Multiserver Retrial Queues with Two Types of Nonpersistent Customers Tuan Phung-Duc Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama, Tokyo 152-8552, Japan Tel: +81-(0)3-5734-3851 E-mail:
[email protected] Abstract We consider M/M/c/K (K ≥ c ≥ 1) retrial queues with two types of nonpersistent customers, which are motivated from modelling of service systems such as call centers. Arriving customers that see the system fully occupied either join the orbit or abandon receiving service forever. After an exponentially distributed time in the orbit, each customer either abandons the system forever or retries to occupy a server again. For the case of K = c = 1, we present an analytical solution for the generating functions in terms of confluent hypegeometric functions. In the general case, the number of customers in the system and that in the orbit form a level-dependent quasi-birth-anddeath (QBD) process whose structure is sparse. Based on this sparse structure, we develop a numerically stable algorithm to compute the joint stationary distribution. We show that the computational complexity of the algorithm is linear to the capacity of the queue. Furthermore, we present a simple fixed point approximation model for the case where the algorithm is time-consuming. Numerical results show various insights into the system behavior.
Keywords: Multiserver retrial queue, Level-dependent QBD, Call center, Resource dimensioning, System planning, Nonpersistent customers
1
1
Introduction
Retrial queues are characterized by the fact that an arriving customer that sees the service facility fully occupied temporarily departs from the system but reattempts for the service after some random amount of time. During consecutive reattempts the customer is said to be in the orbit. Retrial queues have been extensively studying and the literature is vast and rich. Readers are referred to a collection of recent papers [6] and the books [4, 14] for the advances in research of retrial queues. Retrial queues arise in various systems such as telecommunications, computer networks and call centers. The effect of retrials on the performance of call centers is investigated in [1, 5, 18, 19] while retrial phenomenon is also taken into account in various traffic models for cellular communication networks [2, 7, 10, 21, 27, 13]. Queueing models with retrial customers are suitable for many service systems where customers are not willing to waste their time just for waiting. Thus, in case of being blocked, customers may go somewhere else to spent their time and reattempt for service again in a later time. This phenomenon is common in call centers where we often hear the message “the system is very busy at this moment, please wait for a while or call again in a later time”. Retrial phenomenon is also found in cellular networks in which users can easily use the redial function by pushing only one button in their mobile device when a call is not connected. Research of retrial queues is pioneered by Cohen [11] who deals with multiserver models. Analytical solutions for multiserver retrial queues have been obtained for a few special cases. An explicit solution for the joint stationary distribution of the numbers of busy servers and customers in the orbit is obtained only for the M/M/1/1 retrial queue [14] without nonpersistent customer. The joint stationary distribution for M/M/1/1 retrial queue with nonpersistent customer and M/M/2/2 retrial queue without nonpersistent customers are expressed in terms of confluent hypergeometric functions and hypergeometric functions, respectively [14, 16, 24]. We refer to [17, 15, 23, 24] for some efforts in finding analytical solutions for M/M/c/c retrial queues with more than two servers by a generating function approach. Pearce [17] constructs an expression for the joint stationary distribution in terms of generalized con-
2
tinued fractions for an M/M/c/c retrial queue with any c. However, the expression in [17] does not directly yield a numerical algorithm. Gomez-Corral and Ramalhoto [15] derive an analytical solution for the case of three servers under some technical assumption. Using an alternative approach, Phung-Duc et al. [23] show that the joint stationary distribution is expressed in terms of continued fractions and the minimal solution of a three term recurrence relation for the cases of c = 3 and 4. The same authors [24] further derive analytical solutions for the joint stationary distribution of state-dependent M/M/c/c+r retrial queues with Bernoulli abandonment, where c + r ≤ 4. Because applications such as call centers require results for systems with a very large number of servers, it is important to develop numerical algorithms which can analyze largescale models. Since the original multiserver retrial model is not analytically tractable, various approximation models have been developed. Artalejo and Pozo [3] develop a generalized truncation method extending that presented in Falin and Templeton [14]. Neuts and Rao [22] propose an approximation for multiserver retrial queues using a level-independent QBD process with multiple boundary conditions. The main idea of [3, 14, 22] is to approximate the original model by a tractable one with similar dynamics. All of the works [3, 14, 22] involve with a truncation point for the number of customers in the orbit above which the dynamics of the model is approximated. All of the approximate models in [3, 14, 22] try to minimize the truncation point. On the other hand, multiserver retrial queues can be formulated using a level-dependent quasi-birth-and-death process (QBD), whose stationary distribution may be calculated using the algorithm developed by Bright and Taylor [9]. Recently, Phung-Duc et al. [25] develop a simpler and more memory-saving algorithm for level-dependent QBD processes in comparison with that proposed by Bright and Taylor [9]. Phung-Duc et al. [26] further present an efficient method which directly calculates the joint stationary distribution of multiserver retrial queues without nonpersistent customers. The methodology in [26] is based on the algorithm developed in [25] and utilizes the sparse structure in the block matrices of the underlying level-dependent QBD process. It should be noted that [25] does not analyze an approximate model as in [3, 14, 22] but directly calculates the stationary distribution of a censored Markov chain of the original model. In this paper, motivated by modelling of a call center operating in a group of multiple call
3
centers, we propose a multiserver retrial queue with two types of nonpersistent customers. The first type of nonpersistent customer gives up forever if it is blocked upon arrival while the second type of nonpersistent may give up after staying in the orbit. Our model extends many existing ones in the literature [17, 25]. The main contributions of this paper are as follows. First, we present a new model for which an efficient algorithm extending that in [26] is derived. We show that the computational complexity of the algorithm is linear to the capacity of the system and that the algorithm is numerically stable since it manipulates positive numbers. Second, for the slow retrial case where the algorithm is time-consuming, we propose a simple Cohen-type fixed point approximation model which yields accurate numerical results. Third, we prove that the generating functions for a single-server case can be expressed in terms of confluent hypergeometric functions. Finally, various new insights in to the system are explored. The rest of the current paper is organized as follows. In Section 2, we present the model in details and some preliminary results for level-dependent QBD formulation of the model. Main results are given in Section 3, where we present the derivation of an efficient numerical algorithm for the general case, an analytical solution for the single server case and a Cohen-type approximation model. Section 4 is devoted to a detailed implementation of the algorithm presented in Section 3. Section 5 shows numerical examples where some new insights into the behaviors of the system are observed. Finally, Section 6 concludes our paper and presents some future work.
2 2.1
Model Description and Preliminary Results Model description
We first describe the M/M/c/K retrial queue, where there are c servers and a waiting room of K − c waiting positions in front of the servers. Primary customers arrive at the servers according to a Poisson process with rate λ > 0 and the total service rate of all the servers is νi , provided that there are i customers in the system (the servers and the waiting room). We assume that ν0 = 0 ≤ ν1 ≤ ν2 ≤ · · · ≤ νK . This assumption allows us to consider various situations such as the case where customers may abandon after some exponentially distributed waiting time. An arriving primary customer enters the system 4
if there is some idle server or waiting position otherwise the customer either moves to the orbit with probability p or gives up (not joins the orbit) with probability p¯ = 1 − p. A customer that enters the orbit stays there for an exponentially distributed time with a finite positive mean 1/µ. After this time, the customer either retries to enter the servers with probability r or abandons forever with probability r¯ = 1 − r. A retrial customer also enters the system if there is some idle server or waiting position otherwise the customer either joins the orbit again with probability q or gives up forever with probability q¯ = 1 − q. See Figure 1 for the flows of customers. Join the orbit
Abandon forever Primary customers : p = 1 − p Retrial customers : q = 1 − q
Primary customers : p Retrial customers : q
Abandon: 1-r Retry: r
Orbit Size: Infinite Stay for exp. dis. ( µ )
Yes Yes
Redials
System is full?
New calls
Call Agents
Abandon
No
Primary customers : λ
Extra call lines No K-c c servers
Rate :
νi
Figure 1: Retrial Queues with Two Types of Nonpersistent Customers.
