Single Server Markovian Retrial Queues with Multiple Types of Outgoing Calls Hiroyuki Sakurai
Tuan Phung-Duc∗
Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama, Meguro-ku, Tokyo 152-8552, Japan
Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama, Meguro-ku, Tokyo 152-8552, Japan
[email protected]
[email protected]
ABSTRACT We consider single server Markovian retrial queues with multiple types of outgoing calls. Incoming calls arrive at the server according to a Poisson process. The service time of an incoming call follows the exponential distribution. Incoming calls that nd the server busy upon arrival join an orbit and retry after some exponentially distributed time. On the other hand, the server makes an outgoing call after some exponentially distributed idle time. We assume that there are multiple types of outgoing calls whose durations follow the exponential distribution with dierent parameters. For this model, we obtain explicit expressions for the joint stationary distribution of the number of calls in the orbit and the state of the server using the generating function approach. We also obtain simple asymptotic and recursive formulae for the joint stationary distribution. We show a stochastic decomposition property where we prove that the total number of incoming calls in the system (server and orbit) can be expressed by the sum of three independent random variables which have a clear physical meaning.
1. INTRODUCTION In retrial queues, arriving customers that nd the server busy repeat their service after some random time. During consecutive retrials, customers are said to be in a virtual waiting room called orbit. Retrial queues arise from various real life situations as well as telecommunication and network systems [3]. The literature on retrial queues is vast and rich with two books by Falin and Templeton [11] and by Artalejo and Gomez-Corral [3] and a huge number of research papers. Almost papers on retrial queues assume that the server provides service for only incoming customers. However, in various real life situations and especially in a call center context [6, 8], a server not only serves incoming calls but also handles some private work when it is idle [8]. ∗
Corresponding author
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This private work is referred to as an outgoing call and a queue with both incoming and outgoing calls is referred to as a two way communication queueing model in recent literature [4, 5]. Deslauriers et al. [8] consider ve Markovian models for blending call centers where operators not only serve incoming calls but also make calls to outside. However, the retrial behavior of customers is not taken into account in [8]. Falin [9] analyzes an M/G/1 retrial queue with two way communication where service times of incoming and outgoing calls are assumed to follow the same general distribution. Choi et al. [7] extend the analysis to a model in which there is a nite buer for outgoing calls. Martin and Artalejo [13] consider an M/G/1 retrial queue with two way communication and constant retrial rate where the service times of incoming and outgoing calls follow two dierent general distributions. Artalejo and Phung-Duc [4] consider a single server retrial queue with two way communication where incoming calls and outgoing calls follow the exponential distribution with dierent parameters. The authors derive explicit expressions for the generating functions as well as the joint stationary distribution of the number of calls in the orbit and the state of the server. They further derive some asymptotic and recursive formulae for the joint stationary distribution. Artalejo and Phung-Duc [5] extend their analysis to an M/G/1 retrial queue with two way communication which turns out to have a close relation with the priority retrial model [10] where an in nite buer is available for outgoing calls. In all the work above, the are at most one ow of outgoing calls. However, in practice there are various types of outgoing calls whose conversation times may be extremely dierent. In addition, outgoing calls could be considered as the durations that the server breaks down and thus cannot serve incoming calls. From this point of view, there are also various types of breakdowns whose repair times may follow dierent distributions. Thus, modeling all these types of outgoing calls by one exponential distribution as in [4] may aect the accuracy of the performance evaluation. This motivates us to consider the model where various types of outgoing calls follow dierent distributions. The contributions of the current paper are twofold. First, we extend the analysis of Artalejo and Phung-Duc [4] to the model with multiple types of outgoing calls. In this paper, we assume that several types of outgoing calls follow the exponential distribution with dierent parameters. It should be noted that our model can be formulated using a
level-dependent QBD process where a general numerical algorithm based on matrix continued fractions [14] is available. However, explicit solutions for the stationary distribution of level-dependent QBD processes are obtained in only a few special cases. In this paper, using the generating function approach, we show that all the explicit results in Artalejo and Phung-Duc [4] for the model with one type of outgoing calls could be extended for our new model. Second, we study the new model in more depth. In particular, we show that the stationary distribution could be explicitly expressed by some sophisticated formulae for some special cases whose details are omitted in Artalejo and PhungDuc [4]. Our main result is the stochastic decomposition in which we prove that the number of incoming calls in the system is the sum of three independent random variables whose physical meaning is clear. To the best of our knowledge, the decomposition property has not been reported in literature of retrial queues with two way communication [4, 5]. The rest of the current paper is organized as follows. In Section 2, we present the model in details and some preliminary results for the following sections. Section 3 is devoted to our main results where we present explicit formulae for the generating functions, the stationary distribution and the stochastic decomposition property. Furthermore, we derive recursive and asymptotic formulae for the stationary distributions. Section 4 shows to some numerical examples. Finally, Section 5 presents some concluding remarks.
