1 , . . . , ul ) dx du = 1{(x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl }
c(ui )
! Z
∞
u1 −x1
c(u)e
Z
+µ∗
∞
−µ∗ u
Z
u1 −x1 0
du + e
−µ∗ (u1 −x1 )
l+1 Z X i=2
u−(u1 −x1 )
c(u)e
−µ∗ (u−x)
ui−1 −xi−1
ui −xi
c(u) du
dx du = q(n; x1 , . . . , xl ; u1 , . . . , ul ).
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
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From this equation applied at the point (x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul ) it follows that the functions q(n; x1 , . . . , xl ; u1 , . . . , ul ) defined by (2.16) also satisfy (2.15). Hence, these functions solve the system of integral equations (2.12) and (2.13). Since the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) is uniquely determined as the normalized solution of (2.7) and (2.8) this density is given by (2.11) and (2.16). Summarizing we find the representation p(n; x1 , . . . , xl ; u1 , . . . , ul ) = 1{(x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl } g
n−1 Y i=0
! λi
l Y
! c(ui ) e−µ∗ (u1 −x1 ) ,
n > s.
(2.17)
i=1
Remark 2.2. The successful separation of (2.13) into (2.14) and (2.15) corresponds to the fact that for n > s and (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl the partial balance equations p(n; x1 , . . . , xl ; u1 , . . . , ul )=p(n; x1 +(ul −xl ), . . . , xl +(ul −xl ); u1 , . . . , ul )e−µ∗ (ul −xl ) ,
(2.18)
λn p(n; x1 , . . . , xl ; u1 , . . . , ul ) l+1 Z X p(n + 1; x1 , . . . , xi−1 , 0, xi , . . . , xl ; u1 , . . . , ui−1 , u, ui , . . . , ul ) du = i=1
R+
Z
+µ∗
R2+
p(n + 1; x, x1 , . . . , xl ; u, u1 , . . . , ul ) dx du,
(2.19)
hold. Eq. (2.18) means that the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) on the condition that no new service has been started since the last arrival instant only depends on time (by changes of the residual maximal waiting times). Eq. (2.19) means that the intensity of transitions from the state (n; x1 , . . . , xl ; u1 , . . . , ul ) into any state (n + 1; ·; ·) (by an arrival) equals the intensity of transitions from any state (n + 1; ·; ·) into the state (n; x1 , . . . , xl ; u1 , . . . , ul ) (by leaving the queue due to impatience or for starting service). Obviously, from the partial balance equations (2.18) and (2.19) and from the boundary condition (2.8) it follows that the system of integral equations (2.9) holds.
3. Stability condition, occupancy and waiting time distribution As in Section 2 let us assume that the queueing system is stable and that the distribution C(u) of the maximal waiting times is non-defective and has a continuous density c(u). The stationary probability p(n) that n calls are in the system is given by (2.10) for n ≤ s. In case of n > s from (2.17) it follows Z p(n; x1 , . . . , xl ; u1 , . . . , ul ) dx1 · · · dxl du1 · · · dul p(n) = R2l +
=g
n−1 Y i=0
!Z λi
Ωl
l Y i=1
! c(ui ) e−µ∗ (u1 −x1 ) dx1 · · · dxl du1 . . . dul ,
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
where l := (n − s)+ . In view of the definition (2.1) of Ωl the substitution ui = ξi + xi yields ! !Z n−1 l Y Y λi 1{ξ1 ≥ · · · ≥ ξl } c(ξi + xi ) e−µ∗ ξ1 dx1 · · · dxl dξ1 · · · dξl p(n) = g i=0 n−1 Y
=g
! λi
i=0
where
Z
ξ/µ∗
F (ξ ) :=
R2l +
1 (n − s)!
i=1
Z
∞
F (ξ )n−s e−ξ dξ,
n > s,
(3.1)
0
(1 − C(η)) dη,
ξ ∈ R+ .
(3.2)
0
Because of Eqs. (2.10) and (3.1) the normalizing factor g is given by ! s ! Z −1 s+j −1 s−1 jY ∞ X Y X Y 1 ∞ −1 g = λi µi + λi F (ξ )j e−ξ dξ. j ! 0 j =0 i=0 i=j +1 j =0 i=0
(3.3)
Obviously, the stability of the system is equivalent to the finiteness of the right-hand side of (3.3). Therefore, the stability condition reads ! Z s+j −1 ∞ X Y 1 ∞ λi F (ξ )j e−ξ dξ < ∞. (3.4) j ! 0 j =0 i=0 The case of a general distribution C(u) of the maximal waiting times is obtained by considering C(u) as the limit in distribution of a sequence of non-defective distributions Cν (u) with continuous density. From (3.4),(2.10) and (3.1) applied to Cν (u) by arguments of continuity we obtain the following statement. Theorem 3.1. Let the maximal waiting times be i.i.d. with a general distribution C(u). Then, the system is stable iff (3.4) is fulfilled. In case of a stable system for the stationary occupancy distribution it holds ! ! n−1 s Y Y λi µi , n = 0, 1, . . . , s, p(n) = g i=0
p(n) = g
n−1 Y
! λi
i=0
i=n+1
1 (n − s)!
Z
∞
F (ξ )n−s e−ξ dξ,
n = s + 1, s + 2, . . . ,
(3.5)
0
where the normalizing factor g is given by (3.3) and F (ξ ) by (3.2). Let the system be stable, i.e., let (3.4) be satisfied. The cumulative arrival intensity Λ of the calls in the steady state is given by Λ=
∞ X λn p(n).
(3.6)
n=0
Since the sequence of the arrival intensities λn is bounded, Λ is finite. For the investigation of the various waiting time distributions of a typical arriving call we need some notation:
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9
pW pI
probability that a typical arriving call has to wait for service probability that a typical arriving call will leave the system due to impatience later WS (x) = P (WS ≤ x) distribution function of the waiting time WS of a typical arriving call on the condition that it will be served WI (x) = P (WI ≤ x) distribution function of the waiting time WI of a typical arriving call on the condition that it will leave the system due to impatience later W (x) = P (W ≤ x) distribution function of the (unconditional) waiting time W of a typical arriving call In the following we will make use of the Palm-distribution with respect to arrival epochs of calls (cf., e.g., [11, (1.2.6)]) and of the conservation principle for stationary point processes with respect to arrival epochs of calls and epochs where the calls leave the waiting room, respectively. (If a call finds at its arrival a free server, then it leaves the queue immediately, i.e., its arrival epoch coincides with the epoch of leaving the queue.) The corresponding rigorous proofs are not outlined here because the results are well-known. (However, exact proofs can be given, e.g., by means of Campbell’s formula along the lines as, e.g., in [6, Section 6.4, (6.4.1)].) The probability 1 − pW is given by the ratio of the arrival intensity of calls finding at their arrival a free server and of the total arrival intensity Λ, i.e., s−1
1 − pW =
s
1X 1X λn p(n) = µn p(n). Λ n=0 Λ n=1
(3.7)
Note, that by the intensity conservation principle we have that the intensity of calls finding at least one free server at their arrival equals the intensity of served calls leaving behind at least one free server. The latter intensity is just the last sum. The probability 1 − pI is given by the ratio of the intensity λ(S) of arriving calls which will be served (immediately or after waiting in the queue) and of the total arrival intensity Λ. By the conservation principle λ(S) is the intensity of calls leaving the waiting room for starting service. This yields ! ! s−1 ∞ s ∞ X X 1 X 1 X 1 − pI = λn p(n) + µ∗ p(n) = µn p(n) + µ∗ p(n) . Λ n=0 Λ n=1 n=s+1 n=s+1
(3.8)
Note that the expression in the bracket of the last equation is just the intensity of served calls, which by the conservation principle is equal to the intensity of calls leaving the waiting room for service (immediately after their arrival or after waiting). Next, we are interested in the waiting time distributions WS (x), WI (x) and W (x) of a typical arriving call. Let us in the following assume that the queueing system is stable and that the distribution C(u) of the maximal waiting times is non-defective and has a continuous density c(u). For fixed x ∈ R+ the probability P (WS > x) = 1 − WS (x) is just the ratio of the intensity λ(S,x) of arriving calls which will be served and whose waiting times up to service are larger than x to the intensity λ(S) = (1 − pI )Λ of all arriving calls which will be served. By the intensity conservation principle λ(S,x) equals the intensity of time instants where calls leave the waiting room for starting service and which have waited longer than x. Hence, we obtain
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
1 − WS (x) =
Z ∞ X 1 µ∗ 1{x < u1 − x1 }p(n; x1 , . . . , xl ; u1 , . . . , ul ) (1 − pI )Λ n=s+1 R2l + ×dx1 · · · dxl du1 · · · dul .
Because of Eqs. (2.1),(2.17) and (3.2) and using the substitution ui = ξi + xi we find that for n > s Z 1 {x < u1 − x1 }p(n; x1 , . . . , xl ; u1 , . . . , ul ) dx1 · · · dxl du1 · · · dul R2l +
=g
n−1 Y
! λi
i=0
1 (l − 1)!
Z
∞
µ∗ x
F (ξ )l−1 F 0 (ξ )e−ξ dξ,
x ∈ R+ .
In view of (3.5) thus it follows s+j ∞ µ∗ p(s) X Y λi 1 − WS (x) = (1 − pI )Λ j =0 i=s
!
1 j!
Z
∞
µ∗ x
F (ξ )j F 0 (ξ ) e−ξ dξ,
x ∈ R+ .
(3.9)
For finding an explicit expression for WI (x) we proceed analogously to WS (x). The probability P (WI > x) = 1 − WI (x) is the ratio of the intensity λ(I,x) of arriving calls departing from the system due to impatience later and whose time spent in the queue is larger than x to the intensity λ(I ) = pI Λ of all arriving calls leaving due to impatience later. By the conservation principle λ(I,x) equals the intensity of time instants where calls leave the waiting room due to impatience and which have waited longer than x. Hence, we obtain ∞ l Z 1 XX 1{x < ui }p(n; x1 , . . . , xi−1 , 0, xi+1 , . . . , xl ; u1 , . . . , ul ) 1 − WI (x) = pI Λ n=s+1 i=1 R+2l−1
×dx1 · · · dxi−1 dxi+1 · · · dxl du1 · · · dul . In view of (2.1),(2.17) and (3.2) and using the substitution ui = ξi + xi after some algebra we find that for n > s l Z X
d − dx
2l−1 i=1 R+
1{x < ui }p(n; x1 , . . . , xi−1 , 0, xi+1 , . . . , xl ; u1 , . . . , ul ) dx1 · · · dxi−1 !