2.2
Motivation of the model
In almost retrial queue literature, the orbit is an abstracted unit which is not given a clear physical justification. In this paper, we introduce the service function into the orbit. To this end, we assume that the orbit provides some kind of service and a blocked customer may or may not be satisfied with this service. In case of being satisfied, the customer departs from the orbit, otherwise it reattempts to get service at the original service facility. Our model is motivated from modelling of a call center operating in a group of cooperative call centers. In this situation, a blocked call in a call center may be forwarded to another one. The call is lost if the forwarded call center is also busy, otherwise, the call is answered. This behavior corresponds to the first type of nonpersistent customers. An 5
operator in the forwarded call center may not be able to answer perfectly a call forwarded from the original one due to the speciality of each call center. A blocked call either satisfies with the service of the operator and departs or reattempts for service in the original one. This corresponds to the second type of nonpersistent behavior. Another application is found in tourism. We consider a situation where there are a very popular attraction and a group of less popular attractions in a city. Visitors typically wish to visit the very popular attraction. However, if tickets for the very popular attraction are sold out, a visitor may decide to visit a less popular attraction first. If the visitor is satisfied with the less popular attraction or his time is not allowed, he does not go to the very popular attraction again. Otherwise, the visitor may retry to visit the very popular attraction. In this situation, the very popular attraction and the group of less popular attractions correspond to the service facility (servers and waiting positions) and the orbit in our model, respectively.
2.3
Level-dependent QBD formulation
Let X(t) = (C(t), N (t)) (t ≥ 0), where C(t) denotes the total number of customers in the servers and the waiting room and N (t) denotes the number of customers in the orbit, at time t. It is easy to see that the bivariate process {X(t); t ≥ 0} is a Markov chain with the state space {0, 1, . . . , K} × Z+ , where Z+ = {0, 1, 2, . . . }. Throughout the paper, we assume that {X(t)} is ergodic. Lemma 2.1. If q < 1 or r < 1, {X(t)} is always ergodic. If q = r = 1, {X(t)} is ergodic if and only if ρ = (λp)/νK < 1. Proof. A proof of Lemma 2.1 can be obtained by the same method used in [14]. It is easy to confirm that {X(t)} is a level-dependent QBD process where C(t) and N (t) are referred to the phase and the level, respectively. Further, the infinitesimal generator is
6
given by
(0) Q1 (1) Q2
Q = O O .. .
(0) Q0 (1) Q1 (2) Q2
O .. .
O (1) Q0 (2) Q1 (3) Q2 .. .
··· ··· ··· ··· .. .
O O (2) Q0 (3) Q1 .. .
,
(n)
(n)
where O denotes a matrix of an appropriate dimension with zero entries and Q0 , Q1 (n)
(n ∈ Z+ ) and Q2 (n ∈ N = {1, 2, . . . }) are (K + 1) × (K + 1) matrices given by
0 ··· 0 0 0 ··· 0 0 .. . . .. .. = . . . . 0 0 ··· 0 0 0 0 · · · 0 λp nµ¯ r nµr 0 0 nµ¯ r nµr .. .. = . . . . . 0 0 .. .
(n)
Q0
(n)
Q2
0 (n)
b0
ν1 0 = .. . .. . 0
(n) Q1
, 0 .. .
nµ¯ r 0
nµr nµ(¯ r + r¯ q)
···
···
λ
0
(n)
λ
ν2 .. .
b2 ..
.
..
.
··· .. . .. . .. . .. .
···
0
b1
···
(n)
··· .. . .. .
0
··· ..
.
..
.
(n)
bK−1 νK
0 .. . .. .