2. MODEL AND PRELIMINARIES
Figure 1: Behavior of this model.
forms a continuous time Markov chain on the state space: S = {0, 1, 2, 3} × Z+ , where Z+ = {0, 1, 2, . . .}. In what follows, we assume that the Markov chain is ergodic, i.e., the stationary distribution of {X(t)} exists. We will derive the ergodic condition later. Under the ergodic condition, let πi,j = lim P(S(t) = i, N (t) = j), t→∞
i = 0, 1, 2, 3,
j ∈ Z+ ,
denote the joint stationary distribution of the system state. Figure 2 shows the transitions among states.
2.1 Model descriptions We consider single server retrial queues with two way communication and multiple types of outgoing calls. Incoming calls arrive at the server according to a Poisson process with rate λ and request for an exponentially distributed service time with mean 1/ν1 . Incoming calls that nd the server busy join the orbit and repeat their request for service after an exponentially distributed time with mean 1/µ. On the other hand, the server makes an outgoing call after an exponentially distributed idle time. In particular, there are n types of outgoing calls whole durations follow the exponential distribution with n dierent parameters. Our calculations show that the analysis of the case n ≥ 3 is the same as that for the case n = 2. Thus, in this paper we restrict ourselves to the latter, i.e., n = 2. We assume that the durations of outgoing calls of type 1 and type 2 follow the exponential distribution with means 1/ν2 and 1/ν3 , respectively. Furthermore, if the server is idle, it makes an outgoing call of type 1 or type 2 in an exponentially distributed time with mean 1/α2 and 1/α3 , respectively. In order to distinguish these two types of calls, we assume that ν2 > ν3 . Figure 1 shows the behavior of this model.
2.2 Transition diagram and balance equations Let S(t) denote the state of the server at time t ≥ 0, 0, 1, S(t) = 2, 3,
the server is idle, if an incoming call is in service, if an outgoing call of type 1 is in service, if an outgoing call of type 2 is in service.
Let N (t) denote the number of incoming calls in the orbit at time t. It is easy to see that {X(t) = (S(t), N (t)); t ≥ 0}
Figure 2: Transitions among states. It follows from Figure 2 that the system of balance equations for πi,j is given by (λ + α2 + α3 + jµ)π0,j = ν1 π1,j + ν2 π2,j + ν3 π3,j , (1)
(λ + ν1 )π1,j = (j + 1)µπ0,j+1 + λπ0,j + λπ1,j−1 , (2) (λ + ν2 )π2,j = α2 π0,j + λπ2,j−1 , (3) (λ + ν3 )π3,j = α3 π0,j + λπ3,j−1 ,
(4)
for j ∈ Z+ , where πi,−1 = 0 (i = 1, 2, 3). Let Πi (z) denote the partial generating functions of πi,j , i.e., Πi (z) =
∞ ∑
πi,j z j ,
i = 0, 1, 2, 3, |z| ≤ 1.
j=0
Multiplying equations (1)-(4) by z j and taking the sum over
Proof. Equations (12) and (13) immediately follow from (7) and (8). Substituting (9) into (5), we obtain
j yields (λ + α2 + α3 )Π0 (z) +
µzΠ′0 (z)
(5) (λ + ν1 )Π1 (z) = + λΠ0 (z) + λzΠ1 (z), (6) (λ + ν2 )Π2 (z) = α2 Π0 (z) + λzΠ2 (z), (7) (λ + ν3 )Π3 (z) = α3 Π0 (z) + λzΠ3 (z). (8) Summing up equations (5)-(8) and rearranging the result, we obtain = ν1 Π1 (z) + ν2 Π2 (z) + ν3 Π3 (z),
(λ + α2 + α3 )Π0 (z) + λz(Π1 (z) + Π2 (z) + Π3 (z))
µΠ′0 (z)
µΠ′0 (z)(z − 1) = λ(Π1 (z) + Π2 (z) + Π3 (z))(z − 1).
Dividing both sides of the above equation by (z − 1) yields (9) µΠ′0 (z) = λ(Π1 (z) + Π2 (z) + Π3 (z)). We note that equation (9) expresses the balance between the ows going into and out the orbit.
3. STATIONARY DISTRIBUTION In this section, we obtain explicit expressions for the joint stationary distribution using the generating function approach. In what follows, we assume that ν2 > ν3 , ν1 ̸= λ+ν2 and ν1 ̸= λ + ν3 . See Appendix A for the case ν1 = λ + ν3 .
