Noname manuscript No. (will be inserted by the editor)
Two Way Communication Retrial Queues with Multiple Types of Outgoing Calls Hiroyuki Sakurai and Tuan Phung-Duc
October 5, 2014
Abstract In this paper, we start with single server Markovian retrial queues with multiple types of outgoing calls. Incoming calls arrive at system according to a Poisson process. Service times of incoming calls follow the exponential distribution. Incoming calls that find 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 distinct exponential distributions. 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 via 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 number of incoming calls in the system (server and orbit) can be decomposed into the sum of three independent random variables which have a clear physical meaning. We then consider the multiserver model for which we obtain the stability condition and derive some exact formulae by mean value analysis. Finally, we extend the single server model to the case where service time distribution of incoming calls and that of each type of outgoing calls are arbitrary. For this case, we obtain explicit expressions for the partial generating functions and recursive formulae for the joint stationary distribution of the server’s state and the number of calls in the orbit.
1 Introduction In retrial queues, arriving customers that find the server busy repeat their request 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, 18]. The literature on retrial queues is Corresponding author: Tuan Phung-Duc Department of Mathematical and Computing Sciences, Tokyo Institute of Technology Ookayama, Meguro-ku, Tokyo 152-8552, Japan Tel.: +81-(0)3-5734-3851 Fax: +81-(0)3-5734-2752 E-mail:
[email protected]
2
vast and rich with two books by Falin and Templeton [14] and by Artalejo and GomezCorral [3] and a huge number of research papers. Almost all papers on retrial queues assume that the server provides service for only incoming customers [3, 14]. However, in various real life situations and especially in a call center context [7, 10], a server not only serves incoming calls but also handles some private work in idle time [10]. 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. [10] consider five 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 [10]. Falin [12] 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 arbitrary distribution. Choi et al. [8] extend to a model in which there is a finite buffer for outgoing calls. Martin and Artalejo [16] 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 different arbitrary distributions. Dimitriou [11] considers a related M/G/1 retrial queueing model for Long Term Evolution (LTE) networks. Artalejo and Phung-Duc [4] consider a single server retrial queue with two way communication where incoming calls and outgoing calls follow distinct exponential distributions. 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 retrial queues with priority [13] where an infinite buffer is available for outgoing calls. In all the work above, there is at most one flow of outgoing calls. However, in practice there are various types of outgoing calls whose durations may be extremely different. 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 different distributions [17]. Furthermore, from a service point of view, the importance placed on each class of outgoing calls is different. Thus, modeling all these types of outgoing calls by one exponential distribution as in [4] may affect the accuracy of the performance evaluation. This motivates us to consider models where multiple types of outgoing calls follow distinct distributions. The contributions of the current paper are threefold. First, we extend the analysis of Artalejo and Phung-Duc [4] to the model with multiple types of outgoing calls. We assume that several types of outgoing calls follow distinct exponential distributions. 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 is available [20]. However, explicit solutions for the stationary distribution of level-dependent QBD processes are obtained in only a few special cases [6, 19, 21]. 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. 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 a special case whose details are omitted in Artalejo and Phung-Duc [4]. Our main result for this model is the stochastic decomposition in which we prove that the num-
3
ber of incoming calls in the system is decomposed into 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]. This contribution has been presented in [23]. Second, we consider the Markovian model under a multiserver setting. It is wellknown that analytical solutions even for the most simple multiserver retrial queue are complex and are available only for some special cases, i.e., one or two servers [19, 21]. Thus, we focus on the stability condition for the model and explicit expressions for the mean number of busy servers with an incoming call and those with an outgoing call of a particular type. Third, we consider the single server model where the service time distributions of incoming calls and all the classes of outgoing calls are arbitrary. For this model, we obtain explicit expressions for the partial generating functions and recursive formulae for the joint stationary distribution of the server’s state and the number of customers in the orbit using an embedded Markov chain and renewal arguments. These results extend those of Artalejo and Phung-Duc [5], where there is one type of outgoing calls. The rest of the current paper is organized as follows. In Section 2, we present the single server model with exponentially distributed service times in details and some preliminary results for the next 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 presents some explicit results for the multiserver model. Section 5 is devoted to a summary of results for the single server model with arbitrary service time distributions for incoming and outgoing calls. Finally, concluding remarks are presented in Section 6.