,
. 0 λ (n)
bK
(n)
The diagonal components of Q1 are given by (n)
= −(λ + nµ + νi ),
(n)
= −(pλ + nµ(¯ r + r¯ q ) + νK ).
bi
bK
i = 0, 1, 2, . . . , K − 1,
7
Let πi,n = lim Pr{C(t) = i, N (t) = n},
i = 0, 1, 2, . . . , K,
t→∞
n ∈ Z+ ,
denote the joint stationary probability of the number of customers in the system and that in the orbit. Let π n = (π0,n , π1,n , . . . , πK,n ) and π = (π 0 , π 1 , . . . ). The stationary distribution π is the solution of the following system of equations. πQ = 0,
πe = 1,
(1)
where vectors e and 0 denote a column vector and a row vector with an appropriate dimension whose entries are ones and zeros, respectively. Equation (1) is rewritten in a vector form as follows. (n−1)
π n−1 Q0
(n)
(n+1)
+ π n Q1 + π n+1 Q2
= 0,
n ∈ N,
πe = 1.
(2) (3)
The solution of (2) and (3) is given by n ∈ N,
π n = π n−1 R(n) ,
where {R(n) ; n ∈ N} is the minimal nonnegative solution of (n−1)
Q0
(n)
(n+1)
+ R(n) Q1 + R(n) R(n+1) Q2
= O,
n ∈ N,
(4)
and the boundary vector π 0 is the solution of (1)
(0)
π 0 (Q1 + R(1) Q2 ) = 0, π 0 (I + R(1) + R(1) R(2) + . . . )e = 1. Matrix I denotes an identity matrix with an appropriate dimension. Proposition 1. It follows from (4) that R(n) = Rn (R(n+1) ),
8
n ∈ N,
(5)
where Rn (X) is defined as Rn : M −→ M, Rn (X)
(n−1)
=
Q0
(
(n)
(n+1)
−Q1 − XQ2
)−1
,
n ∈ N.
Here, M denotes a set of (K + 1) × (K + 1) matrices in which Rn (·) is well defined. Proposition 2 (Proposition 2.4 in [25]). For k, n ∈ N, we have (n)
lim Rk = R(n) ,
k→∞ (n)
where Rk
is defined as follows. (n)
(n)
Rk = Rn ◦ Rn+1 ◦ · · · ◦ Rn+k−1 (O),
R0 = O,
n, k ∈ N,
where f ◦ g(·) = f (g(·)).
3 3.1
Main Results Derivation of the numerical algorithm
As is presented in Section 2.3, the stationary distribution can be obtained if we have the rate matrices and the solution for the boundary equation (5) at level 0. In the computation of the rate matrix by Proposition 1, a crucial step is the calculation of R(n) provided that R(n+1) is given. In this section, we present an efficient procedure for this calculation. The computational complexity of the proposed method is only O(K). Indeed, the first K (n)
rows of Rk
(n−1)
and R(n) are all zeros due to the special structure of Q0
. Therefore, the
computation of R(n) is reduced to that of r (n) , where r (n) denote the last rows of R(n) . Let r
(n)
( =
(n) (n) (n) r0 , r1 , . . . , rK
) ,
n ∈ N.
Remark 1. The sparse structure of the rate matrices is also used by Liu and Zhao [20], who exploit the special structure of R(n) to derive explicit solutions for the M/M/c/c retrial queues with c = 1, 2, and some asymptotic results for the general case. In this paper, the sparse structure of the rate matrices is used in order to reduce the computational complexity 9
of a numerical algorithm. Theorem 3.1. For n, k ∈ N, r (n) is expressed in terms of r (n+1) by (n)
ri (n)
(n)
= αi
(n) (n)
i = 0, 1, . . . , K − 1,
+ βi r K ,
(n)
(6)
(n)
where {αi , βi ; i = K − 1, K − 2, . . . , 1} and rK are determined as follows. (n)
(n)
αK
(n)
= 0,
(n)
(n)
(n)
(n)
rek
(n)
(n)
βK−1 = −
(n+1)
bK + reK λ
,
(n)
+ νi+1 αi+1 , i = K − 1, K − 2, . . . , 1, λ (n) (n) (n) (n+1) bi βi + νi+1 βi+1 + rei = − , i = K − 1, K − 2, . . . , 1, λ
αi−1 = − βi−1
(n)
αK−1 = −p,
βK = 1, bi α i
is defined by (23)–(25) in Appendix A, and (n)
(n)
(n)
b0 α0 + ν1 α1
(n)
rK = −
(n) (n)
(n)
(n+1)
b0 β0 + ν1 β1 + re0
.