3.1 Generating function First, we derive explicit expressions for the partial generating functions. Theorem 3.1. For |z| ≤ 1, explicit expressions for the partial generating functions are given as follows: 1−ρ 1 + σ2 + σ3 ( ) D1 ( ) D2 ( ) D3 µ µ µ 1−ρ 1 − θ2 1 − θ3 × 1 − ρz 1 − θ2 z 1 − θ3 z
Π0 (z) =
D − µ1
D − µ2
D − µ3
= π0,0 (1 − ρz) (1 − θ2 z) (1 − θ3 z) , (10) ( ) λ + C12 + C13 C2 C3 Π1 (z) = + + ν1 − λz λ + ν2 − λz λ + ν3 − λz × Π0 (z), (11) α2 Π0 (z), (12) Π2 (z) = λ + ν2 − λz α3 Π3 (z) = Π0 (z), (13) λ + ν3 − λz
where λα2 λα3 , C13 = − , ν1 − (λ + ν2 ) ν1 − (λ + ν3 ) λα2 λα3 C2 = , C3 = , ν1 − (λ + ν2 ) ν1 − (λ + ν3 ) λα3 λα2 − , D1 = λ − ν1 − (λ + ν2 ) ν1 − (λ + ν3 ) α2 (ν1 − ν2 ) α3 (ν1 − ν3 ) D2 = , D3 = , ν1 − (λ + ν2 ) ν1 − (λ + ν3 ) λ α2 α3 ρ= , σ2 = , σ3 = , ν1 ν2 ν3 λ λ θ2 = , θ3 = , λ + ν2 λ + ν3 C12 = −
and π0,0 =
D1 D2 D3 1−ρ (1 − ρ) µ (1 − θ2 ) µ (1 − θ3 ) µ . (14) 1 + σ2 + σ3
= ν1 Π1 (z) + ν2 Π2 (z) + ν3 Π3 (z).
Substituting (12) and (13) into the above equation and rearranging the result, we nd that λα2 λ Π0 (z) + Π0 (z) ν1 − λz (ν1 − λz)(λ + ν2 − λz) λα3 + Π0 (z) (ν1 − λz)(λ + ν3 − λz) ( ) λ + C12 + C13 C2 C3 = + + ν1 − λz λ + ν2 − λz λ + ν3 − λz × Π0 (z),
Π1 (z) =
where C12 , C13 , C2 and C3 satisfy λα3 λα2 + (ν1 − λz)(λ + ν2 − λz) (ν1 − λz)(λ + ν3 − λz) C12 + C13 C2 C3 + + . = ν1 − λz λ + ν2 − λz λ + ν3 − λz
Substituting (11)-(13) into (9) yields λ (Π1 (z) + Π2 (z) + Π3 (z)) µ ( ) λ + C12 + C13 α 2 + C2 α 3 + C3 = + + ν1 − λz λ + ν2 − λz λ + ν3 − λz λ × Π0 (z). µ
Π′0 (z) =
De ning D1 = λ+C12 +C13 , D2 = α2 +C2 , D3 = α3 +C3 , we obtain the dierential equation: Π′0 (z) λ = Π0 (z) µ
(
D1 D2 D3 + + ν1 − λz λ + ν2 − λz λ + ν3 − λz
)
.
The solution of this dierential equation is given by ( Π0 (z) = Π0 (1) (
) D1 µ ν1 − λ ν1 − λz ) D2 (
) D3 µ µ ν2 ν3 λ + ν2 − λz λ + ν3 − λz ( ) D1 ( ) D2 ( ) D3 µ µ µ 1−ρ 1 − θ2 1 − θ3 = Π0 (1) . 1 − ρz 1 − θ2 z 1 − θ3 z
×
It follows from (11)-(13) that (
Π1 (1) =
λ λα2 λα3 + + ν1 − λ (ν1 − λ)ν2 (ν1 − λ)ν3
Π2 (1) =
) Π0 (1),
α3 α2 Π0 (1), Π3 (1) = Π0 (1). ν2 ν3
Furthermore, from the normalization condition: Π0 (1) + Π1 (1) + Π2 (1) + Π3 (1) = 1,
we obtain Π0 (1) =
1−ρ . 1 + σ2 + σ3
Corollary 3.2. It follows from (14) that the necessary and su cient for the ergodicity of {X(t); t ≥ 0} is ρ < 1.
3.2 Stochastic decomposition In this section, we consider the stochastic decomposition property for the number of incoming calls in the system (the server and the orbit). In particular, we prove that the number of incoming calls in the system can be decomposed into three independent random variables which have a clear physical justi cation. An intuitive interpretation of this observation is as follows. Our model can be considered as a vacation queue where the vacation corresponds the period where the server is idle while some incoming calls are available in the orbit. This type of decomposition property has been reported in the literature [2, 11]. Thus, we expect that the number of incoming calls is the sum of the number of calls in the orbit under the condition that the server is idle and the number of incoming calls in the conventional M/M/1 queue without retrial but with outgoing call. The conventional M/M/1 queue with outgoing call and without retrial can be further considered as a vacation model where the vacation corresponds to the period where the server is not serving an incoming call. Therefore, we also expect another decomposition property for this model. We will rigorously prove these observations in Theorem 3.3. Let Nµ,α (t) denote the number of incoming calls in the system and let Rµ,α (t) denote the number of incoming calls in the system under the condition that the server is idle. The subscript (µ, α) where α = (α1 , α2 ), expresses the dependence of the parameters.