×dxi+1 · · · dxl du1 · · · dul
=g
n−1 Y
! λi
i=0
1 c(x) (l − 1)!
Z
∞
µ∗ x
F (ξ )l−1 e−ξ dξ,
x ∈ R+ ,
and by integration it follows that l Z X i=1
2l−1 R+
1{x < ui }p(n; x1 , . . . , xi−1 , 0, xi+1 , . . . , xl ; u1 , . . . , ul )dx1 · · · dxi−1 dxi+1 · · · dxl
×du1 · · · dul = µ∗ g
n−1 Y i=0
! λi
1 (l − 1)!
Z
∞
µ∗ x
F (ξ )l−1 (F 0 (µ∗ x) − F 0 (ξ )) e−ξ dξ,
x ∈ R+ .
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
11
In view of (3.5) thus we obtain s+j ∞ µ∗ p(s) X Y λi 1 − WI (x) = pI Λ j =0 i=s
!
1 j!
Z
∞
µ∗ x
F (ξ )j (F 0 (µ∗ x) − F 0 (ξ )) e−ξ dξ,
x ∈ R+ .
(3.10)
The waiting time distribution W (x) is given by W (x) = (1 − pI )WS (x) + pI WI (x),
x ∈ R+ .
(3.11)
The case of a general distribution C(u) of the maximal waiting times is obtained again by considering C(u) as the limit in distribution of a sequence of non-defective distributions Cν (u) with continuous density. From (3.9), (3.10) and (3.11) applied to Cν (u) by arguments of continuity we obtain the following statement. Theorem 3.2. Let the system be stable with a general distribution C(u) of the i.i.d. maximal waiting times. Then, for the waiting time distributions it holds (3.9), (3.10) and (3.11), where the probability p(s) that precisely s calls are in the system, the probability pI that a typical arriving call will leave the system due to impatience later and the cumulative arrival intensity Λ in the steady state are given by (3.5), (3.8) and (3.6), respectively. Especially from (3.9) and (3.10) by integration over x ∈ R+ we obtain the mean waiting times ! Z s+j −1 ∞ p(s) X Y 1 ∞ λi F (ξ )j (ξ − 1)e−ξ dξ, (3.12) E WS = (1 − pI )Λ j =1 i=s j! 0 ∞
p(s) X E WI = pI Λ j =1
!
s+j −1
Y
λi
i=s
1 j!
Z
∞
F (ξ )j (j + 1 − ξ )e−ξ dξ.
(3.13)
0
From Little’s formula for the (unconditional) mean waiting time EW of a typical arriving call we find EW =
∞ 1 X (n − s)p(n). Λ n=s+1
(3.14)
4. Application: performance analysis of an ACD system with outbound calls and impatient inbound calls For many services and businesses there is a need to match incoming calls with agents, e.g. in telebanking, teleshopping, information services, mail orders, etc. Automatic call distributor systems (ACD systems) are controlled by a software allowing to manage the incoming calls such that a group of ACD agents can handle a high volume of incoming calls. For a description of ACD systems see [10,21,22,24]. The ACD feature provides statistical reporting tools in addition to call queueing. ACD historical reports allow to identify times when incoming calls abandon after long waits in the queue because too few agents are staffed, or times where agents are idle. One might wish to use these data to determine an optimal operational strategy with respect to minimal cost (or maximal revenue) and various service quality requirements, e.g. for the abandon probability and for quantiles of waiting times. There are various operational strategies for
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
Fig. 2. Automatic call distributor system: combined inbound and outbound call center with impatient inbound calls, s agents and k waiting places, where n denotes the number of inbound and served outbound calls in the system.
managing the congestion level in such ACD systems. Based on the traffic the number of agents to meet a certain service varies in time. Also, by programming the telephone switching system, the service provider can temporarily deactivate or activate some of the available lines. As pointed out by So and Tang [24] there is in general a tradeoff between various performance measures, e.g. between the number of ‘abandoned’ and ‘blocked’ calls, i.e., calls which are rejected by the system when all lines are occupied. Thus, it is a non-trivial problem to determine the optimal operational strategy. For optimizing ACD systems, cf., e.g. [5], it is necessary to have stable and fast algorithms for computing the relevant performance measures. In this section we apply the results of Section 3 to a performance analysis of an ACD system providing phone service (inbound) and calling customers directly (outbound, telemarking). Combined inbound and outbound call centers were first analyzed by Dumas et al. [10]. At the call center, cf. Fig. 2, consisting of a finite positive number s of agents and a finite positive number k of waiting places (i.e., s + k lines), there arrive calls from outside (inbound calls) according to a Poisson process of intensity λ. An arriving inbound call requires an exponentially distributed service time with parameter µ. If there is at least one free line, i.e. agent or waiting place, then the call will be accepted; otherwise it gets lost. An accepted call will be served immediately by one of the idle agents if there is anyone idle. If all agents are busy then the call begins to wait until one of the agents becomes idle; the queueing discipline is FCFS. But the inbound calls are impatient, i.e., they are lost if the waiting time exceeds a random time (maximal waiting time). The maximal waiting times are assumed to be i.i.d. with a general distribution C(u) = P (I ≤ u), where I denotes a typical maximal waiting time. The maximal waiting times may correspond to real maximal waiting times or/and to a special management for those inbound calls whose waiting time exceeds an acceptable amount of time. Of special interest is the case that I = min(X, τ ), where X is exponentially distributed describing the individual maximal waiting time of an inbound call and τ is deterministic describing the technical maximal waiting time of the system. The time τ may be a deterministic time after that the inbound call is routed automatically by the system to another call center of the company or will be handled by special agents in order to improve the grade-of-service. To improve the efficiency of the call center, underutilized agents may also be employed to dial outbound calls. Specifically, we introduce a parameter, a ∈ {0, 1, . . . , s}, such that if more than a agents are idle
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
13
and no calls are in the queue, then one of the idle agents will dial an outbound call rather than wait for an arrival of an inbound call. Thus, at each time instant with probability one at least s − a agents are busy, cf. [10]. We assume that there is an infinite reservoir of possible outbound calls (list of customers) and that the service times of the outbound calls are exponentially distributed with the same parameter µ as the inbound calls. For the ACD system the following performance characteristics are of interest: Λin Λout pB
rate of accepted inbound calls rate of dialed outbound calls probability that a typical arriving inbound call finds no free waiting place or agent (blocking probability, 1 − pB is the acceptance probability) pW probability that a typical accepted inbound call has to wait for service probability that a typical accepted inbound call will get lost due to impatience later (impatience pI probability) WS (x) waiting time distribution of a typical accepted inbound call up to its service on the condition that it will be served WI (x) waiting time distribution of a typical accepted inbound call on the condition that it will leave the system due to impatience later W (x) (unconditional) waiting time distribution of a typical accepted inbound call The ACD system may be modeled by an M(n)/M(n)/s + GI queueing system, where n corresponds to the number of inbound and served outbound calls in the ACD system, cf. Fig. 1. The state dependent arrival intensity λn is given by λn = 1{n < s + k}λ,
n = 0, 1, . . . ,
(4.1)
and the state dependent service intensity µn by µn = 1{n > s − a}min(n, s)µ,
n = 0, 1, . . . ,
(4.2)
especially we have µ∗ = sµ. (Note that the time instants where the service of an outbound call is started does not correspond to changes of the system state. However, for finding the relevant performance measures this does not matter as seen below.) In view of Theorem 3.1 the system is stable and for the stationary occupancy distribution it holds (λ/µ)n , s − a ≤ n ≤ s, n! Z ∞ 1 (λF (ξ ))n−s e−ξ dξ, p(n) = g λs (n − s)! 0
p(n) = g s!µs
p(n) = 0
s < n ≤ s + k,
elsewhere,
where F (ξ ) is given by (3.2). From (3.3),(4.1) and (4.2) we obtain the normalizing factor Z s−1 k X X (λ/µ)j 1 ∞ (λF (ξ ))j e−ξ dξ. + λs g −1 = s!µs j ! j ! 0 j =s−a j =0
(4.3)
(4.4)
For the ACD system p(n) is the stationary probability that precisely n inbound and served outbound calls are in the system. The cumulative arrival intensity Λ in the M(n)/M(n)/s + GI queueing system in
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
the steady state corresponds to the rate Λin of accepted inbound calls in the ACD system. Hence, from (3.6),(4.1) and (4.3) it follows Λin = Λ = λ
s+k−1 X
p(n) = (1 − p(s + k))λ.
(4.5)
n=s−a
The blocking probability pB for inbound calls is given by the PASTA-property, i.e., pB = p(s + k),
Λin = Λ = (1 − pB )λ.
(4.6)
The service of an outbound call is started if precisely n = s − a agents are busy, no calls are in the queue and if one of the service times just finishes. The probability of being in the state n = s − a is p(s − a) and for the ACD system the cumulative intensity of finishing service in this state is (s − a)µ. Therefore, for the rate Λout of dialed outbound calls it follows Λout = (s − a)µp(s − a).
(4.7)
From (3.7), (4.1), (4.2), (4.3) and (4.6) we obtain the probability pW that an accepted inbound call has to wait for service s s−1 X X 1 µ 1 − pW = p(n) = np(n), 1 − pB n=s−a (1 − pB )λ n=s−a+1
(4.8)
and from (3.8), (4.2), (4.3) and (4.6) the probability pI that an accepted inbound call will get lost due to impatience later s+k X µ 1 − pI = min(n, s)p(n). (1 − pB )λ n=s−a+1
(4.9)
From (3.9), (4.1) and (4.6) for the waiting time distribution of an accepted inbound call up to its service on the condition that it will be served it follows Z k−1 X µ∗ p(s) 1 ∞ 1 − WS (x) = (λF (ξ ))j F 0 (ξ )e−ξ dξ, x ∈ R+ . (4.10) (1 − pB )(1 − pI ) j =0 j ! µ∗ x From (3.10), (4.1) and (4.6) for the waiting time distribution of an accepted inbound call on the condition that it will leave the system due to impatience later we find Z k−1 µ∗ p(s) X 1 ∞ (λF (ξ ))j (F 0 (µ∗ x) − F 0 (ξ ))e−ξ dξ, x ∈ R+ . (4.11) 1 − WI (x) = (1 − pB )pI j =0 j ! µ∗ x From (3.11), (4.10) and (4.11) for the unconditional waiting time distribution of an accepted inbound call we obtain Z k−1 X 1 ∞ µ∗ p(s) 0 1 − W (x) = F (µ∗ x) (λF (ξ ))j e−ξ dξ, x ∈ R+ . (4.12) 1 − pB j ! µ∗ x j =0 The corresponding expectations result from (4.1),(4.3),(4.6),(3.12),(3.13) and (3.14), respectively.