2 Model and Preliminaries 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 system according to a Poisson process with rate λ and request for an exponentially distributed service time with mean 1/ν1 . Incoming calls that find 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 n distinct exponential distributions. Our calculations show that the analysis of the case n ≥ 3 is the same as that for the case n = 2. Thus, for this model 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 distributions 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 or 1/α3 , respectively. In order to distinguish these two types of calls, we assume that ν2 > ν3 . Figure 1 shows the flows of calls in this model.
4
Fig. 1 Flow of customers.
2.2 Transition diagram and balance equations Let S(t) denote the state of the server at time t ≥ 0,
0, if the server is idle, 1, if an incoming call is in service, S(t) = 2, if an outgoing call of type 1 is in service, 3,
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} 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. 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).
5
Fig. 2 Transitions among states.
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 j yields (λ + α2 + α3 )Π0 (z) + µzΠ0′ (z) = ν1 Π1 (z) + ν2 Π2 (z) + ν3 Π3 (z),
(5)
(λ + ν1 )Π1 (z) = µΠ0′ (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 µΠ0′ (z)(z − 1) = λ(Π1 (z) + Π2 (z) + Π3 (z))(z − 1). Dividing both sides of the above equation by (z − 1) yields µΠ0′ (z) = λ(Π1 (z) + Π2 (z) + Π3 (z)).
(9)
We note that equation (9) expresses the balance between the flows 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 .
6
3.1 Generating function First, we derive explicit expressions for the partial generating functions. Theorem 1 For |z| ≤ 1, explicit expressions for the partial generating functions are given as follows: 1−ρ Π0 (z) = 1 + σ2 + σ3
(
= π0,0 (1 − ρz)−
(
1−ρ 1 − ρz
D1 µ
) Dµ1 (
(1 − θ2 z)−
1 − θ2 1 − θ2 z
D2 µ
) Dµ2 (
(1 − θ3 z)−
1 − θ3 1 − θ3 z
D3 µ
,
λ + C12 + C13 C2 C3 + + ν1 − λz λ + ν2 − λz λ + ν3 − λz α2 Π2 (z) = Π0 (z), λ + ν2 − λz α3 Π3 (z) = Π0 (z), λ + ν3 − λz
) Dµ3
(10)
)
Π1 (z) =
Π0 (z)
(11) (12) (13)
where λα2 λα3 , C13 = − , ν1 − (λ + ν2 ) ν1 − (λ + ν3 ) λα2 λα3 C2 = , C3 = , ν1 − (λ + ν2 ) ν1 − (λ + ν3 ) λα2 λα3 D1 = λ − − , ν1 − (λ + ν2 ) ν1 − (λ + ν3 ) α (ν − ν2 ) α (ν − ν3 ) D2 = 2 1 , D3 = 3 1 , ν1 − (λ + ν2 ) ν1 − (λ + ν3 ) λ α α ρ= , σ2 = 2 , σ3 = 3 , 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 . 1 + σ2 + σ3
(14)
Proof Equations (12) and (13) immediately follow from (7) and (8). Substituting (9) into (5), we obtain (λ + α2 + α3 )Π0 (z) + λz(Π1 (z) + Π2 (z) + Π3 (z)) = ν1 Π1 (z) + ν2 Π2 (z) + ν3 Π3 (z). Substituting (12) and (13) into the above equation and rearranging the result, we find that λα2 λ Π0 (z) + Π0 (z) ν1 − λz (ν1 − λz)(λ + ν2 − λz) λα3 + Π0 (z) (ν1 − λz)(λ + ν3 − λz)
Π1 (z) =
(
=
λ + C12 + C13 C2 C3 + + ν1 − λz λ + ν2 − λz λ + ν3 − λz
)
Π0 (z),
7
where C12 , C13 , C2 and C3 satisfy λα2 λα3 C + C13 C2 C3 + = 12 + + . (ν1 − λz)(λ + ν2 − λz) (ν1 − λz)(λ + ν3 − λz) ν1 − λz λ + ν2 − λz λ + ν3 − λz Substituting (11)-(13) into (9) yields Π0′ (z) = =
λ µ
λ (Π1 (z) + Π2 (z) + Π3 (z)) µ
(
λ + C12 + C13 α 2 + C2 α3 + C3 + + ν1 − λz λ + ν2 − λz λ + ν3 − λz
) Π0 (z).