We show that the generating functions for N∞,0 (t), Rµ,α (t) and V∞,α (t) are (17), (18) and (19), respectively. First, because N∞,0 (t) is the number of incoming calls in the M/M/1 queue, it is clear that the generating function of N∞,0 (t) is given by (17)． Second, we compute the generating function of Rµ,α (t). Indeed, ] [ ] [ E z Rµ,α (t) = E z N (t) S(t) = 0 Π0 (z) Π0 (z) = P(S(t) = 0) Π0 (1) ( ) D1 ( ) D2 ( ) D3 µ µ µ 1−ρ 1 − θ2 1 − θ3 = , 1 − ρz 1 − θ2 z 1 − θ3 z =
which is consistent with (18). Finally, we compute the generating function for V∞,α (t). Let N∞,α (t) denote the number of incoming calls in the M/M/1 queue with outgoing calls but without retrials. We have [ ] [ ] E z V∞,α (t) = E z N∞,α (t) S(t) ̸= 1
1 1 − P(S(t) = 1) ] ∑ [ N × E z ∞,α (t) S(t) = i P(S(t) = i) =
i̸=1
] ∑ [ N 1 E z ∞,α (t) S(t) = i Πi (1) 1 − Π1 (1) i̸=1 [ ] 1 = E z N∞,α (t) S(t) = 0 1 + σ2 + σ3 [ ] σ2 + E z N∞,α (t) S(t) = 2 1 + σ2 + σ3 [ ] σ3 + E z N∞,α (t) S(t) = 3 . 1 + σ2 + σ3
Theorem 3.3. In the steady state, we have Nµ,α (t) = N∞,0 (t) + Rµ,α (t) + V∞,α (t),
=
(15)
where the equality in (15) means equal in distribution. The notations in the right hand side of (15) are de ned as follows. • N∞,0 (t): number of incoming calls in the M/M/1 queue. • V∞,α (t): the number of incoming calls arrive at the
M/M/1 queue with outgoing call and without retrial during a vacation period (i.e., under the condition that the server is not serving an incoming call). Proof. Assuming that (15) is established, the generating function of Nµ,α (t) satis es [ ] [ ] E z Nµ,α (t) = E z N∞,0 (t)+Rµ,α (t)+V∞,α (t) [ ] [ ] [ ] = E z N∞,0 (t) E z Rµ,α (t) E z V∞,α (t) . (16)
We will prove this formula. Let Π(z) denote the left hand side of (16). We have Π(z) ≡ Π0 (z) + zΠ1 (z) + Π2 (z) + Π3 (z) ( ) λz = Π0 (z) 1 + ν1 − λz ( ) α2 α3 × 1+ + λ + ν2 − λz λ + ν3 − λz 1−ρ = (17) 1 − ρz ) D1 ( ) D2 ( ) D3 ( µ µ µ 1−ρ 1 − θ2 1 − θ3 × (18) 1 − ρz 1 − θ2 z 1 − θ3 z ( ) ν2 ν3 1 1 + σ2 + σ3 . (19) × 1 + σ2 + σ3 λ + ν2 − λz λ + ν3 − λz
It should be noted that under the condition S(t) = 0, we have N∞,α (t) = 0. On the other hand, under the condition: S(t) = 2 or 3, N∞,α (t) corresponds to the number of incoming calls that arrive during the remaining service time of the outgoing call on service. [ ] E z N∞,α (t) S(t) = i =
νi , λ + νi − λz
i = 2, 3.
Thus, we have [ ] E z V∞,α (t) =
1 1 + σ2 + σ3 ( ) ν3 ν2 + σ3 , × 1 + σ2 λ + ν2 − λz λ + ν3 − λz
which is consistent with (19).
3.3
Explicit and recursive expressions
In this section, we present the explicit expressions for the stationary distribution. Because these formulae are obtained directly from the generating functions presented in Section 3.1, we omit the proof. We refer to [4] for a detailed proof for the model with one type of outgoing calls.