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
15
Remark 4.1. For k → ∞ we obtain the case of a waiting room of unlimited size. From (4.4) for the normalizing factor it follows: Z ∞ s−1 X (λ/µ)j −1 s s eλF (ξ )−ξ dξ. +λ g = s!µ j ! 0 j =s−a The stability condition corresponds to the finiteness of the right-hand side of this equation, because of (3.2) being equivalent to (λ/µ∗ )limu→∞ (1 − C(u)) < 1, cf. [2]. Finally, we consider the special case that the typical maximal waiting time I is the minimum of a constant and an exponentially distributed time, i.e., I = min(X, τ ), where X is exponentially distributed with parameter α describing the individual maximal waiting time of an inbound call and τ is deterministic describing the technical maximal waiting time of the ACD system. The distribution of I is given by C(u) = 1 − 1{u < τ }e−αu ,
u ∈ R+ .
(4.13)
Remark 4.2. For the distribution (4.13) of the maximal waiting times the limiting case k → ∞ with λn ≡ λ for n ≥ s and µn = min(n, s)µ for n ≥ 0 was earlier analyzed in [16]. For several performance measures, basing on the waiting time vector process, in this case formulae are available. However, they do not include the occupancy distribution. Remark 4.3. The distribution (4.13) of the maximal waiting times includes the following two special cases. For τ → ∞ the process N (t) of the number of calls in the M(n)/M(n)/s + GI system converges to a birth death process which can be analyzed in the usual manner. For α → 0 we obtain a system with constant maximal waiting times; for earlier results we refer to [13]. For the special distribution (4.13) of the maximal waiting times stable and fast numerical algorithms for the performance measures considered were derived from the above given formulae and implemented. As an example we consider the following basic parameter set: – mean service time 1/µ: 120 s, – mean individual maximal waiting time 1/α: 90 s, – deterministic maximal waiting time τ : 60 s. In Table 1 the offered load λ/µ is 10 Erl, in Table 2 the offered load is 100 Erl. The number s of agents, the number k of waiting places and the outbound parameter a vary in these tables. The blocking probability pB , the impatience probability pI , the expectations EWS and EWI of the conditional waiting time distributions and the rate of dialed outbound calls Λout are presented. In the last four lines of the tables, the corresponding performance measures for an ACD system without outbound calls are given. The numerical results show the impact of the operational strategy (outbound parameter a) on the system performance. If the outbound parameter a is chosen not too small, then the performance measures are not significantly worse than corresponding performance measures for the system without outbound calls (given in the last four lines of the tables), but the efficiency of the call center is improved by dialing outbound calls. The numerical results illustrate the tradeoff between the blocking probability pB and the impatience probability pI if k varies. But there is no obvious relation between the mean conditional waiting times
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
Table 1 Blocking probability pB , impatience probability pI , the expectations EWS and EWI of the conditional waiting time distributions and the rate of dialed outbound calls Λout for an offered load of 10 Erl s
k
a
λ/µ
1/µ
1/α
τ
pB
pI
EWS
EWI
Λout
8 12 16 20 8 12 16 20 8 12 16 20 8 12 16 20
3 3 3 3 6 6 6 6 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 6 6 6 6 8 12 16 20
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
0.137 0.049 0.016 0.006 0.024 0.006 0.001 0.000 0.131 0.034 0.006 0.001 0.131 0.031 0.003 0.000
0.170 0.061 0.024 0.011 0.254 0.088 0.031 0.013 0.162 0.042 0.009 0.002 0.162 0.039 0.005 0.000
11.472 4.729 1.955 0.884 15.696 6.341 2.446 1.037 10.769 3.174 0.723 0.173 10.758 2.931 0.388 0.023
22.286 14.258 9.988 7.687 26.739 17.738 11.960 8.769 22.286 14.258 9.988 7.687 22.286 14.258 9.988 7.687
0.003 0.015 0.039 0.067 0.002 0.015 0.038 0.067 0.000 0.004 0.019 0.045 0.000 0.000 0.000 0.000
Table 2 Blocking probability pB , impatience probability pI , the expectations EWS and EWI of the conditional waiting time distributions and the rate of dialed outbound calls Λout for an offered load of 100 Erl s
k
a
λ/µ
1/µ
1/α
τ
pB
pI
EWS
EWI
Λout
90 100 110 120 90 100 110 120 90 100 110 120 90 100 110 120
15 15 15 15 30 30 30 30 15 15 15 15 15 15 15 15
10 10 10 10 10 10 10 10 20 20 20 20 90 100 110 120
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
0.037 0.012 0.003 0.001 0.002 0.000 0.000 0.000 0.037 0.010 0.002 0.000 0.036 0.010 0.001 0.000
0.081 0.042 0.018 0.007 0.111 0.050 0.020 0.007 0.079 0.036 0.011 0.002 0.079 0.035 0.009 0.001
7.365 3.681 1.551 0.608 10.155 4.442 1.702 0.634 7.146 3.117 0.913 0.200 7.138 3.053 0.776 0.102
6.568 5.328 4.307 3.501 8.817 6.466 4.827 3.726 6.568 5.328 4.307 3.501 6.568 5.328 4.307 3.501
0.006 0.025 0.067 0.127 0.005 0.024 0.066 0.127 0.000 0.004 0.022 0.065 0.000 0.000 0.000 0.000
EWS and EWI . Comparing Tables 1 and 2 shows us the well-known effect that larger systems are more efficient than small ones. Finally, we remark that the derived numerical algorithms have been implemented in the form of a software tool for cost optimizing ACD systems over several (quality of service) constraints.
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
17
Acknowledgements This work was supported by a grant from the Siemens AG. References [1] F. Baccelli, P. Boyer, G. Hebuterne, Single server queues with impatient customers, Adv. Appl. Probab. 16 (1984) 887–905. [2] F. Baccelli, G. Hebuterne, On queues with impatient customers, in: F.J. Kylstra (Ed.), Performance ’81, North-Holland, Amsterdam, 1981, pp. 159–179. [3] D.Y. Barrer, Queueing with impatient customers and indifferent clerks, Oper. Res. 5 (1957) 644–649. [4] D.Y. Barrer, Queueing with impatient customers and ordered service, Oper. Res. 5 (1957) 650–656. [5] A. Brandt, M. Brandt, G. Spahl, D. Weber, Modelling and optimization of call distribution systems, Proc. 15th Int. Teletraffic Congress (ITC 15), Washington, DC, USA, 1997, pp. 133–144. [6] A. Brandt, P. Franken, B. Lisek, Stationary Stochastic Models, Akademie-Verlag, Berlin; Wiley, Chichester, 1990. [7] S.M. Brodi, On an integro-differential equation for systems with τ -waiting (in Ukrainian), Dop. AN URSR 6 (1959) 571–573. [8] S.M. Brodi, On a service problem (in Russian), Tr. V. Vsesojuznogo sove˘sc˘ anija po teorii verojatnostej i matemati˘ceskoj statistike, Izd-vo AN Arm. SSR, Erevan (1960) 143–147. [9] D.J. Daley, General customer impatience in the queue GI/G/1, J. Appl. Probab. 2 (1965) 186–205 . [10] G. Dumas, M. Perkins, C. White, Improving efficiency of PBX-based call centers: Combining inbound and outbound agents with automatic call sharing, Proc. 15th Int. Switching Symp., Berlin, 1995, pp. 346–350. [11] P. Franken, D. König, U. Arndt, V. Schmidt, Queues and Point Processes, Akademie-Verlag, Berlin; Wiley, Chichester, 1982. [12] B.W. Gnedenko, D. König, Handbuch der Bedienungstheorie, vol. II, Akademie-Verlag, Berlin, 1984. [13] B.W. Gnedenko, I.N. Kowalenko, Einführung in die Bedienungstheorie (1st ed. in Russian, Nauka, Moscow, 1966). Akademie-Verlag, Berlin, 1974. [14] R.B. Haugen, Queueing systems with several input streams and time out, Telektronikk 2 (1978) 100–106. [15] R.B. Haugen, E. Skogan, Queueing systems with stochastic time out, IEEE Trans. Commun. COM-28 (1980) 1984–1989. [16] O.M. Jurkevi˘c, On the investigation of many-server queueing systems with bounded waiting time (in Russian), Izv. Akad. Nauk SSSR Techni˘ceskaja kibernetika 5 (1970) 50–58. [17] O.M. Jurkevi˘c, On many-server systems with stochastic bounds for the waiting time (in Russian), Izv. Akad. Nauk SSSR Techni˘ceskaja kibernetika 4 (1971) 39–46. [18] O.M. Jurkevi˘c, On many-server systems with general service time distribution of ”impatient customers” (in Russian), Izv. Akad. Nauk SSSR Techni˘ceskaja kibernetika 6 (1973) 80–89. [19] A. Mandelbaum, N. Shimkin, A model for rational abandonments from invisible queues, Unpublished paper, Technion, Haifa, 1998. [20] C. Palm, Methods of judging the annoyance caused by congestion, Tele No. 2 (1953) 1–20. [21] M. Perry, Performance modelling of automatic call distributors, Ph.D. Thesis, North Carolina State University, 1991. [22] M. Perry, A. Nilsson, Performance modelling of automatic call distributors: Assignable grade of service staffing, Proc. 14th Int. Switching Symp., Yokohama, 1992, pp. 294–298. [23] M. Singh, Steady state behaviour of serial queueing processes with impatient customers, Math. Operationsforsch. Ser. Statist. 15(2) (1984) 289–298. [24] K.C. So, C. Tang, Operational strategies for managing congestion in service systems. Working paper, Graduate School of Management University of California, Irvine, CA, 1993. [25] J. Teghem, Use of discrete transforms for the study of a GI /M/s queue with impatient customer phenomena, Z. Oper. Res. 23 (1979) 95–106. [26] B. Wallstrøm, A queueing system with time-outs and random departure, Proc. ITC 8, Melbourne, 1976, paper 231.