Defining D1 = λ+C12 +C13 , D2 = α2 +C2 , D3 = α3 +C3 , we obtain the differential equation: Π0′ (z) λ = Π0 (z) µ
(
D1 D2 D3 + + ν1 − λz λ + ν2 − λz λ + ν3 − λz
) .
The solution of this differential equation is given by
( Π0 (z) = Π0 (1)
( = Π0 (1)
ν1 − λ ν1 − λz
1−ρ 1 − ρz
) Dµ1 (
) Dµ1 (
ν2 λ + ν2 − λz
1 − θ2 1 − θ2 z
) Dµ2 (
) Dµ2 (
1 − θ3 1 − θ3 z
ν3 λ + ν3 − λz
) Dµ3
) Dµ3
.
It follows from (11)-(13) that
( Π1 (1) =
λ λα2 λα3 + + ν1 − λ (ν1 − λ)ν2 (ν1 − λ)ν3
Π2 (1) =
) Π0 (1),
α α2 Π0 (1), Π3 (1) = 3 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 2 It follows from (14) that the necessary and sufficient for the ergodicity of {X(t); t ≥ 0} is ρ < 1.
8
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 the sum of three independent random variables which have a clear physical justification. 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, 14]. 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 M/M/1 queue without retrials but with outgoing calls. The M/M/1 queue with outgoing calls and without retrials 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. 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 in the steady state. The subscript (µ, α) where α = (α1 , α2 ), expresses the dependence of the parameters. Theorem 3 In the steady state, we have
d
Nµ,α (t)=N∞,0 (t) + V∞,α (t) + Rµ,α (t),
(15)
d
where = in (15) means equal in distribution. The notations in the right hand side of (15) are defined as follows. – N∞,0 (t): number of incoming calls in the conventional M/M/1 queue, i.e., without retrials and outgoing calls. – V∞,α (t): the number of incoming calls arrive at the M/M/1 queue with outgoing calls and without retrials during a vacation period (i.e., under the condition that the server is not serving an incoming call). – Rµ,α (t): number of incoming calls in the system under the condition that the server is idle.
Proof Assuming that (15) is established, the generating function of Nµ,α (t) satisfies
[
]
[
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)
9
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 ν1 − λz
= Π0 (z) 1 + =
1−ρ 1 − ρz
(
)(
1+
α2 α3 + λ + ν2 − λz λ + ν3 − λz
)
(17)
)
D1 µ
×
1−ρ 1 − ρz
×
1 1 + σ2 + σ3
(
(
1 − θ2 1 − θ2 z
1 + σ2
)
D2 µ
(
1 − θ3 1 − θ3 z
)
D3 µ
(18)
ν3 ν2 + σ3 λ + ν2 − λz λ + ν3 − λz
) .
(19)
We show that the generating functions for N∞,0 (t), Rµ,α (t) and V∞,α (t) are given by (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) Π (z) = 0 P(S(t) = 0) Π0 (1)
( =
1−ρ 1 − ρz
) Dµ1 (
1 − θ2 1 − θ2 z
) Dµ2 (
1 − θ3 1 − θ3 z
) Dµ3 ,
which is consistent with (18). Third, 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 =
]
] ∑ [ N (t) 1 E z ∞,α S(t) = i P(S(t) = i) 1 − P(S(t) = 1) i̸=1
] ∑ [ N (t) 1 = E z ∞,α 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 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. ] [ νi , i = 2, 3. E z N∞,α (t) S(t) = i = λ + νi − λz
10
Thus, we have
[
]
E z V∞,α (t) =
1 1 + σ2 + σ3
( 1 + σ2
ν2 ν3 + σ3 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 4 For j ∈ Z+ , we have π0,j = π0,0
j j−k ∑ ∑ { ( D1 ) ρk ( D2 ) θℓ ( D3 ) 2
µ
k=0 ℓ=0
π1,j
k!
µ
ℓ
ℓ!
µ
j−k−ℓ
(20)
( )j−k j ∑ 1 λ = (λπ0,k + (k + 1)µπ0,k+1 ) λ + ν1 λ + ν1 k=0
( =
k
}
θ3j−k−ℓ , (j − k − ℓ)!