Theorem 3.4. For j ∈ Z+ , we have ) k( ) ℓ j j−k { ( ∑ ∑ D1 ρ D2 θ2 µ k k! µ ℓ ℓ! k=0 ℓ=0 ( ) } θ3j−k−ℓ D3 × , (20) µ j−k−ℓ (j − k − ℓ)! ( )j−k j λ 1 ∑ (λπ0,k + (k + 1)µπ0,k+1 ) = λ + ν1 λ + ν1 k=0 ( ) j C12 + C13 ∑ = ρ+ π0,k ρj−k ν1
π0,j = π0,0
π1,j
Corollary 3.6. (Theorem VI.12, p. 434 in [12]). ∑∞ Let rna and rb denote the convergent radii of a(z) = n=0 an z ∑ n and b(z) = ∞ n=0 bn z , respectively. If ra > rb ≥ 0 then the convergent radius of g(z) = a(z)b(z) is rb and we have [z n ]g(z) ∼ a(rb )bn ,
where [z n ]g(z) is the coe cient of z n in the series expansion of g(z) and xn ∼ yn is de ned by limn→∞ xn /yn = 1. Corollary 3.7. (Corollary 2.1 in [4]). For a, γ > 0, we have [z n ](1 − γz)−a ∼
k=0
j C2 ∑ C3 ∑ π0,k θ2j−k + π0,k θ3j−k , λ + ν2 λ + ν3 j
+
k=0
(21)
where Γ(z) denotes the Gamma function de ned by ∫
α2 ∑ π0,k θ2j−k , λ + ν2
(22)
j α3 ∑ = π0,k θ3j−k . λ + ν3
(23)
k=0
where
{ 1, (x)j = x(x + 1) · · · (x + j − 1),
j = 0, j ∈ N = {1, 2, . . . },
Theorem 3.5. =
π2,j
=
π3,j
=
π1,j
=
Corollary 3.8. (Proposition 2.3 in [4]). If an ∼ a ˜n ,
bn ∼ ˜bn ,
lim
n→∞
a ˜n = 0, ˜bn
then an + bn ∼ ˜bn .
denotes the Pochhammer symbol. Substituting parameters into (14) and (20)-(23), we can obtain numerical result for the stationary distribution. However, the computation may be numerically unstable since D1 , D2 and D3 may be negative. In Theorem 3.5 below, we present simple recursive formulae which yield a numerically stable procedure for calculating the stationary distribution. π0,j
e−t ta−1 dt.
0
k=0
π , 3,j
∞
Γ(a) =
k=0
j
π2,j =
na−1 γ n , Γ(a)
λ(π1,j−1 + π2,j−1 + π3,j−1 ) , j ∈ N, jµ α2 π0,j + λπ2,j−1 , j ∈ Z+ , λ + ν2 α3 π0,j + λπ3,j−1 , j ∈ Z+ , λ + ν3 λ(π0,j + π1,j−1 + π2,j + π3,j ) , j ∈ Z+ , ν1
Using Corollaries 3.6, 3.7 and 3.8, we obtain Theorem 3.9 as follows. Theorem 3.9. (i) If θ2 < θ3 < ρ, we have ( π0,j
π1,j
∼ π0,0
∼ π0,0
1−
θ2 ρ
) − D2 ( µ 1− ( ) Γ Dµ1
( D1 1 −
θ2 ρ
(
(25)
π2,j
∼ π2,0 (
(26) (27)
where πi,−1 = 0 (i = 1, 2, 3) and π0,0 is given by (14). Proof. It is easy to see that (25) and (26) are followed from (3) and (4). Furthermore, (24) is obtained by comparing the coe cients of z j−1 (j ∈ N) in both sides of (9). Finally, substituting (24) into (2) and arranging the result yields (27). Remark 1. Theorem 3.5 implies a recursive algorithm for the stationary distribution where π0,0 is explicitly given by (14). It should be noted that the algorithm can be implemented in both numerical and symbolic manners.
3.4 Asymptotic analysis
π3,j
∼ π3,0
1−
θ2 ρ
θ2 ρ
µ
j
µ
(
D1 µ
1 − θρ3 ) +1
)− D2 −1 ( µ 1− ( ) Γ Dµ1 ) − D2 ( µ 1− ( ) Γ Dµ1
θ3 ρ
θ3 ρ
D1 µ
−1 j
ρ ,
(28)
) − D3 µ
j
D1 µ
ρj ,
(29)
)− D3 µ
j
D1 µ
−1 j
D1 µ
−1 j
ρ , (30)
)− D3 −1 µ
j
ρ . (31)
(ii) If θ2 < ρ < θ3 or ρ < θ2 < θ3 , we have π0,j
π1,j
π2,j
∼
∼
∼
( 1− π0,0
∼
) − D1 ( µ 1− ( ) Γ Dµ3
( 1− π2,0 ( 1− π3,0
θ2 θ3
)− D2 µ
j
D3 µ
−1 j θ3 ,
)− D1 ( ) − D2 µ µ 1 − θθ23 D3 ( ) j µ θ3j , (λ + ν3 )Γ Dµ3 + 1
π0,0
We observe from explicit expressions in Theorem 3.4 that π3,j
ρ θ3
( C3 1 −
πi,j (i = 0, 1, 2, 3) depends on j in a complex manner. In this
section, we derive some simpler asymptotic formulae showing the order of j in the expressions for πi,j (i = 0, 1, 2, 3). To this end, we rst list the following corollaries.
1−
)− D3
)− D2 (
ν1 Γ
(24)
θ3 ρ
ρ θ3
ρ θ3
ρ θ3
) − D1 ( µ 1− ( ) Γ Dµ3
θ2 θ3
(32)
(33)
)− D2 −1 µ
j
D3 µ
−1 j θ3 ,
) − D1 ( )− D2 µ µ 1 − θθ32 D3 ( ) j µ θ3j . D3 Γ µ +1
(34)
(35)
right hand side of the above formula as follows.