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18 Andreas Brandt was born in Berlin in 1955. He received his Ph.D. degrees in mathematics from the Humboldt-University of Berlin in 1983. From 1983 until 1989 he was research assistant and from 1990– 1992 head of the group Applied Probability at the Department of Mathematics. Since 1992 he has been Professor for Operations Research at the Humboldt-University of Berlin. His main research interests are in stochastic processes, in particular point processes, and their applications. Over the last years he has worked closely with several industrial partners. The projects address the performance analysis of telecommunication and computer systems as well as reliability analysis of technical systems.
Manfred Brandt received his Ph.D. degrees in mathematics from the Humboldt-University of Berlin in 1973. Since 1989 he has been a Privatdozent at the Humboldt-University of Berlin. His main research interests are in pure complex analysis and since the eighties, also in the performance analysis of computer and teletraffic systems as well as queueing theory. He works closely with industrial partners and has implemented various algorithms. Since 1994 he has been a member of the scientific staff at the KonradZuse-Zentrum for Information Technology in Berlin.
On the M(n)/M(n)/s queue with impatient calls Andreas Brandt a,∗ , Manfred Brandt b,1 a
Wirtschaftswissenschaftliche Fakultät, Humboldt-Universität zu Berlin, Spandauer Str. 1, D-10178 Berlin, Germany b Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustr. 7, D-14195 Berlin, Germany Received 25 June 1997; received in revised form 28 August 1998
Abstract The paper is concerned with the analysis of an s-server queueing system wherein the calls may leave the system due to impatience. The individual maximal waiting times are assumed to be i.i.d. and arbitrarily distributed. The arrival and cumulative service rates may depend on the number n of calls in the system, but the service rate is assumed to be constant for n > s. For this system, denoted by M(n)/M(n)/s + GI , we derive a system of integral equations for the vector of the residual maximal waiting times of the waiting calls and their original maximal waiting times. By solving these equations explicitly we obtain the stability condition and for the steady state of the system, the occupancy distribution and various waiting time distributions. The results are also new for special cases analyzed in earlier papers. As an application of the M(n)/M(n)/s + GI system we give a performance analysis of an automatic call distributor system (ACD system) of finite capacity with outbound and impatient inbound calls; numerical results are given for the case of maximal waiting times as the minimum of constant and exponentially distributed times. ©1999 Elsevier Science B.V. All rights reserved. Keywords: M(n)/M(n)/s queue with impatient calls; State dependent arrivals and services; Finite capacity; Integral equations; Occupancy distribution; Waiting time distribution; ACD system
1. Introduction In this paper we consider an s-server queueing system with an unlimited waiting room with FCFS queueing discipline and where the calls waiting in the queue for service are impatient, cf. Fig. 1. The arrival and service processes are allowed to be state dependent with respect to the number n of calls in the system, but the cumulative service rate is assumed to be constant for n > s. We assume that the sequence of the arrival rates λn is bounded and that λn > 0 for n ≥ 0 or that there exists a positive integer k such that λn > 0 for 0 ≤ n < s + k and λn ≡ 0 for n ≥ s + k. Concerning the cumulative rate µn of finishing service we assume that µ0 = 0 and µn ≡ µ∗ > 0 for n > s. Each call arriving at the system has a maximal waiting time I . If the offered waiting time W o (i.e., the time which a call would have to ∗ 1
Corresponding author. E-mail: [email protected] E-mail: [email protected].
0166-5316/99/$ – see front matter ©1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 9 8 ) 0 0 0 4 2 - X
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
Fig. 1. The M(n)/M(n)/s + GI system with impatient calls and state dependent arrival and service rates, where n denotes the number of calls in the system, W o the offered waiting time and I the maximal waiting time.
wait for accessing a server if it were sufficiently patient) exceeds I , then the call departs from the system after having waited time I . The maximal waiting times are assumed to be i.i.d. with a general distribution C(u) := P (I ≤ u), u ∈ R+ which may be defective, i.e., P (I = ∞) > 0 is not excluded. This system is denoted by M(n)/M(n)/s + GI , where the first M(n) denotes the arrival process, the second M(n) the service process depending on min(n, s + 1) only, i.e., on the number of busy servers and additionally whether there are calls waiting for service. The symbol GI stands for the i.i.d. maximal waiting times. Remark 1.1. Note that if λn > 0 for 0 ≤ n < s + k and λn ≡ 0 for n ≥ s + k, then we have the case of a limited waiting room with k waiting places. In case of λn ≡ λ > 0, µn = min(n, s)µ for n ≥ 0 the model corresponds to a M/M/s + GI system, cf. [2]. The more general case λn ≡ λ > 0 for n ≥ s and µn = min(n, s)µ for n ≥ 0 is treated in [17]. Relations to the results of Baccelli and Hebuterne [2] and Jurkevi˘c [17], which are most relevant papers to ours, and further references are given below. The paper is organized as follows. In Section 2 a system of integral equations for the density of the stationary vector process of the number of calls, residual maximal waiting times and original maximal waiting times of calls waiting for service in the system is derived. By a separation approach corresponding to partial balance, the integral equations are solved explicitly; the densities are of an elementary structure. In Section 3 we derive the distribution of the number of calls in the system (occupancy distribution), the stability condition and performance measures such as the impatience probability, the cumulative arrival rate and the waiting time distributions of the served calls as well as of the calls leaving due to impatience. In Section 4 an application of the results to a performance analysis of an automatic call distributor system (ACD system) with outbound calls and impatient inbound calls is given. In case of maximal waiting times as the minimum of constant and exponentially distributed times some numerical results are presented. The results of the paper can also be applied to more complicated ACD systems directly (e.g. to ACD systems with integrated voice-mail-server) or indirectly for constructing system approximations. A number of papers have dealt with impatience phenomena. It seems that Palm [20] was the first to deal with the impatience problem. Barrer [3,4] analyzed the M/M/1+D system. Brodi [7,8] derived and solved for the M/M/1 + D system the corresponding integro differential equation. The general GI /GI /1 + GI system is treated in [9]. The many-server M/M/s +D system was analyzed by Gnedenko and Kowalenko [13] by giving an explicit solution of the system of integro differential equations for the work load of the s servers, which yields formulae for the performance measures. This method was successfully further applied by Jurkevi˘c [16] to maximal waiting times as the minimum of a constant and an exponentially distributed time and [17] for the general M/M/s + GI system where the arrival intensities may depend on the number of busy servers (cf. also [12, p. 270]). Later, independently Haugen and Skogan [15] and Baccelli and Hebuterne [2] derived results for the M/M/s + GI system, too. Haugen and Skogan [15] started with a result of Wallstrøm [26], cf. also [14], for an s-server system with several Poisson input streams where each call type has an individual constant maximal waiting time and where the calls in the
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system (waiting and served) are exponentially impatient. By an appropriate limiting construction they obtained a formula for the waiting time distribution for the M/M/s + GI system and a generalization of this by allowing departures for the case of a limitation of the total time spent in the system (queue and server). In [2] the Kolmogorov equations are derived and solved for the virtual offered waiting time of a call. By means of this quantity formulae for the relevant performance measures are derived. However, the distribution of the total number of calls in the M/M/s + GI system and in the more general model in [17] was not treated in the references mentioned above. Thus, the results concerning the occupancy distribution given in Section 3 are new – as far as we know – also for the special case of an M/M/s + GI system (treated in [2]) and of the model in [17]. Note that our model includes the case of a finite waiting room, whereas this case is excluded in the models mentioned above. In the literature, there are several other known mechanisms where calls leave the system due to impatience: If the call can calculate its prospective waiting time at its arrival instant, then it leaves immediately if this time exceeds its maximal waiting time. This strategy yields a better utilization of the waiting places because they will not be occupied by calls which later abandon due to impatience. Also impatience may act on the sojourn times (waiting time plus service time). In this case not all work is useful because a call may leave the system due to impatience during its service. For references and other more general models with impatience mechanism we refer to [1,2,18,19,23,25] and the references therein.
2. A system of integral equations and its solution Throughout this section we assume that the queueing system is stable and that the distribution C(u) of the maximal waiting times is non-defective and has a continuous density c(u). If n calls are in the system then l := (n − s)+ calls are waiting in the queue for service. (The notation l := (n − s)+ will be used also in the following.) We number the waiting calls according to their positions in the queue. By the FCFS discipline the first call in the queue will be potentially the next for service. Let N (t) L(t) := (N (t) − s)+ (X1 (t), . . . , XL(t) (t)) (I1 (t), . . . , IL(t) (t))
number of calls in the system at time t number of waiting calls at time t vector of the residual maximal waiting times of waiting calls ordered according to their positions in the queue at time t vector of the original maximal waiting times of the waiting calls ordered according to their positions in the queue at time t stationary distribution of the number of calls in the system
p(n) := P (N (t) = n) P (n; x1 , . . . , xl ; u1 , . . . , ul ) := P (N (t) = n; X1 (t) ≤ x1 , . . . , Xl (t) ≤ xl ; I1 (t) ≤ u1 , . . . , Il (t) ≤ ul ) stationary distribution on N(t) = n, where l := (n − s)+ . Obviously, for fixed n > s the support of P (n; x1 , . . . , xl ; u1 , . . . , ul ) is contained in Ωl := {(x1 , . . . , xl ; u1 , . . . , ul ) ∈ R2l + : u1 − x1 ≥ · · · ≥ ul − xl ≥ 0}.