Proof. A proof is given in Appendix A. (n)
(n)
Remark 2. Because {αK−i , βK−i ; i = 0, 1, . . . , K} grow fast with i, we need some proper scaling while calculating these values. Corollary 1. The solution of the system of equations: (0)
(1)
x0 (Q1 + R(1) Q2 ) = 0,
x0 e = 1,
(7)
is given by xK,0 = xi,0 =
1 (0) (0) β0 + β1 (0) βi xK,0 ,
(0)
+ · · · + βK−1 + 1
,
i = 0, 1, . . . , K − 1,
where x0 = (x0,0 , x1,0 , . . . , xK,0 ). Proof. A proof of Corollary 1 is given in Appendix B. (n)
(n)
(n)
Because {αi , βi ; i = 0, 1, . . . , K} are big and the order of αi we confirm that the computation of
(n) ri
(n)
and βi
is the same,
(i = 0, 1, . . . , K − 1) by (6) is numerically unstable. 10
(n)
Instead of using (6), we use the following theorem to determine ri provided that
(n) rK
(i = 0, 1, . . . , K − 1)
is given. (n)
(n)
(i = 0, 1, . . . , K − 1) can be determined by
Theorem 3.2. If rK is given, then ri (n)
(n) ri
=
(n)
νi+1 ri+1 + Di
i = K − 1, K − 2, . . . , 0,
,
(n)
Bi (n)
(8)
(n)
where the sequences {Bi , Di ; i = 0, 1, . . . , K − 1} are recursively defined by (n)
(n)
(n+1) (n) rK ,
D0 = re0
B0 = λ + nµ, and (n)
Bi
= λ + nµ + νi −
(n)
λνi
(n)
, (n)
Di
Bi−1
(n+1) (n) rK
= rei
+
λDi−1 (n)
.
(9)
Bi−1
for i = 1, 2, . . . , K − 1. Furthermore, we have (n)
Bi
> λ,
(n)
Di
≥ 0,
i = 0, 1, . . . , K − 1.
(10)
Proof. A proof of Theorem 3.2 is presented in Appendix C. Remark 3. This result is a natural extension of Theorem 2 in Phung-Duc et al. [26]. The result also implies the numerical stability of the algorithm. Definition 3.3. Let rn denote a function such that n ∈ N,
rn (x) = Lr (Rn (X(x)) , (
where X(x) =
O x
) .
In the above, x is a row vector with an appropriate dimension and Lr(Y ) denotes the last row of matrix Y . (n)
Corollary 2. Let r k (n)
r 0 = 0,
(n)
denote the last row of Rk (n)
(n+1)
(n, k ∈ N). We then have
r k = rn (r k−1 ) = · · · = rn ◦ rn+1 ◦ · · · ◦ rn+k−1 (0), 11
(n)
and limk→∞ r k = r (n) . (n)
Remark 4. The computation of r k is based on Theorems 3.1 and 3.2.
3.2
Analytical Solution for Single Server Case
The system of balance equations for πi,j for any c and K is given as follows. (λ + νi + jµ)πi,j = λπi−1,j + νi+1 πi+1,j + (j + 1)µrπi−1,j+1 + (j + 1)µ¯ rπi,j+1 , i = 0, 1, . . . , K − 1,
(11)
(λp + νK + jµ(¯ r + r¯ q ))πK,j = λπK−1,j + (j + 1)µrπK−1,j+1 + λpπK,j−1 + (j + 1)µ(¯ r + r¯ q )πK,j+1 ,
i = K.
(12)
We define the generating function πi (z) (i = 0, 1, 2, . . . , K) as πi (z) =
∞ ∑
πi,j z j ,
i = 0, 1, . . . , K.