Remark 2. In case (i), π0,j , π2,j and π3,j have the same order while π1,j has a higher order. On the other hand, in case (ii), π0,j and π2,j have the same order while π1,j and π3,j have the same higher order. Furthermore, the orders of π2,j and π3,j are dierent in case (ii) implying that two types of outgoing calls could not be merged into one exponential distribution as in Theorem 3.6 in [4].
[z j ]
Proof. We derive asymptotic formulae for πi,j in case (i). It follows from (10) that D
− µ1
Π0 (z) = π0,0 (1 − ρz)
D
− µ2
(1 − θ2 z)
[z j ]
D
− µ3
(1 − θ3 z)
(36)
= π0,0 a(z)b(z)c(z),
where
[z j ] D
− µ1
a(z) = (1−ρz)
D
, b(z) = (1−θ2 z)
− µ2
D
− µ3
, c(z) = (1−θ3 z)
Let ra , rb , rc denote the convergent radii of a(z), b(z) and c(z), respectively. We investigate the order of these radii. It follows from θ2 < θ3 < ρ that λ + ν2 > λ + ν3 > ν1 leading to D1 > 0 and ra = 1/ρ. D2 and D3 are either positive or negative. If ν1 < ν2 then D2 > 0 and rb = 1/θ2 while if ν1 ≥ ν2 then D2 ≤ 0 and rb = ∞. Similarly, if ν1 < ν3 , we have D3 > 0 and rc = 1/θ3 while if ν1 ≥ ν3 then D3 ≤ 0 and rc = ∞. In any case, we have ra < rb , ra < rc . Therefore, applying Corollary 3.6 to (36), we obtain [z j ]b(z)c(z)a(z) ∼ b(1/ρ)[z j ]c(z)a(z)
(37)
∼ b(1/ρ)c(1/ρ)[z j ]a(z).
Furthermore, applying Corollary 3.7 to a(z) yields [z ]a(z) ∼ j
j
D1 µ
Γ
(
−1 j D1 µ
ρ ).
(38)
From (36)-(38), we obtain (28). We derive asymptotic formula for π2,j and π3,j . We obtain the following formulae for Π2 (z) and Π3 (z) by substituting (10) into (12) and (13) and using (25) and (26) with j = 0. D
− µ1
Π2 (z) = π2,0 (1 − ρz)
D
− µ1
Π3 (z) = π3,0 (1 − ρz)
D
− µ2 −1 (1
(1 − θ2 z)
D
− µ2
(1 − θ2 z)
D
− µ3
− θ3 z)
,
D
− µ3 −1 .
(1 − θ3 z)
Thus, we obtain (30) and (31) by the same arguments as used in the derivation of the asymptotic formula for π0,j . We derive an asymptotic formula for π1,j . Substituting (10) into (11), we obtain D D D D1 − 1 −1 − 2 − 3 π0,0 (1 − ρz) µ (1 − θ2 z) µ (1 − θ3 z) µ ν1 D D D C2 − 1 − 2 −1 − 3 π0,0 (1 − ρz) µ (1 − θ2 z) µ + (1 − θ3 z) µ λ + ν2 D D D C3 − 1 − 2 − 3 −1 + π0,0 (1 − ρz) µ (1 − θ2 z) µ (1 − θ3 z) µ . λ + ν3
Π1 (z) =
Using the same arguments as used for Π0 (z), we obtain the asymptotic formulae for the coe cients of each term in the
.
D D D D1 − 1 −1 − 2 − 3 (1 − ρz) µ (1 − θ2 z) µ (1 − θ3 z) µ ν1 ( )− D2 ( )− D3 µ µ θ2 θ3 1 − 1 − D1 ρ ρ D1 ( ) ∼ j µ ρj , ν1 Γ Dµ1 + 1
(39)
D D D C2 − 1 − 2 −1 − 3 (1 − ρz) µ (1 − θ2 z) µ (1 − θ3 z) µ λ + ν2 ( )− D2 −1 ( )− D3 µ µ 1 − θρ3 1 − θρ2 D1 C2 −1 j ( ) j µ ρ , (40) ∼ D1 λ + ν2 Γ µ D D D C3 − 1 − 2 − 3 −1 (1 − ρz) µ (1 − θ2 z) µ (1 − θ3 z) µ λ + ν3 ( ) − D2 ( )− D3 −1 µ µ θ3 θ2 1 − 1 − D1 ρ ρ C3 −1 j ( ) ∼ j µ ρ . (41) D1 λ + ν3 Γ µ
Furthermore, because lim
j→∞
j
D1 µ
j
−1 j
D1 µ
ρ
= 0,
ρj
we obtain (29) from (39)-(41). The case (ii) is proved by the same method as that for case (i). Thus we omit a detailed proof.
4.