(2.1)
In view of the assumptions on C(u) the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) :=
∂ 2l P (n; x1 , . . . , xl ; u1 , . . . , ul ) ∂x1 · · · ∂xl ∂u1 · · · ∂ul
(2.2)
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
is continuous on Ωl ; from a representation of this density given at the end of this section its continuity on Ωl follows, too. In case of n ≤ s we have the balance equations (λn + µn )p(n) = 1{n > 0}λn−1 p(n − 1) + µn+1 p(n + 1), n = 0, 1, . . . , s − 1, Z Z p(s + 1; 0; u) du + µ∗ p(s + 1; x; u) dx du. (λs + µs )p(s) = λs−1 p(s − 1) + R+
R2+
(2.3) (2.4)
In view of µ0 = 0 by summing the first n + 1 equations of (2.3) we obtain λn p(n) = µn+1 p(n + 1),
n = 0, 1, . . . , s − 1.
Therefore, (2.4) is equivalent to Z Z p(s + 1; 0; u) du + µ∗ λs p(s) = R+
R2+
(2.5)
p(s + 1; x; u) dx du.
(2.6)
In case of n > s and (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl , which in view of (2.1) especially implies 0 ≤ xl ≤ ul , we have the balance conditions p(n; x1 , . . . , xl ; u1 , . . . , ul ) = p(n; x1 + h, . . . , xl + h; u1 , . . . , ul )(1 − hλn − hµ∗ ) l+1 Z X p(n + 1; x1 , . . . , xi−1 , 0, xi , . . . , xl ; u1 , . . . , ui−1 , u, ui , . . . , ul ) du +h i=1
Z
+hµ∗
R+
R2+
p(n + 1; x, x1 , . . . , xl ; u, u1 , . . . , ul ) dx du + o(h),
h > 0, xl < ul ,
p(n; x1 , . . . , xl−1 , ul ; u1 , . . . , ul ) = λn−1 p(n − 1; x1 , . . . , xl−1 ; u1 , . . . , ul−1 )c(ul ).
(2.7) (2.8)
In the following from (2.7) and (2.8) we shall derive an equivalent system of integral equations for the density p(n; x1 , . . . , xl ; u1 , . . . , ul ). Let n > s and (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl with xl < ul be fixed. For t ∈ [0, ul − xl ] we define Z ∞ p(n; x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul ) ϕ(t) := p(n; x1 + t, . . . , xl + t; u1 , . . . , ul )e−µ∗ t + λn ×e−µ∗ ξ dξ −
l+1 Z X i=1
t
∞Z t
R+
p(n + 1; x1 + ξ, . . . , xi−1 + ξ, 0, xi + ξ, . . . , xl + ξ ;
u1 , . . . , ui−1 , u, ui , . . . , ul )e−µ∗ ξ dudξ − µ∗
Z t
∞Z
R2+
p(n + 1; x, x1 + ξ, . . . , xl + ξ ;
u, u1 , . . . , ul )e−µ∗ ξ dx du dξ. By continuity of the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) on Ωl the function ϕ(t) is also continuous, and for t ∈ [0, ul − xl ), h ∈ (0, ul − xl − t] by some algebra and applying (2.7) at the point (x1 + t, . . . , xl + t; u1 , . . . , ul ) we find ϕ(t + h) − ϕ(t) = o(h), t ∈ [0, ul − xl ), h ∈ (0, ul − xl − t]. By continuity of ϕ(t)
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
5
therefore we conclude that ϕ(t) is constant, especially it follows the relation ϕ(0) = ϕ(ul − xl ). In view of the definition of ϕ(t), since the support of the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) is contained in Ωl and because of the boundary condition (2.8) thus we obtain the following system of integral equations for n > s and (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl : Z p(n; x1 , . . . , xl ; u1 , . . . , ul ) + λn
R+
p(n; x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul )e−µ∗ ξ dξ
= λn−1 p(n − 1; x1 + ul − xl , . . . , xl−1 + ul − xl ; u1 , . . . , ul−1 )c(ul )e−µ∗ (ul −xl ) l+1 Z X + p(n + 1; x1 + ξ, . . . , xi−1 + ξ, 0, xi + ξ, . . . , xl + ξ ; u1 , . . . , ui−1 , u, ui , . . . , ul ) i=1
×e
−µ∗ ξ
R2+
Z
du dξ + µ∗
R3+
p(n + 1; x, x1 + ξ, . . . , xl + ξ ; u, u1 , . . . , ul )e−µ∗ ξ dx du dξ.
(2.9)
On the other hand, from the system of integral equations (2.9) the balance conditions (2.7) and (2.8) may be derived. In the following we shall solve the system of equations (2.5),(2.6) and (2.9). From (2.5) we find p(n) = g
n−1 Y
! λi
i=0
s Y
! µi ,
n ≤ s,
(2.10)
i=n+1
where g > 0 is a normalizing factor. Remark 2.1. In case of µn > 0 for n = 1, 2, . . . , s the representation p(n) = g ∗
n−1 Y
λi , µ i+1 i=0
g∗ = g
s Y
µi ,
i=1
is possible for n ≤ s, too, being more closely to the birth death process notations. In view of the representation (2.10) of p(n) for n ≤ s, in case of n ≥ s we choose the substitution p(n; x1 , . . . , xl ; u1 , . . . , ul ) = g
n−1 Y
! λi q(n; x1 , . . . , xl ; u1 , . . . , ul ).
(2.11)
i=0
This substitution is verified by the fact that obviously we have p(n; x1 , . . . , xl ; u1 , . . . , ul ) ≡ 0 in case of λ0 · · · λn−1 = 0. From (2.10) and (2.11) it follows q(s) = 1. Hence, Eq. (2.6) now reads Z
Z 1=
R+
q(s + 1; 0; u) du + µ∗
R2+
q(s + 1; x; u) dx du,
and for n > s, (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl the system of integral equations (2.9) reads
(2.12)
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
Z
q(n; x1 , . . . , xl ; u1 , . . . , ul ) + λn
R+
q(n; x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul )e−µ∗ ξ dξ
= q(n − 1; x1 + ul − xl , . . . , xl−1 + ul − xl ; u1 , . . . , ul−1 )c(ul )e−µ∗ (ul −xl ) l+1 Z X +λn q(n + 1; x1 + ξ, . . . , xi−1 + ξ, 0, xi + ξ, . . . , xl + ξ ; u1 , . . . , ui−1 , u, ui , . . . , ul ) R2+
i=1
×e
−µ∗ ξ
Z
du dξ + λn µ∗
R3+
q(n + 1; x, x1 + ξ, . . . , xl + ξ ; u, u1 , . . . , ul )e−µ∗ ξ dx du dξ. (2.13)
In case of a finite waiting room with k waiting places, i.e., λn > 0 for 0 ≤ n < s + k and λn ≡ 0 for n ≥ s + k, Eqs. (2.13) are relevant only for s < n ≤ s + k. The structure of (2.12) and (2.13) suggests to conjecture that q(n; x1 , . . . , xl ; u1 , . . . , ul ) is independent on λn . Assuming this independence, then for n > s, (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl from (2.13) it follows q(n; x1 , . . . , xl ; u1 , . . . , ul ) = q(n − 1; x1 + ul − xl , . . . , xl−1 + ul − xl ; u1 , . . . , ul−1 )c(ul )e−µ∗ (ul −xl ) , Z R+
(2.14)
q(n; x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul )e−µ∗ ξ dξ
=
l+1 Z X 2 i=1 R+
×e
−µ∗ ξ
q(n + 1; x1 + ξ, . . . , xi−1 + ξ, 0, xi + ξ, . . . , xl + ξ ; u1 , . . . , ui−1 , u, ui , . . . , ul ) Z du dξ + µ∗
R3+
q(n + 1; x, x1 + ξ, . . . , xl + ξ ; u, u1 , . . . , ul )e−µ∗ ξ dx du dξ. (2.15)
In view of q(s) = 1 and of the definition (2.1) of Ωl from (2.14) by induction over n > s we get ! l Y c(ui ) e−µ∗ (u1 −x1 ) , n > s. q(n; x1 , . . . , xl ; u1 , . . . , ul ) = 1{(x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl } i=1
(2.16)
Since the distribution C(u) of the maximal waiting times is non-defective by integration it follows that the function q(s + 1; x1 ; u1 ) defined by (2.16) satisfies (2.12). Moreover, using the convention xl+1 = ul+1 = 0 for the functions q(n; x1 , . . . , xl ; u1 , . . . , ul ) defined by (2.16) for n > s, (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl we obtain l+1 Z X q(n + 1; x1 , . . . , xi−1 , 0, xi , . . . , xl ; u1 , . . . , ui−1 , u, ui , . . . , ul ) du i=1
R+
Z
+µ∗ ×
R2+
l Y i=1
q(n + 1; x, x1 , . . . , xl ; u, u1 , . . . , ul ) dx du = 1{(x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl }
c(ui )
! Z
∞
u1 −x1
c(u)e
Z
+µ∗
∞
−µ∗ u
Z
u1 −x1 0
du + e
−µ∗ (u1 −x1 )
l+1 Z X i=2
u−(u1 −x1 )
c(u)e
−µ∗ (u−x)
ui−1 −xi−1
ui −xi
c(u) du
dx du = q(n; x1 , . . . , xl ; u1 , . . . , ul ).
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
7
From this equation applied at the point (x1 + ξ, . . . , xl + ξ ; u1 , . . . , ul ) it follows that the functions q(n; x1 , . . . , xl ; u1 , . . . , ul ) defined by (2.16) also satisfy (2.15). Hence, these functions solve the system of integral equations (2.12) and (2.13). Since the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) is uniquely determined as the normalized solution of (2.7) and (2.8) this density is given by (2.11) and (2.16). Summarizing we find the representation p(n; x1 , . . . , xl ; u1 , . . . , ul ) = 1{(x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl } g
n−1 Y i=0
! λi
l Y
! c(ui ) e−µ∗ (u1 −x1 ) ,
n > s.