ρ+
π2,j =
π3,j =
C12 + C13 ν1
α2 λ + ν2
j ∑
)∑ j
π0,k ρj−k +
k=0
j j C2 ∑ C3 ∑ π0,k θ2j−k + π0,k θ3j−k , (21) λ + ν2 λ + ν3 k=0
k=0
π0,k θ2j−k ,
(22)
j α3 ∑ π0,k θ3j−k , λ + ν3
(23)
k=0
k=0
{
where (x)j =
1,
j = 0,
x(x + 1) · · · (x + j − 1),
j ∈ N = {1, 2, . . . },
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 5 below, we present simple recursive formulae which yield a numerically stable procedure for calculating the stationary distribution. Theorem 5 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
π0,j =
(24)
π2,j
(25)
π3,j π1,j
(26) (27)
11
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 coefficients 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 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 We observe from explicit expressions in Theorem 4 that 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 first list the following corollaries. Corollary 6 (Theorem VI.12, p. 434 in Let ra and rb denote the convergent ∑∞ ∑[15]). ∞ radii of a(z) = n=0 an z n and b(z) = n=0 bn z n , 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 coefficient of z n in the series expansion of g(z) and xn ∼ yn is defined by limn→∞ xn /yn = 1. Corollary 7 (Corollary 2.1 in [4]). For a, γ > 0, we have [z n ](1 − γz)−a ∼
na−1 γ n , Γ (a)
where Γ (z) denotes the Gamma function defined by
∫
∞
Γ (a) =
e−t ta−1 dt.
0
Corollary 8 (Proposition 2.3 in [4]). If an ∼ a ˜n ,
bn ∼ ˜bn ,
lim
n→∞
a ˜n = 0, ˜bn
then an + bn ∼ ˜bn . Using Corollaries 6, 7 and 8, we obtain Theorem 9 as follows.
12
Theorem 9 (i) If θ2 < θ3 < ρ, we have
(
1−
π0,j ∼ π0,0
θ2 ρ
)− Dµ2 ( ( Γ
(
π1,j ∼ π0,0
D1 1 −
θ2 ρ
1−
D1 µ
1−
π2,j ∼ π2,0
θ2 ρ
1−
π3,j ∼ π3,0
θ2 ρ
j
(
1−
)
D1 µ
(
D1 µ
)
Γ
1−
D1 µ
θ3 ρ
1−
)− Dµ2 ( (
θ3 ρ
D1 µ
−1 j
ρ ,
(28)
)− Dµ3 j
D1 µ
ρj ,
(29)
+1
)− Dµ2 −1 ( Γ
(
)− Dµ3
)− Dµ2 (
ν1 Γ
(
)
θ3 ρ
)
θ3 ρ
)− Dµ3 j
D1 µ
−1 j
D1 µ
−1 j
ρ ,
(30)
)− Dµ3 −1 j
ρ .
(31)
(ii) If θ2 < ρ < θ3 or ρ < θ2 < θ3 , we have
(
1−
π0,j ∼ π0,0
ρ θ3
)− Dµ1 ( ( Γ
(
π1,j ∼ π0,0
C3 1 −
ρ θ3
1−
D3 µ
)
)− Dµ1 ( (
(λ + ν3 )Γ
(
1−
π2,j ∼ π2,0
ρ θ3
(
1−
π3,j ∼ π3,0
ρ θ3
D3 µ
1−
D3 µ
)
)− Dµ1 ( ( Γ
1−
D3 µ
)− Dµ2 j
1−
)− Dµ1 ( Γ
(
θ2 θ3
θ2 θ3
D3 µ
−1 j θ3 ,
)− Dµ2
)
j
D3 µ
+1
θ2 θ3
θ2 θ3
(32)
θ3j ,
(33)
)− Dµ2 −1 j
D3 µ
−1 j θ3 ,
(34)
)− Dµ2
)
j
D3 µ
+1
θ3j .
(35)
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 different 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]. Proof We derive asymptotic formulae for πi,j in case (i). It follows from (10) that Π0 (z) = π0,0 (1 − ρz)−
D1 µ
(1 − θ2 z)−
D2 µ
(1 − θ3 z)−
D3 µ
= π0,0 a(z)b(z)c(z),
(36)
where a(z) = (1 − ρz)−
D1 µ
,
b(z) = (1 − θ2 z)−
D2 µ
,
c(z) = (1 − θ3 z)−
D3 µ
.