(13)
j=0
From (11), (12) and (13), we obtain the following system of differential equations. (λ + νi )πi (z) + µzπi0 (z) = λπi−1 (z) + νi+1 πi+1 (z) 0 + µrπi−1 (z) + µ¯ rπi0 (z),
i = 0, 1, . . . , K − 1,
(14)
0 0 (λp + νK )πK (z) + µ(¯ r + r¯ q )zπK (z) = λπK−1 (z) + µrπK−1 (z) + λpzπK (z) 0 + µ(¯ r + r¯ q )πK (z),
i = K.
(15)
Summing up (14) with i = 0, 1, . . . , K − 1 and (15) and then rearranging the result yields λpπK (z) =
K−1 ∑
0 (z). r + r¯ q )πK µπi0 (z) + µ(¯
i=0
12
(16)
For the case of K = c = 1, equations (14) and (15) become λπ0 (z) + µzπ00 (z) = ν1 π1 (z) + µ¯ rπ00 (z),
(17)
(λ + ν1 )π1 (z) + µ(¯ r + r¯ q )zπ10 (z) = λπ0 (z) + λzπ1 (z) + µrπ00 (z) + µ(¯ r + r¯ q )π10 (z).
(18)
From (17), we can express ν1 π1 (z) in terms of π0 (z). Substituting this expression into (16) with K = 1 yields a differential equation for π0 (z) as follows. (a1 z − a2 )π000 (z) + (a3 − a4 z)π00 (z) − a5 π0 (z) = 0,
(19)
where a1 = µ2 (¯ r + r¯ q ),
a2 = µ2 (¯ r + r¯ q )¯ r,
a3 = µ[ν1 + (λ + µ)(¯ r + r¯ q ) + λp¯ r],
a4 = λµp,
a5 = λ2 p.
Let x = a1 z − a2 , which is equivalent to z = (a2 + x)/a1 . Furthermore, let p(x) = π0 (z). We have p0 (x) = π00 (z)
1 , a1
p00 (x) = π000 (z)
1 . a21
Equation (19) is transformed to a21 xp00 (x) + (a3 a1 − a4 a2 − a4 x)p0 (x) − a5 p(x) = 0. Furthermore, let
( q(x) = p
a21 x a4
(20)
) .
Equation (20) is further transformed to the confluent hypergeometric differential equation xq 00 (x) + (β − x)q 0 (x) − αq(x) = 0, where α=
λ a5 = , a4 µ
β=
a3 a1 − a4 a2 ν1 + (λ + µ)(¯ r + r¯ q) = . 2 a1 µ(¯ r + r¯ q)
13
Because q(x) is regular at x = 0 (q(0) = π0 (¯ r)), we have q(x) = CΦ(α, β, x), where C denotes a constant number and Φ(α, β, x) =
∞ ∑ (α)n xn n=0
(β)n n!
,
denotes the confluent hypergeometric function. Here, (ϕ)n (−∞ < ϕ < ∞, n ∈ Z+ ) is the Pochhammer, whose definition is given by { (ϕ)n =
1, ϕ(ϕ + 1) · · · (ϕ + n − 1),
n = 0, n ∈ N.
As a result, we obtain π0 (z) = p(a1 z − a2 ) = CΦ (α, β, γ(z − r¯)) , where γ=
a4 pλ = . a1 µ(¯ r + r¯ q)
Thus, π0 (1) = CΦ (α, β, γr). We have ν1 π1 (z) = λπ0 (z) + µ(z − r¯)π00 (z), α = λπ0 (z) + Cµ γ(z − r¯)Φ(α + 1, β + 1, γ(z − a)), β = λπ0 (z) + Cµα [Φ(α + 1, β, γ(z − r¯)) − Φ(α, β, γ(z − r¯))] , = CλΦ(α + 1, β, γ(z − r¯)), which shows that π1 (z) = C%Φ(α + 1, β, γ(z − r¯)), where % = λ/ν1 . Because π0 (1) + π1 (1) = 1, we obtain C = (Φ(α, β, γr) + %Φ(α + 1, β, γr))−1 . Remark 5. When r = 1, we have π0,0 = C. We confirm that these results are consistent 14
with those presented in [24]. It follows from the generating functions that the stationary distribution is given by π0,j = C
∞ ∑ (α)i i=j
where i Cj =
3.3
(β)i
i! j!(i−j)!
i
i−j
γ i Cj (−¯ r)
,
π1,j = C%
∞ ∑ (α + 1)i i=j
(β)i
γ i i Cj (−¯ r)i−j ,
j ∈ Z+ ,
for i ≥ j.