NUMERICAL EXAMPLES
In this section, we show some numerical results to show the feasibility of the recursive formulae in Theorem 3.5 and verify the accuracy of the asymptotic formulae in Theorem 3.9. Using Theorem 3.5, we can exactly compute the stationary probabilities πi,j . On the other hand, asymptotic formulae are dierent for cases (i) and (ii) as is presented in Theorem 3.9. We x µ = 1, ν1 = 1, α2 = 1, α3 = 0.8. First, we consider the parameter set: λ = 0.9, ν2 = 2.5, ν3 = 1 for which ρ = 0.9, θ2 = 0.26, θ3 = 0.47 and (28)-(31) are obtained. Figure 3 presents both exact and asymptotic values for the joint stationary distribution πi,j against j . We observe that the curve of exact value and that of the corresponding asymptotic formula become consistent as j increases. We further observe from Figure 3 that π1,j is greater than π0,j , π2,j , π3,j when j is large. This fact veri es our asymptotic formulae where we observe that the order of π1,j is bigger than that of πi,j with i ̸= 1. Second, we consider the case λ = 0.1, ν2 = 0.1, ν3 = 0.01 for which ρ = 0.1, θ2 = 0.5, θ3 = 0.91 and asymptotic formulae (32)-(35) are obtained. We also observe from Figure 4 that the curves by asymptotic formulae are consistent with that of exact value when j becomes large. We also observe that π1,j , π3,j are greater than π0,j , π2,j when j is large enough. This fact veri es the asymptotic formulae for case (ii).
5.
CONCLUDING REMARKS
In this paper, we have considered Markovian single server retrial queues with multiple types of outgoing calls. Using the generating function approach, we have shown the stochastic decomposition property for the number of incoming calls in the system. We have derived explicit and re-
Probability
0.1
1
π0,j exact π2,j exact π0,j asymptotic π2,j asymptotic
0.1 Probability
1
0.01 0.001 0.0001
0.001
0 5 10 15 20 25 30 Number of customers in the orbit (ρ = 0.1) 1
π1,j exact π3,j exact π1,j asymptotic π3,j asymptotic
0.1 Probability
Probability
0.1
0.01
0.0001
0 5 10 15 20 25 30 Number of customers in the orbit (ρ = 0.9) 1
π0,j exact π2,j exact π0,j asymptotic π2,j asymptotic
0.01 0.001 0.0001
cursive formulae for the joint stationary distribution of the number of retrial calls and the state of the server. We also have obtained asymptotic formulae for the joint stationary distribution. Some numerical examples have been provided to show the feasibility of recursive formulae and verify the asymptotic formulae. It is interesting to extend the analysis to an M/G/1 retrial queueing model with multiply types of outgoing calls whose durations follow dierent general distributions. We are planing to investigate this topic in the future.
6. REFERENCES [1] J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000-2009, Mathematical and Computer Modelling, 51 (2010), 1071-1081. [2] J. R. Artalejo and G. I. Falin, Stochastic decomposition for retrial queues, Top, 2 (1994), 329-342. [3] J. R. Artalejo and A. Gomez-Corral, Retrial Queueing Systems: a Computational Approach, Springer, 2008. [4] J. R. Artalejo and T. Phung-Duc, Markovian retrial queues with two way communication, Journal of Industrial and Management Optimization, Vol. 8, No. 4 (2012), 781-806. [5] J. R. Artalejo and T. Phung-Duc, Single server retrial queues with two way communication, Applied Mathematical Modelling, Vol. 37, No. 4 (2013), 1811-1822. [6] S. Bhulai and G. Koole, A queueing model for call
0.01 0.001 0.0001
0 5 10 15 20 25 30 Number of customers in the orbit (ρ = 0.9)
Figure 3: # of calls in the orbit for case θ2 < θ3 < ρ.
π1,j exact π3,j exact π1,j asymptotic π3,j asymptotic
0 5 10 15 20 25 30 Number of customers in the orbit (ρ = 0.1)
Figure 4: # of calls in the orbit for case ρ < θ2 < θ3 .
[7] [8]
[9] [10] [11] [12] [13] [14]
blending in call centers, IEEE Transactions on Automatic Control, 48 (2003), 1434-1438. B. D. Choi, K. B. Choi and Y. W. Lee, M/G/1 retrial queueing systems with two types of calls and nite capacity, Queueing systems, 19 (1995), 215-229. A. Deslauriers, P. L'Ecuyer, J. Pichitlamken, A. Ingolfsson and A. N. Avramidis, Markov chain models of a telephone call center with call blending, Computers & Operations Research, 34 (2007), 1616-1645. G. I. Falin, Model of coupled switching in presence of recurrent calls, Engineering Cybernetics Review, 17 (1979), 53-59. G. I. Falin, J. R. Artalejo and M. Martin, On the single server retrial queue with priority customers, Queueing systems, 14 (1993), 439-455. G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, 1997. P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009. M. Martin and J. R. Artalejo, Analysis of an M/G/1 queue with two types of impatient units, Advances in Applied Probability, 27 (1995), 840-861. T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, A simple algorithm for the rate matrices of level-dependent QBD processes, Proceedings of the 5th International Conference on Queueing Theory and Network Applications, (2010), 46-52.