13
Let ra , rb and 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 6 to (36), we obtain
[z j ]b(z)c(z)a(z) ∼ b(1/ρ)[z j ]c(z)a(z) ∼ b(1/ρ)c(1/ρ)[z j ]a(z).
(37)
Furthermore, applying Corollary 7 to a(z) yields
[z j ]a(z) ∼
j
D1 µ
Γ
(
−1 j
ρ
D1 µ
).
(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 by using (25) and (26) with j = 0.
Π2 (z) = π2,0 (1 − ρz)−
D1 µ
(1 − θ2 z)−
D2 µ
−1
Π3 (z) = π3,0 (1 − ρz)−
D1 µ
(1 − θ2 z)−
D2 µ
(1 − θ3 z)−
(1 − θ3 z)−
D3 µ
D3 µ
−1
,
.
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
Π1 (z) =
D1 D2 D3 D1 π0,0 (1 − ρz)− µ −1 (1 − θ2 z)− µ (1 − θ3 z)− µ ν1
D1 D2 D3 C2 π0,0 (1 − ρz)− µ (1 − θ2 z)− µ −1 (1 − θ3 z)− µ λ + ν2 D1 D2 D3 C3 + π0,0 (1 − ρz)− µ (1 − θ2 z)− µ (1 − θ3 z)− µ −1 . λ + ν3
+
14
Using the same arguments as used for Π0 (z), we obtain the asymptotic formulae for the coefficients of each term in the right hand side of the above formula as follows. [z j ]
D1 D2 D3 D1 (1 − ρz)− µ −1 (1 − θ2 z)− µ (1 − θ3 z)− µ ν1
(
∼ [z j ]
D1 ν1
1−
θ2 ρ
)− Dµ2 ( ( Γ
[z j ]
C2 λ + ν2
1−
)− Dµ3
)
j
D1 µ
ρj ,
(39)
+1
θ2 ρ
)− Dµ2 −1 ( ( Γ
D1 µ
1−
)
θ3 ρ
)− Dµ3
j
D1 µ
−1 j
ρ ,
(40)
D1 D2 D3 C3 (1 − ρz)− µ (1 − θ2 z)− µ (1 − θ3 z)− µ −1 λ + ν3
(
∼
D1 µ
θ3 ρ
D1 D2 D3 C2 (1 − ρz)− µ (1 − θ2 z)− µ −1 (1 − θ3 z)− µ λ + ν2
(
∼
1−
C3 λ + ν3
1−
θ2 ρ
)− Dµ2 ( ( Γ
1−
D1 µ
)
θ3 ρ
)− Dµ3 −1
j
D1 µ
−1 j
ρ .
(41)
Furthermore, because lim
j→∞
j
D1 µ
j
−1 j
D1 µ
ρ
= 0,
ρj
we obtain (29) from (39)-(41). Case (ii) is proved by the same method as that for case (i). Thus we omit a detailed proof.
3.5 Numerical examples In this section, we present some numerical results to show the feasibility of the recursive formulae in Theorem 5 and to verify the accuracy of the asymptotic formulae in Theorem 9. Using Theorem 5, we can exactly compute the stationary probabilities πi,j . On the other hand, asymptotic formulae are different for cases (i) and (ii) as in Theorem 9. We fix 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 verifies 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 those of corresponding exact ones when j is large. We also observe that π1,j and π3,j are greater than π0,j and π2,j when j is large enough. This fact verifies the asymptotic formulae for case (ii).
15
Probability
0.1
π0,j exact π2,j exact π0,j asymptotic π2,j asymptotic
0.01 0.001
1 0.1 Probability
1
0.0001
π1,j exact π3,j exact π1,j asymptotic π3,j asymptotic
0.01 0.001 0.0001
0 5 10 15 20 25 30 Number of customers in the orbit (ρ = 0.9)
0 5 10 15 20 25 30 Number of customers in the orbit (ρ = 0.9)
Fig. 3 The number of calls in the orbit for case θ2 < θ3 < ρ.
Probability
0.1
π0,j exact π2,j exact π0,j asymptotic π2,j asymptotic
0.01 0.001 0.0001 0 5 10 15 20 25 30 Number of customers in the orbit (ρ = 0.1)
1 0.1 Probability
1
π1,j exact π3,j exact π1,j asymptotic π3,j asymptotic
0.01 0.001 0.0001 0 5 10 15 20 25 30 Number of customers in the orbit (ρ = 0.1)
Fig. 4 The number of calls in the orbit for case ρ < θ2 < θ3 .