(2.17)
i=1
Remark 2.2. The successful separation of (2.13) into (2.14) and (2.15) corresponds to the fact that for n > s and (x1 , . . . , xl ; u1 , . . . , ul ) ∈ Ωl the partial balance equations p(n; x1 , . . . , xl ; u1 , . . . , ul )=p(n; x1 +(ul −xl ), . . . , xl +(ul −xl ); u1 , . . . , ul )e−µ∗ (ul −xl ) ,
(2.18)
λn p(n; x1 , . . . , xl ; u1 , . . . , ul ) l+1 Z X p(n + 1; x1 , . . . , xi−1 , 0, xi , . . . , xl ; u1 , . . . , ui−1 , u, ui , . . . , ul ) du = i=1
R+
Z
+µ∗
R2+
p(n + 1; x, x1 , . . . , xl ; u, u1 , . . . , ul ) dx du,
(2.19)
hold. Eq. (2.18) means that the density p(n; x1 , . . . , xl ; u1 , . . . , ul ) on the condition that no new service has been started since the last arrival instant only depends on time (by changes of the residual maximal waiting times). Eq. (2.19) means that the intensity of transitions from the state (n; x1 , . . . , xl ; u1 , . . . , ul ) into any state (n + 1; ·; ·) (by an arrival) equals the intensity of transitions from any state (n + 1; ·; ·) into the state (n; x1 , . . . , xl ; u1 , . . . , ul ) (by leaving the queue due to impatience or for starting service). Obviously, from the partial balance equations (2.18) and (2.19) and from the boundary condition (2.8) it follows that the system of integral equations (2.9) holds.
3. Stability condition, occupancy and waiting time distribution As in Section 2 let us assume that the queueing system is stable and that the distribution C(u) of the maximal waiting times is non-defective and has a continuous density c(u). The stationary probability p(n) that n calls are in the system is given by (2.10) for n ≤ s. In case of n > s from (2.17) it follows Z p(n; x1 , . . . , xl ; u1 , . . . , ul ) dx1 · · · dxl du1 · · · dul p(n) = R2l +
=g
n−1 Y i=0
!Z λi
Ωl
l Y i=1
! c(ui ) e−µ∗ (u1 −x1 ) dx1 · · · dxl du1 . . . dul ,
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
where l := (n − s)+ . In view of the definition (2.1) of Ωl the substitution ui = ξi + xi yields ! !Z n−1 l Y Y λi 1{ξ1 ≥ · · · ≥ ξl } c(ξi + xi ) e−µ∗ ξ1 dx1 · · · dxl dξ1 · · · dξl p(n) = g i=0 n−1 Y
=g
! λi
i=0
where
Z
ξ/µ∗
F (ξ ) :=
R2l +
1 (n − s)!
i=1
Z
∞
F (ξ )n−s e−ξ dξ,
n > s,
(3.1)
0
(1 − C(η)) dη,
ξ ∈ R+ .
(3.2)
0
Because of Eqs. (2.10) and (3.1) the normalizing factor g is given by ! s ! Z −1 s+j −1 s−1 jY ∞ X Y X Y 1 ∞ −1 g = λi µi + λi F (ξ )j e−ξ dξ. j ! 0 j =0 i=0 i=j +1 j =0 i=0
(3.3)
Obviously, the stability of the system is equivalent to the finiteness of the right-hand side of (3.3). Therefore, the stability condition reads ! Z s+j −1 ∞ X Y 1 ∞ λi F (ξ )j e−ξ dξ < ∞. (3.4) j ! 0 j =0 i=0 The case of a general distribution C(u) of the maximal waiting times is obtained by considering C(u) as the limit in distribution of a sequence of non-defective distributions Cν (u) with continuous density. From (3.4),(2.10) and (3.1) applied to Cν (u) by arguments of continuity we obtain the following statement. Theorem 3.1. Let the maximal waiting times be i.i.d. with a general distribution C(u). Then, the system is stable iff (3.4) is fulfilled. In case of a stable system for the stationary occupancy distribution it holds ! ! n−1 s Y Y λi µi , n = 0, 1, . . . , s, p(n) = g i=0
p(n) = g
n−1 Y
! λi
i=0
i=n+1
1 (n − s)!
Z
∞
F (ξ )n−s e−ξ dξ,
n = s + 1, s + 2, . . . ,
(3.5)
0
where the normalizing factor g is given by (3.3) and F (ξ ) by (3.2). Let the system be stable, i.e., let (3.4) be satisfied. The cumulative arrival intensity Λ of the calls in the steady state is given by Λ=
∞ X λn p(n).
(3.6)
n=0
Since the sequence of the arrival intensities λn is bounded, Λ is finite. For the investigation of the various waiting time distributions of a typical arriving call we need some notation:
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9
pW pI
probability that a typical arriving call has to wait for service probability that a typical arriving call will leave the system due to impatience later WS (x) = P (WS ≤ x) distribution function of the waiting time WS of a typical arriving call on the condition that it will be served WI (x) = P (WI ≤ x) distribution function of the waiting time WI of a typical arriving call on the condition that it will leave the system due to impatience later W (x) = P (W ≤ x) distribution function of the (unconditional) waiting time W of a typical arriving call In the following we will make use of the Palm-distribution with respect to arrival epochs of calls (cf., e.g., [11, (1.2.6)]) and of the conservation principle for stationary point processes with respect to arrival epochs of calls and epochs where the calls leave the waiting room, respectively. (If a call finds at its arrival a free server, then it leaves the queue immediately, i.e., its arrival epoch coincides with the epoch of leaving the queue.) The corresponding rigorous proofs are not outlined here because the results are well-known. (However, exact proofs can be given, e.g., by means of Campbell’s formula along the lines as, e.g., in [6, Section 6.4, (6.4.1)].) The probability 1 − pW is given by the ratio of the arrival intensity of calls finding at their arrival a free server and of the total arrival intensity Λ, i.e., s−1
1 − pW =
s
1X 1X λn p(n) = µn p(n). Λ n=0 Λ n=1
(3.7)
Note, that by the intensity conservation principle we have that the intensity of calls finding at least one free server at their arrival equals the intensity of served calls leaving behind at least one free server. The latter intensity is just the last sum. The probability 1 − pI is given by the ratio of the intensity λ(S) of arriving calls which will be served (immediately or after waiting in the queue) and of the total arrival intensity Λ. By the conservation principle λ(S) is the intensity of calls leaving the waiting room for starting service. This yields ! ! s−1 ∞ s ∞ X X 1 X 1 X 1 − pI = λn p(n) + µ∗ p(n) = µn p(n) + µ∗ p(n) . Λ n=0 Λ n=1 n=s+1 n=s+1
(3.8)
Note that the expression in the bracket of the last equation is just the intensity of served calls, which by the conservation principle is equal to the intensity of calls leaving the waiting room for service (immediately after their arrival or after waiting). Next, we are interested in the waiting time distributions WS (x), WI (x) and W (x) of a typical arriving call. Let us in the following assume that the queueing system is stable and that the distribution C(u) of the maximal waiting times is non-defective and has a continuous density c(u). For fixed x ∈ R+ the probability P (WS > x) = 1 − WS (x) is just the ratio of the intensity λ(S,x) of arriving calls which will be served and whose waiting times up to service are larger than x to the intensity λ(S) = (1 − pI )Λ of all arriving calls which will be served. By the intensity conservation principle λ(S,x) equals the intensity of time instants where calls leave the waiting room for starting service and which have waited longer than x. Hence, we obtain
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
1 − WS (x) =
Z ∞ X 1 µ∗ 1{x < u1 − x1 }p(n; x1 , . . . , xl ; u1 , . . . , ul ) (1 − pI )Λ n=s+1 R2l + ×dx1 · · · dxl du1 · · · dul .
Because of Eqs. (2.1),(2.17) and (3.2) and using the substitution ui = ξi + xi we find that for n > s Z 1 {x < u1 − x1 }p(n; x1 , . . . , xl ; u1 , . . . , ul ) dx1 · · · dxl du1 · · · dul R2l +
=g
n−1 Y
! λi
i=0
1 (l − 1)!
Z
∞
µ∗ x
F (ξ )l−1 F 0 (ξ )e−ξ dξ,
x ∈ R+ .
In view of (3.5) thus it follows s+j ∞ µ∗ p(s) X Y λi 1 − WS (x) = (1 − pI )Λ j =0 i=s
!
1 j!
Z
∞
µ∗ x
F (ξ )j F 0 (ξ ) e−ξ dξ,
x ∈ R+ .
(3.9)
For finding an explicit expression for WI (x) we proceed analogously to WS (x). The probability P (WI > x) = 1 − WI (x) is the ratio of the intensity λ(I,x) of arriving calls departing from the system due to impatience later and whose time spent in the queue is larger than x to the intensity λ(I ) = pI Λ of all arriving calls leaving due to impatience later. By the conservation principle λ(I,x) equals the intensity of time instants where calls leave the waiting room due to impatience and which have waited longer than x. Hence, we obtain ∞ l Z 1 XX 1{x < ui }p(n; x1 , . . . , xi−1 , 0, xi+1 , . . . , xl ; u1 , . . . , ul ) 1 − WI (x) = pI Λ n=s+1 i=1 R+2l−1
×dx1 · · · dxi−1 dxi+1 · · · dxl du1 · · · dul . In view of (2.1),(2.17) and (3.2) and using the substitution ui = ξi + xi after some algebra we find that for n > s l Z X
d − dx
2l−1 i=1 R+
1{x < ui }p(n; x1 , . . . , xi−1 , 0, xi+1 , . . . , xl ; u1 , . . . , ul ) dx1 · · · dxi−1 !
×dxi+1 · · · dxl du1 · · · dul
=g
n−1 Y
! λi
i=0
1 c(x) (l − 1)!
Z
∞
µ∗ x
F (ξ )l−1 e−ξ dξ,
x ∈ R+ ,
and by integration it follows that l Z X i=1
2l−1 R+
1{x < ui }p(n; x1 , . . . , xi−1 , 0, xi+1 , . . . , xl ; u1 , . . . , ul )dx1 · · · dxi−1 dxi+1 · · · dxl
×du1 · · · dul = µ∗ g
n−1 Y i=0
! λi
1 (l − 1)!
Z
∞
µ∗ x
F (ξ )l−1 (F 0 (µ∗ x) − F 0 (ξ )) e−ξ dξ,
x ∈ R+ .
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
11
In view of (3.5) thus we obtain s+j ∞ µ∗ p(s) X Y λi 1 − WI (x) = pI Λ j =0 i=s
!
1 j!
Z
∞
µ∗ x
F (ξ )j (F 0 (µ∗ x) − F 0 (ξ )) e−ξ dξ,
x ∈ R+ .
(3.10)
The waiting time distribution W (x) is given by W (x) = (1 − pI )WS (x) + pI WI (x),
x ∈ R+ .