APPENDIX A. SPECIAL CASES We consider only the case ν1 = λ + ν3 but the case ν1 = λ+ν2 is the same. It should be noted that (15) and (24)-(27) are also established for these special cases. Theorem A.1. For |z| ≤ 1,
[ ] 1−ρ α2 ρˆ ρ exp − 1 + σ2 + σ3 µ ( ) D1 [ ]( ) D3 µ µ 1−ρ α2 ρ2 z 1 − θ3 × exp 1 − ρz µ(1 − ρz) 1 − θ3 z [ ] 2 D1 D α ρ z 2 − − 3 = π0,0 (1 − ρz) µ exp (1 − θ3 z) µ , (42) µ(1 − ρz) ( ) λ + C13 λα2 C3 Π1 (z) = + + Π0 (z), ν1 − λz (ν1 − λz)2 λ + ν3 − λz α2 Π2 (z) = Π0 (z), ν1 − λz α3 Π3 (z) = Π0 (z), λ + ν3 − λz Π0 (z) =
where λα3 λα3 , C3 = , ν1 − (λ + ν3 ) ν1 − (λ + ν3 ) α3 (ν1 − ν3 ) λα3 D1 = λ + α2 − , D3 = , ν1 − (λ + ν3 ) ν1 − (λ + ν3 ) λ ρ λ ρ= , ρˆ = , θ3 = , ν1 1−ρ λ + ν3 α2 α3 σ2 = , σ3 = , ν2 ν3 [ ] D1 D3 1−ρ α2 ρˆ ρ π0,0 = (1 − ρ) µ exp − (1 − θ3 ) µ . 1 + σ2 + σ3 µ C13 = −
Proof. This theorem is obtained after a minor modi cation of the derivation in Section 3.1. Theorem A.2. For j ∈ Z+ , π0,j
π1,j
k=0
j λα2 ∑ C3 ∑ π0,k (j − k + 1)ρj−k + π0,k θ3j−k , 2 ν1 λ + ν3 j
k=0
α2 ∑ π0,k ρj−k , ν1 j
π2,j =
k=0
π3,j =
j α3 ∑ π0,k θ3j−k . λ + ν3 k=0
[
exp
] ∑ ( )k ∞ α2 ρ2 z 1 α2 ρ2 z = µ(1 − ρz) k! µ(1 − ρz) k=0 ( ( ))k ∞ ∑ 1 α2 ρ 1 = −1 k! µ 1 − ρz k=0 (∞ )k ( ) ∞ k ∑ ∑ 1 α2 ρ j = (ρz) k! µ j=1 k=0 )k ( )k (∑ ∞ ∞ ∑ 1 α2 ρ2 z j = (ρz) k! µ j=0 k=0 ( )k ∑ ∞ ∞ ∑ 1 α2 ρ2 z j =1+ j+k−1 Ck−1 (ρz) k! µ j=0 k=1 ( )k ∑ ∞ ∞ ∑ 1 α2 ρ j+k =1+ j+k−1 Ck−1 (ρz) k! µ j=0 k=1 ( )k ∑ ∞ ∞ ∑ 1 α2 ρ j =1+ j−1 Ck−1 (ρz) k! µ k=1 j=k ( j ( )k ) ∞ ∑ ∑ 1 α2 ρ =1+ ρj z j . j−1 Ck−1 k! µ j=1 k=1
It follows from (42) that
[ ] D D α2 ρ2 z − 1 − 3 Π0 (z) = π0,0 (1 − ρz) µ exp (1 − θ3 z) µ µ(1 − ρz) (∞ ( )( ∞ ( ) ∑ D1 ) ρ j j ∑ D3 ) θ j j 3 = π0,0 z z µ j j! µ j j! j=0 j=0 { ( j ( )k ) } ∞ ∑ ∑ 1 α2 ρ ρj z j , × 1+ C j−1 k−1 k! µ j=1 k=1
) k( ) ℓ j−1 j−k−1 { ( ∑ ∑ ρ D3 θ3 D1 = µ k k! µ ℓ ℓ! k=0 ℓ=0 (j−k−ℓ ( )m ) } ∑ 1 α2 ρ × ρj−k−ℓ j−k−ℓ−1 Cm−1 m! µ m=1 ( ) ( ) j ∑ θ3j−k D1 ρ k D3 + , (43) µ k k! µ j−k (j − k)! k=0 ( )j−k j 1 ∑ λ = (λπ0,k + (k + 1)µπ0,k+1 ) λ + ν1 λ + ν1 k=0 ( ) j C13 ∑ π0,k ρj−k = ρ+ ν1 +
Proof. We prove (43) from which the expressions for πi,j (i = 1, 2, 3) are straightforward. First, we derive the Taylor expansion at z = 0 for the exponential part in (42) as follows.
k=0
leading to (43). Other formulae are obtained by the same manner as is presented in [4].