Approximation of slow retrial cases
Let ν k = (ν0 , ν1 , . . . , νk ) (k = 0, 1, . . . , K) denote the vector of departure rates. Let B(λ, ν k ) denote the blocking probability of the loss system with arrival rate λ, capacity k and service rate νi (i = 0, 1, . . . , k) provided that there are i customers in the system. We then have λK ∏K
B(λ, ν K ) =
1+
∑Ki=1 i=1
νi ∏i λ
.
i
j=1
(21)
νi
We assume that the retrial flow could be seen as a Poisson process under a slow retrial rate. This observation is referred to as retrial customers see time average (see Artalejo and Gomez-Corral [4]). Let Λ denote the rate of this Poisson process for which a fixed point equation is derived. Under this assumption, the arrival to the loss system is compounded by two Poisson processes. The overflow from the loss system is the input for the orbit which is seen as an M/M/∞ system. Furthermore, the departure rate from this M/M/∞ system is again the retrial flow to the loss system. As a result, we obtain the following fixed point equation for the retrial rate Λ. Λ = r(λp + Λq)B(λ + Λ, ν K ), which is a variant of the Cohen-type equation [11] for the retrial rate. We determine Λ by the following iteration. Λ0 = 0,
Λn+1 = r(λp + Λn q)B(λ + Λn , ν K ),
n ∈ Z+ .
The iteration stops when |Λn+1 − Λn | < and we approximate Λ = Λn+1 and the blocking probability by B(λ + Λ, ν K ). 15
Remark 6. It should be noted that the blocking probability B(λ, ν K ) can be efficiently computed using a recursion similar to that used in the Erlang B formula. Indeed, a simple transformation yields, 1 νK 1 =1+ . B(λ, ν K ) λ B(λ, ν K−1 ) From this equation, we obtain B(λ, ν 0 ) = 1,
B(λ, ν K ) =
λB(λ, ν K−1 ) . νK + λB(λ, ν K−1 )
The implementation of this recursion is simpler than that by (21).
4 4.1
Implementation of the algorithm Truncation point
Because the stationary distribution {π n ; n ∈ Z+ } is expressed in terms of {r n ; n ∈ Z+ } which does not have a close form expression, we need to truncate the level at some truncation point N0 . The truncation point N0 should be large enough such that the tail probability after N0 is small enough to be disregarded. In particular, given an > 0, it is desired that we can find N0 such that
∞ ∑
π n e < .
n=N0
Since explicit result for retrial queue could be obtained for M/M/1/1 retrial queue without nonpersistent customers, we use this model in order to determine the truncation point. We consider an M/M/1/1 retrial queue without nonpersistent customer where the arrival rate, service rate and retrial rate are λ0 , ν and µ, respectively. The arrival rate λ0 is determined by λ0 /ν = λ/νK , which means that the traffic intensity of the single server retrial queue is equal to that of the multiserver model. Let p0,n and p1,n denote the joint stationary probability that the server is idle and busy, respectively and the number of customers in the orbit is n. We then have (see [14, 24]), p0,n
λ0 %n = (1 − %) µ +1 n!
(
λ0 µ
) , n
p1,n
λ0 %n+1 (1 − %) µ +1 = n!
16
(
) λ0 +1 , µ n
n ∈ Z+ .
We choose the truncation point N0 such that N0 = inf{n |
n ∑
(p0,i + p1,i ) > 1 − }.