4 Multiserver Markovian Model In this section, we consider a model of c identical servers. Incoming calls arrive at the system according to a Poisson process with rate λ and request for an exponentially distributed service time with mean 1/ν1 . The behavior of a server is the same as that in the single server model in Section 3. In particular, if a server is idle, it makes an outgoing call of type 1 and type 2 in exponentially distributed times with mean 1/α2 and 1/α3 , respectively. The durations of outgoing calls of type 1 and 2 follow exponential distributions with mean 1/ν2 and 1/ν3 , respectively. An incoming call that sees all the servers occupied joins the orbit and retries in an exponentially distributed time with mean 1/µ. For this system, we obtain the following explicit results. First, we prove that the stability condition is simply λ < cν1 . Second, we derive explicit expressions for the mean number of busy servers with an incoming call and those with an outgoing call of types 1 and type 2, respectively. Let S1 (t), S2 (t) and S3 (t) denote the number of busy servers with an incoming call, and those with an outgoing call of type 1 and type 2 at time t, respectively. Let N (t) denote the number of customers in the orbit at time t. It is easily to see that {X(t) = (N (t), S1 (t), S2 (t), S3 (t)); t ≥ 0} forms a Markov chain in the state space {(n, i, j, k) | n ∈ Z+ , i + j + k ≤ c}. Theorem 10 The necessary and sufficient condition for the stability of {X(t); t ≥ 0} is λ < cν1 .
16
Proof The transition rates of {X(t); t ≥ 0} are given as follows. i) i + j + k < c.
(n, i, j, k) → (n, i + 1, j, k), (n, i, j, k) → (n, i, j + 1, k), (n, i, j, k) → (n, i, j, k + 1), (n, i, j, k) → (n, i − 1, j, k),
λ, (c − i − j − k)α1 , (c − i − j − k)α2 , iν1 ,
(n, i, j, k) → (n, i, j − 1, k), jν2 , (n, i, j, k) → (n, i, j, k − 1), kν3 , (n, i, j, k) → (n − 1, i + 1, j, k), nµ. ii) i + j + k = c.
(n, i, j, k) → (n + 1, i, j, k),
(n, i, j, k) → (n, i − 1, j, k),
λ, iν1 ,
(n, i, j, k) → (n, i, j − 1, k), jν2 , (n, i, j, k) → (n, i, j, k − 1),
kν3 .
We apply the Tweedie’s criterion [24] to show the sufficient condition for the stability. To this end, we consider test function f (n, i, j, k) = n + a1 i + a2 j + a3 k. The mean drift h(s) associated with f (·) is given by h(s) =
∑
qs,r (f (r) − f (s)),
s ∈ S,
r ∈S where qs,r is the transition rate from state s to state r. i) i + j + k < c h(n, i, j, k) = λ + (c − i − j − k)α1 (a2 + a3 ) − iν1 a1 − jν2 a2 − kν3 a3 + nµ(a1 − 1).
(42)
ii) i + j + k = c h(n, i, j, k) = λ − cν1 a1 + j(ν1 a1 − ν2 a2 ) + k(ν1 a1 − ν3 a3 ). It follows from (42) that we need to choose 0 < a1 < 1 in order for h(n, i, j, k) < 0 for (n, i, j, k) ∈ S except for a finite number of states. Furthermore, we choose a2 and a3 such that ν1 a1 − ν2 a2 = 0, ν1 a1 − ν3 a3 = 0. Thus, for the case i+j+k = c, we have h(n, i, j, k) = λ−cν1 a1 . In order for h(n, i, j, k) < 0 except for a finite number of states, we need to choose a1 such that λ − cν1 a1 < 0 which is possible if λ/(cν1 ) < a1 < 1 or λ < cν1 . Thus λ < cν1 is the sufficient condition for the stability due to Tweedie [24]. The necessary condition follows from Little’s law E[S1 (t)] = λ/ν1 < c which is consistent with the model with one type of outgoing calls [22]. Indeed, we consider the system of the c servers. Since incoming calls are not lost in our model, they are eventually served and leave the system of c servers. The arrival rate of incoming calls (not necessarily Poisson) to this system is λ. The mean sojourn time of an incoming call in this system is the mean service time and is given by 1/ν1 . It then follows from Little’s law that E[S1 (t)] 1 = , ν1 λ leading to the announced result.