(3.11)
The case of a general distribution C(u) of the maximal waiting times is obtained again by considering C(u) as the limit in distribution of a sequence of non-defective distributions Cν (u) with continuous density. From (3.9), (3.10) and (3.11) applied to Cν (u) by arguments of continuity we obtain the following statement. Theorem 3.2. Let the system be stable with a general distribution C(u) of the i.i.d. maximal waiting times. Then, for the waiting time distributions it holds (3.9), (3.10) and (3.11), where the probability p(s) that precisely s calls are in the system, the probability pI that a typical arriving call will leave the system due to impatience later and the cumulative arrival intensity Λ in the steady state are given by (3.5), (3.8) and (3.6), respectively. Especially from (3.9) and (3.10) by integration over x ∈ R+ we obtain the mean waiting times ! Z s+j −1 ∞ p(s) X Y 1 ∞ λi F (ξ )j (ξ − 1)e−ξ dξ, (3.12) E WS = (1 − pI )Λ j =1 i=s j! 0 ∞
p(s) X E WI = pI Λ j =1
!
s+j −1
Y
λi
i=s
1 j!
Z
∞
F (ξ )j (j + 1 − ξ )e−ξ dξ.
(3.13)
0
From Little’s formula for the (unconditional) mean waiting time EW of a typical arriving call we find EW =
∞ 1 X (n − s)p(n). Λ n=s+1
(3.14)
4. Application: performance analysis of an ACD system with outbound calls and impatient inbound calls For many services and businesses there is a need to match incoming calls with agents, e.g. in telebanking, teleshopping, information services, mail orders, etc. Automatic call distributor systems (ACD systems) are controlled by a software allowing to manage the incoming calls such that a group of ACD agents can handle a high volume of incoming calls. For a description of ACD systems see [10,21,22,24]. The ACD feature provides statistical reporting tools in addition to call queueing. ACD historical reports allow to identify times when incoming calls abandon after long waits in the queue because too few agents are staffed, or times where agents are idle. One might wish to use these data to determine an optimal operational strategy with respect to minimal cost (or maximal revenue) and various service quality requirements, e.g. for the abandon probability and for quantiles of waiting times. There are various operational strategies for
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
Fig. 2. Automatic call distributor system: combined inbound and outbound call center with impatient inbound calls, s agents and k waiting places, where n denotes the number of inbound and served outbound calls in the system.
managing the congestion level in such ACD systems. Based on the traffic the number of agents to meet a certain service varies in time. Also, by programming the telephone switching system, the service provider can temporarily deactivate or activate some of the available lines. As pointed out by So and Tang [24] there is in general a tradeoff between various performance measures, e.g. between the number of ‘abandoned’ and ‘blocked’ calls, i.e., calls which are rejected by the system when all lines are occupied. Thus, it is a non-trivial problem to determine the optimal operational strategy. For optimizing ACD systems, cf., e.g. [5], it is necessary to have stable and fast algorithms for computing the relevant performance measures. In this section we apply the results of Section 3 to a performance analysis of an ACD system providing phone service (inbound) and calling customers directly (outbound, telemarking). Combined inbound and outbound call centers were first analyzed by Dumas et al. [10]. At the call center, cf. Fig. 2, consisting of a finite positive number s of agents and a finite positive number k of waiting places (i.e., s + k lines), there arrive calls from outside (inbound calls) according to a Poisson process of intensity λ. An arriving inbound call requires an exponentially distributed service time with parameter µ. If there is at least one free line, i.e. agent or waiting place, then the call will be accepted; otherwise it gets lost. An accepted call will be served immediately by one of the idle agents if there is anyone idle. If all agents are busy then the call begins to wait until one of the agents becomes idle; the queueing discipline is FCFS. But the inbound calls are impatient, i.e., they are lost if the waiting time exceeds a random time (maximal waiting time). The maximal waiting times are assumed to be i.i.d. with a general distribution C(u) = P (I ≤ u), where I denotes a typical maximal waiting time. The maximal waiting times may correspond to real maximal waiting times or/and to a special management for those inbound calls whose waiting time exceeds an acceptable amount of time. Of special interest is the case that I = min(X, τ ), where X is exponentially distributed describing the individual maximal waiting time of an inbound call and τ is deterministic describing the technical maximal waiting time of the system. The time τ may be a deterministic time after that the inbound call is routed automatically by the system to another call center of the company or will be handled by special agents in order to improve the grade-of-service. To improve the efficiency of the call center, underutilized agents may also be employed to dial outbound calls. Specifically, we introduce a parameter, a ∈ {0, 1, . . . , s}, such that if more than a agents are idle
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
13
and no calls are in the queue, then one of the idle agents will dial an outbound call rather than wait for an arrival of an inbound call. Thus, at each time instant with probability one at least s − a agents are busy, cf. [10]. We assume that there is an infinite reservoir of possible outbound calls (list of customers) and that the service times of the outbound calls are exponentially distributed with the same parameter µ as the inbound calls. For the ACD system the following performance characteristics are of interest: Λin Λout pB
rate of accepted inbound calls rate of dialed outbound calls probability that a typical arriving inbound call finds no free waiting place or agent (blocking probability, 1 − pB is the acceptance probability) pW probability that a typical accepted inbound call has to wait for service probability that a typical accepted inbound call will get lost due to impatience later (impatience pI probability) WS (x) waiting time distribution of a typical accepted inbound call up to its service on the condition that it will be served WI (x) waiting time distribution of a typical accepted inbound call on the condition that it will leave the system due to impatience later W (x) (unconditional) waiting time distribution of a typical accepted inbound call The ACD system may be modeled by an M(n)/M(n)/s + GI queueing system, where n corresponds to the number of inbound and served outbound calls in the ACD system, cf. Fig. 1. The state dependent arrival intensity λn is given by λn = 1{n < s + k}λ,
n = 0, 1, . . . ,
(4.1)
and the state dependent service intensity µn by µn = 1{n > s − a}min(n, s)µ,
n = 0, 1, . . . ,
(4.2)
especially we have µ∗ = sµ. (Note that the time instants where the service of an outbound call is started does not correspond to changes of the system state. However, for finding the relevant performance measures this does not matter as seen below.) In view of Theorem 3.1 the system is stable and for the stationary occupancy distribution it holds (λ/µ)n , s − a ≤ n ≤ s, n! Z ∞ 1 (λF (ξ ))n−s e−ξ dξ, p(n) = g λs (n − s)! 0
p(n) = g s!µs
p(n) = 0
s < n ≤ s + k,
elsewhere,
where F (ξ ) is given by (3.2). From (3.3),(4.1) and (4.2) we obtain the normalizing factor Z s−1 k X X (λ/µ)j 1 ∞ (λF (ξ ))j e−ξ dξ. + λs g −1 = s!µs j ! j ! 0 j =s−a j =0
(4.3)
(4.4)
For the ACD system p(n) is the stationary probability that precisely n inbound and served outbound calls are in the system. The cumulative arrival intensity Λ in the M(n)/M(n)/s + GI queueing system in
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
the steady state corresponds to the rate Λin of accepted inbound calls in the ACD system. Hence, from (3.6),(4.1) and (4.3) it follows Λin = Λ = λ
s+k−1 X
p(n) = (1 − p(s + k))λ.
(4.5)
n=s−a
The blocking probability pB for inbound calls is given by the PASTA-property, i.e., pB = p(s + k),
Λin = Λ = (1 − pB )λ.
(4.6)
The service of an outbound call is started if precisely n = s − a agents are busy, no calls are in the queue and if one of the service times just finishes. The probability of being in the state n = s − a is p(s − a) and for the ACD system the cumulative intensity of finishing service in this state is (s − a)µ. Therefore, for the rate Λout of dialed outbound calls it follows Λout = (s − a)µp(s − a).
(4.7)
From (3.7), (4.1), (4.2), (4.3) and (4.6) we obtain the probability pW that an accepted inbound call has to wait for service s s−1 X X 1 µ 1 − pW = p(n) = np(n), 1 − pB n=s−a (1 − pB )λ n=s−a+1
(4.8)
and from (3.8), (4.2), (4.3) and (4.6) the probability pI that an accepted inbound call will get lost due to impatience later s+k X µ 1 − pI = min(n, s)p(n). (1 − pB )λ n=s−a+1
(4.9)
From (3.9), (4.1) and (4.6) for the waiting time distribution of an accepted inbound call up to its service on the condition that it will be served it follows Z k−1 X µ∗ p(s) 1 ∞ 1 − WS (x) = (λF (ξ ))j F 0 (ξ )e−ξ dξ, x ∈ R+ . (4.10) (1 − pB )(1 − pI ) j =0 j ! µ∗ x From (3.10), (4.1) and (4.6) for the waiting time distribution of an accepted inbound call on the condition that it will leave the system due to impatience later we find Z k−1 µ∗ p(s) X 1 ∞ (λF (ξ ))j (F 0 (µ∗ x) − F 0 (ξ ))e−ξ dξ, x ∈ R+ . (4.11) 1 − WI (x) = (1 − pB )pI j =0 j ! µ∗ x From (3.11), (4.10) and (4.11) for the unconditional waiting time distribution of an accepted inbound call we obtain Z k−1 X 1 ∞ µ∗ p(s) 0 1 − W (x) = F (µ∗ x) (λF (ξ ))j e−ξ dξ, x ∈ R+ . (4.12) 1 − pB j ! µ∗ x j =0 The corresponding expectations result from (4.1),(4.3),(4.6),(3.12),(3.13) and (3.14), respectively.
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
15
Remark 4.1. For k → ∞ we obtain the case of a waiting room of unlimited size. From (4.4) for the normalizing factor it follows: Z ∞ s−1 X (λ/µ)j −1 s s eλF (ξ )−ξ dξ. +λ g = s!µ j ! 0 j =s−a The stability condition corresponds to the finiteness of the right-hand side of this equation, because of (3.2) being equivalent to (λ/µ∗ )limu→∞ (1 − C(u)) < 1, cf. [2]. Finally, we consider the special case that the typical maximal waiting time I is the minimum of a constant and an exponentially distributed time, i.e., I = min(X, τ ), where X is exponentially distributed with parameter α describing the individual maximal waiting time of an inbound call and τ is deterministic describing the technical maximal waiting time of the ACD system. The distribution of I is given by C(u) = 1 − 1{u < τ }e−αu ,
u ∈ R+ .