17
Theorem 11 In the steady state, we have
E[S1 (t)] =
λ = ρ, ν1
E[S2 (t)] =
(c − ρ)σ1 , 1 + σ1 + σ2
E[S3 (t)] =
(c − ρ)σ2 , 1 + σ1 + σ2
where ρ = λ/ν1 , σ1 = α1 /ν1 and σ2 = α2 /ν2 . Proof The mean number of idle servers is given by E[I(t)] = c − E[S1 (t)] − E[S2 (t)] − E[S3 (t)]. Thus, the arrival rates for outgoing calls of type 1 and 2 are E[I(t)]α2 and E[I(t)]α3 , respectively. We also consider the system of only c servers. The mean sojourn times of outgoing calls of type 1 and 2 are their mean service times given by 1/ν2 and 1/ν3 , respectively. Thus, Little’s law yields
E[Si (t)] 1 = , νi E[I(t)]αi
i = 2, 3,
or equivalently
E[Si (t)] αi = , i = 2, 3. νi E[I(t)] Noting that E[S1 (t)] = λ/ν1 and solving these equations yields the desired results. σi =
Remark 3 It should be noted that the proof of Theorem 11 requires neither Poisson arrivals nor exponentially distributed service times and idle times. Thus, Theorem 11 is valid as long as Little’s law is established. 5 M/G/1 retrial queue with n types of outgoing calls In this section, we consider the M/G/1 retrial queues with n types of arbitrarily distributed outgoing calls. Incoming calls arrive at the system according to a Poisson process with rate λ. If the server is idle it makes an outgoing call of type i in an exponentially distributed time with mean 1/αi (i = 2, 3, . . . , n + 1). Thus, when the server is idle, the next state is determined by the competition of the arrival of a new incoming call, the arrival of an incoming call from the orbit and the arrivals of n types of outgoing calls. The distribution function of incoming calls is B1 (x). The duration of an outgoing call of type i (i = 1, 2, . . . , n) is a random variable with the distribution function Bi+1 (x) (i = 1, 2, . . . , n). We denote the Laplace-Stieltjes transform and the kth moment of Bl (x) by βl (s) and βlk for l = 1, 2, . . . , n, n + 1. 5.1 Embedded Markov chain We consider the embedded Markov chain representing the number of customers in the orbit at the service completion epoch of either an incoming call or an outgoing call. It is easy to see that the one step transition probabilities of this Markov chain are given as follows. iµ pi,i−1 = k10 , i ≥ 1, ∑n+1 λ + m=2 αm + iµ pi,j =
∑n+1
λ
λ+
∑n+1
+ iµ
iµ
+ λ+ pi,j = 0,
m=2 αm
k1j−i +
∑n+1
m=2 αm + iµ i − j ≥ 2,
j−i m=2 αm km ∑n+1 λ + m=2 αm + iµ
k1j−i+1 ,
0 ≤ i ≤ j,
18
where, kℓj = have
∫∞ 0
e−λx
(λx)j j! dBℓ (x)
(ℓ = 1, 2, . . . , n + 1 and j ∈ Z+ ). Furthermore, we
Kℓ (z) =
∞ ∑
kℓj z j = βℓ (λ − λz).
j=0
Remark 4 The stability condition of the embedded Markov chain is ρ = λβ11 < 1 which can be proved by the same manner as in Artalejo and Phung-Duc [5]. Let πi (i ∈ Z+ ) denote the limiting ∑∞ probability of the embedded Markov chain at state i. Furthermore, let Π(z) = i=0 πi z i denote the generating function of {πi ; i ∈ Z+ }. Using the same derivation as used in [5] yields the following expression.
(
Π(z) =
λ 1+ ×
1−ρ
)
∑n+1
m=2 σm +
λ(1 − z)β1 (λ − λz) +
[ 1 × exp − µ
∫
1
z
(∑
n+1 m=2 αm
)
(1 − ρ)
∑n+1
m=2 αm (β1 (λ − λz) − zβm (λ − λz)) β1 (λ − λz) − z
]
∑n+1
λ(1 − β1 (λ − λu)) + m=2 αm (1 − βm (λ − λu)) du , β1 (λ − λu) − u
1 where ρ = λβ11 and σm = αm βm (m = 2, 3, . . . , n + 1).
5.2 Computation of stationary probabilities In this section, we suggest a computational procedure for calculating the limiting probability of the embedded Markov chain as well as the joint stationary probability of the state of the server and the number of customers in the orbit. In particular, we transform the balance equations in order to have the following recursion. π0 = Π(0) =
( λ+
( λ 1+
∑n+1
1 × exp − µ πk =
1
m=2 αm (1 − ρ)
)
m=2 σm +
[
∫
pk,k−1
1 0
)
∑n+1
(∑ n+1
)
m=2 αm (1 − ρ)
]
∑n+1
λ(1 − β1 (λ − λu)) + m=2 αm (1 − βm (λ − λu)) du , β1 (λ − λu) − u
π0 1 −
k−1 ∑ j=0
p0,j +
k−1 ∑ i=1
πi 1 −
k−1 ∑ j=i−1
pi,j ,
(43)
k ∈ N. (44)
It should be noted that these recursive formulae are numerically stable since they involve only positive terms. Furthermore, using the renewal theory arguments [9] as used in [5] we obtain the following procedure for the joint stationary probability of the state of the server and the number of customers in the orbit at an arbitrary time. In particular, let Pi,j (i = 0, 1, 2 . . . , n + 1, j ∈ Z+ ) denote the probability that the state of the server is i and the number of incoming calls in the orbit is j at an arbitrary time. Here, i = 0 denotes idle server and i = 1 expresses that the server is busy with
19
an incoming call while i = k (k = 2, 3, . . . , n + 1) represents that the server is busy with an outgoing call of type k. We have
1+
∑n+1
m=2 σm
(∑ n+1
(
1+
j ∑ ℓ=0
)
∑n+1
∑n+1
m=2 αm
(∑ n+1
Pi,j = λ +
∑n+1
,
m=2 αm
+ jµ
m=2 σm
λ
λ+
πj
λ+
m=2 αm (1 − ρ)
P1,j = λ +
m=2 αm (1 − ρ)
P0,j = λ +
×
)
(∑ n+1
+ ℓµ
)
ˆj−ℓ πℓ k 1
+
ℓ=1
j ∑ m=2 αm (1 − ρ) ∑n+1 1 + m=2 σm ℓ=0
)
j+1 ∑
ℓµ
λ+
∑n+1
m=2 αm
αi
λ+
∑n+1
m=2 αm + ℓµ
+ ℓµ
ˆj−ℓ+1 πℓ k 1
,
ˆj−ℓ , πℓ k i
for i = 2, 3, . . . n + 1, where {πi ; i ∈ Z+ } has been calculated by the recursion in (43) and (44).
5.3 Partial generating functions From the formulae for the joint stationary probabilities, we obtain the formulae for ∑∞ the partial generating functions, i.e., Pi (z) = j=0 Pi,j z j (i = 0, 1, 2, . . . , n + 1) after some tedious calculations as follows. P0 (z) =
1−ρ
1+
∑n+1
σm
m=2 [ ∫
1 × exp − µ
1 z
1 − β1 (λ − λz) P1 (z) = β1 (λ − λz) − z Pi (z) =
]
∑n+1
λ(1 − β1 (λ − λu)) + m=2 αm (1 − βm (λ − λu)) du , β1 (λ − λu) − u
(
∑n+1
m=2 αm (1 − βm (λ − λz))
1+
λ(1 − z)
αi (1 − βi (λ − λz)) P0 (z), λ(1 − z)
)
P0 (z),
i = 2, 3, . . . , n + 1.
5.4 Tail asymptotic We obtain the following light tailed asymptotic results by the same technique as used in [5] as follows. We make the following assumptions A1 and A2.
{
A1.
∫
∞
γ1 = sup t ∈ R
0
There exist θ1 ∈
1,
(
A2.
} etx dB1 (x) < ∞ λ + γ1 λ
> 0.
(45)
) such that β1 (λ − λθ1 ) = θ1 .
(46)
20
It should be noted that A2 is established under the stability condition ρ < 1. Furthermore, we assume that there exists a θi > 1 such that
lim βi (λ − λz)(z − θi ) < +∞.
lim βi (λ − λz) = +∞,
z↑θi
z↑θi
For simplification, let
∑n+1
λ(1 − β1 (λ − λz)) + m=2 αm (1 − βm (λ − λz)) f (z) = , β1 (λ − λz) − z
|z|