(4.13)
Remark 4.2. For the distribution (4.13) of the maximal waiting times the limiting case k → ∞ with λn ≡ λ for n ≥ s and µn = min(n, s)µ for n ≥ 0 was earlier analyzed in [16]. For several performance measures, basing on the waiting time vector process, in this case formulae are available. However, they do not include the occupancy distribution. Remark 4.3. The distribution (4.13) of the maximal waiting times includes the following two special cases. For τ → ∞ the process N (t) of the number of calls in the M(n)/M(n)/s + GI system converges to a birth death process which can be analyzed in the usual manner. For α → 0 we obtain a system with constant maximal waiting times; for earlier results we refer to [13]. For the special distribution (4.13) of the maximal waiting times stable and fast numerical algorithms for the performance measures considered were derived from the above given formulae and implemented. As an example we consider the following basic parameter set: – mean service time 1/µ: 120 s, – mean individual maximal waiting time 1/α: 90 s, – deterministic maximal waiting time τ : 60 s. In Table 1 the offered load λ/µ is 10 Erl, in Table 2 the offered load is 100 Erl. The number s of agents, the number k of waiting places and the outbound parameter a vary in these tables. The blocking probability pB , the impatience probability pI , the expectations EWS and EWI of the conditional waiting time distributions and the rate of dialed outbound calls Λout are presented. In the last four lines of the tables, the corresponding performance measures for an ACD system without outbound calls are given. The numerical results show the impact of the operational strategy (outbound parameter a) on the system performance. If the outbound parameter a is chosen not too small, then the performance measures are not significantly worse than corresponding performance measures for the system without outbound calls (given in the last four lines of the tables), but the efficiency of the call center is improved by dialing outbound calls. The numerical results illustrate the tradeoff between the blocking probability pB and the impatience probability pI if k varies. But there is no obvious relation between the mean conditional waiting times
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
Table 1 Blocking probability pB , impatience probability pI , the expectations EWS and EWI of the conditional waiting time distributions and the rate of dialed outbound calls Λout for an offered load of 10 Erl s
k
a
λ/µ
1/µ
1/α
τ
pB
pI
EWS
EWI
Λout
8 12 16 20 8 12 16 20 8 12 16 20 8 12 16 20
3 3 3 3 6 6 6 6 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 6 6 6 6 8 12 16 20
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
0.137 0.049 0.016 0.006 0.024 0.006 0.001 0.000 0.131 0.034 0.006 0.001 0.131 0.031 0.003 0.000
0.170 0.061 0.024 0.011 0.254 0.088 0.031 0.013 0.162 0.042 0.009 0.002 0.162 0.039 0.005 0.000
11.472 4.729 1.955 0.884 15.696 6.341 2.446 1.037 10.769 3.174 0.723 0.173 10.758 2.931 0.388 0.023
22.286 14.258 9.988 7.687 26.739 17.738 11.960 8.769 22.286 14.258 9.988 7.687 22.286 14.258 9.988 7.687
0.003 0.015 0.039 0.067 0.002 0.015 0.038 0.067 0.000 0.004 0.019 0.045 0.000 0.000 0.000 0.000
Table 2 Blocking probability pB , impatience probability pI , the expectations EWS and EWI of the conditional waiting time distributions and the rate of dialed outbound calls Λout for an offered load of 100 Erl s
k
a
λ/µ
1/µ
1/α
τ
pB
pI
EWS
EWI
Λout
90 100 110 120 90 100 110 120 90 100 110 120 90 100 110 120
15 15 15 15 30 30 30 30 15 15 15 15 15 15 15 15
10 10 10 10 10 10 10 10 20 20 20 20 90 100 110 120
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
0.037 0.012 0.003 0.001 0.002 0.000 0.000 0.000 0.037 0.010 0.002 0.000 0.036 0.010 0.001 0.000
0.081 0.042 0.018 0.007 0.111 0.050 0.020 0.007 0.079 0.036 0.011 0.002 0.079 0.035 0.009 0.001
7.365 3.681 1.551 0.608 10.155 4.442 1.702 0.634 7.146 3.117 0.913 0.200 7.138 3.053 0.776 0.102
6.568 5.328 4.307 3.501 8.817 6.466 4.827 3.726 6.568 5.328 4.307 3.501 6.568 5.328 4.307 3.501
0.006 0.025 0.067 0.127 0.005 0.024 0.066 0.127 0.000 0.004 0.022 0.065 0.000 0.000 0.000 0.000
EWS and EWI . Comparing Tables 1 and 2 shows us the well-known effect that larger systems are more efficient than small ones. Finally, we remark that the derived numerical algorithms have been implemented in the form of a software tool for cost optimizing ACD systems over several (quality of service) constraints.
A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18
17
Acknowledgements This work was supported by a grant from the Siemens AG. References [1] F. Baccelli, P. Boyer, G. Hebuterne, Single server queues with impatient customers, Adv. Appl. Probab. 16 (1984) 887–905. [2] F. Baccelli, G. Hebuterne, On queues with impatient customers, in: F.J. Kylstra (Ed.), Performance ’81, North-Holland, Amsterdam, 1981, pp. 159–179. [3] D.Y. Barrer, Queueing with impatient customers and indifferent clerks, Oper. Res. 5 (1957) 644–649. [4] D.Y. Barrer, Queueing with impatient customers and ordered service, Oper. Res. 5 (1957) 650–656. [5] A. Brandt, M. Brandt, G. Spahl, D. Weber, Modelling and optimization of call distribution systems, Proc. 15th Int. Teletraffic Congress (ITC 15), Washington, DC, USA, 1997, pp. 133–144. [6] A. Brandt, P. Franken, B. Lisek, Stationary Stochastic Models, Akademie-Verlag, Berlin; Wiley, Chichester, 1990. [7] S.M. Brodi, On an integro-differential equation for systems with τ -waiting (in Ukrainian), Dop. AN URSR 6 (1959) 571–573. [8] S.M. Brodi, On a service problem (in Russian), Tr. V. Vsesojuznogo sove˘sc˘ anija po teorii verojatnostej i matemati˘ceskoj statistike, Izd-vo AN Arm. SSR, Erevan (1960) 143–147. [9] D.J. Daley, General customer impatience in the queue GI/G/1, J. Appl. Probab. 2 (1965) 186–205 . [10] G. Dumas, M. Perkins, C. White, Improving efficiency of PBX-based call centers: Combining inbound and outbound agents with automatic call sharing, Proc. 15th Int. Switching Symp., Berlin, 1995, pp. 346–350. [11] P. Franken, D. König, U. Arndt, V. Schmidt, Queues and Point Processes, Akademie-Verlag, Berlin; Wiley, Chichester, 1982. [12] B.W. Gnedenko, D. König, Handbuch der Bedienungstheorie, vol. II, Akademie-Verlag, Berlin, 1984. [13] B.W. Gnedenko, I.N. Kowalenko, Einführung in die Bedienungstheorie (1st ed. in Russian, Nauka, Moscow, 1966). Akademie-Verlag, Berlin, 1974. [14] R.B. Haugen, Queueing systems with several input streams and time out, Telektronikk 2 (1978) 100–106. [15] R.B. Haugen, E. Skogan, Queueing systems with stochastic time out, IEEE Trans. Commun. COM-28 (1980) 1984–1989. [16] O.M. Jurkevi˘c, On the investigation of many-server queueing systems with bounded waiting time (in Russian), Izv. Akad. Nauk SSSR Techni˘ceskaja kibernetika 5 (1970) 50–58. [17] O.M. Jurkevi˘c, On many-server systems with stochastic bounds for the waiting time (in Russian), Izv. Akad. Nauk SSSR Techni˘ceskaja kibernetika 4 (1971) 39–46. [18] O.M. Jurkevi˘c, On many-server systems with general service time distribution of ”impatient customers” (in Russian), Izv. Akad. Nauk SSSR Techni˘ceskaja kibernetika 6 (1973) 80–89. [19] A. Mandelbaum, N. Shimkin, A model for rational abandonments from invisible queues, Unpublished paper, Technion, Haifa, 1998. [20] C. Palm, Methods of judging the annoyance caused by congestion, Tele No. 2 (1953) 1–20. [21] M. Perry, Performance modelling of automatic call distributors, Ph.D. Thesis, North Carolina State University, 1991. [22] M. Perry, A. Nilsson, Performance modelling of automatic call distributors: Assignable grade of service staffing, Proc. 14th Int. Switching Symp., Yokohama, 1992, pp. 294–298. [23] M. Singh, Steady state behaviour of serial queueing processes with impatient customers, Math. Operationsforsch. Ser. Statist. 15(2) (1984) 289–298. [24] K.C. So, C. Tang, Operational strategies for managing congestion in service systems. Working paper, Graduate School of Management University of California, Irvine, CA, 1993. [25] J. Teghem, Use of discrete transforms for the study of a GI /M/s queue with impatient customer phenomena, Z. Oper. Res. 23 (1979) 95–106. [26] B. Wallstrøm, A queueing system with time-outs and random departure, Proc. ITC 8, Melbourne, 1976, paper 231.
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A. Brandt, M. Brandt / Performance Evaluation 35 (1999) 1–18 Andreas Brandt was born in Berlin in 1955. He received his Ph.D. degrees in mathematics from the Humboldt-University of Berlin in 1983. From 1983 until 1989 he was research assistant and from 1990– 1992 head of the group Applied Probability at the Department of Mathematics. Since 1992 he has been Professor for Operations Research at the Humboldt-University of Berlin. His main research interests are in stochastic processes, in particular point processes, and their applications. Over the last years he has worked closely with several industrial partners. The projects address the performance analysis of telecommunication and computer systems as well as reliability analysis of technical systems.
Manfred Brandt received his Ph.D. degrees in mathematics from the Humboldt-University of Berlin in 1973. Since 1989 he has been a Privatdozent at the Humboldt-University of Berlin. His main research interests are in pure complex analysis and since the eighties, also in the performance analysis of computer and teletraffic systems as well as queueing theory. He works closely with industrial partners and has implemented various algorithms. Since 1994 he has been a member of the scientific staff at the KonradZuse-Zentrum for Information Technology in Berlin.