0
4.2
The rate matrices
b (n) Recently, Phung-Duc et al. [25] propose an algorithm to compute an approximation R b(n) , to R(n) . Based on Corollary 2, we modify the algorithm in [25] to efficiently compute r b (n) . In Algorithm 1, {kl ; l ∈ Z+ } is a strictly increasing sequence which is the last row of R of non-negative integers, and ||x||∞ denotes the infinity norm of vector x defined by ||x||∞ = max |xj |, j
where xj represents the jth entry of x. Table 1: Computation of r (n) . Begin Algorithm 1 (n) (n) (n) Input: {Q0 , Q1 , Q2 , kn ; n ∈ Z+ , }. Output: {b r (n) }. l = 1; (n) (n) Compute r k1 and r k0 using Corollary 2 and Theorems 3.1 and 3.2. (n) (n) while ||r kl − r kl−1 ||∞ > do l := l + 1; (n) (n) Compute r kl and r kl−1 using Corollary 2 and Theorems 3.1 and 3.2. end b(n) := r (n) r kl ; End Algorithm 1
Corollary 3. The computational complexity of each step in Algorithm 1 and of the boundary equation (33) is O(K). Proof. This corollary is a direct consequence of Theorems 3.1 and 3.2. Remark 7. The computational complexity for the rate matrices and the boundary equations is O(K 3 ) if we use the original algorithm presented in [25]. 17
4.3
Stationary distribution
In the general case, no closed form for {π n ; n ∈ N} exists. Therefore, we present an algorithm to compute an approximation {b π n ; n = 0, 1, . . . , N0 } to the stationary distribution {π n ; n ∈ N}, where N0 is a natural number chosen in Section 4.1. Table 2 shows the details of the algorithm, which is modified from Algorithm 3 in [25]. In Table 2, xn is given by xn = (x0,n , x1,n , . . . , xK,n ),
n = 0, 1, . . . , N0 ,
which corresponds to π n . We also use b (n) = xK,n−1 r b(n) xn−1 R to simplify the algorithm. Table 2: The stationary distribution. Begin Algorithm 2 Input: λ, µ, ν, c, K, {kn ; n ∈ N}, , N0 . Output: {b π n ; n = 0, 1, . . . , N0 }. (N0 ) b Compute r using Algorithm 1 with {kn } and . for n = 1 to N0 − 1 do b(N0 −n) = rN0 −n (b r r (N0 −n+1) ); end Compute x0 by Corollary 1. for n = 1 to N0 do b(n) ; xn = xK,n−1 r end for n = 0 to N0 do b n := ∑N0xn ; π n=0 xn e end End Algorithm 2
5 5.1
Performance Measures and Numerical Examples Performance measures
In this section, we derive some performance measures of our model. 18
• Let PB denote the blocking probability that all the servers and waiting spaces are occupied.
∞ ∑
PB =
πK,j .
j=0
• Let E[C] denote the average number of customers in the system. We have E[C] =
K ∑ ∞ ∑
πi,j i.
i=0 j=0
• Let E[N ] denote the average number of customers in the orbit, i.e, E[N ] =
∞ ∑ K ∑
πi,j j.
j=0 i=0
• Let E[Q] denote the average number of waiting customers in the buffer, i.e., E[Q] =
K ∑ ∞ ∑
πi,j (i − c).
i=c j=0
• Let Copt denote the minimal number of servers such that PB < while other parameters are given in advance. We call Copt the optimal number of servers satisfying a QoS constrain PB < . • For a special case where c = K, r = 1 and νi = i (i = 0, 1, . . . , c), we have the following relations (see [14]). E[N ] =
λq + λ(p − q)PB − qE[C] . µ(1 − q)
In numerical examples presented in the next section, the above infinite sums are truncated at N0 and πi,j is replaced by the approximation π bi,j obtained by our algorithm. b E[N b ] denote the approximations to PB , E[C], E[N ]. We define the Let PbB , E[C], relative error Er by the following formula b b λq + λ(p − q)PB − qE[C] b b ]. Er = E[N ] − /E[N µ(1 − q)
19
5.2
Parameter setting
We use the same = 10−10 for determining the truncation point N0 in Section 4.1, the calculation of the rate vector r N0 in Algorithm 1 and the fixed point approximation in Section 3.3. Furthermore, we use kn = 2n − 1 in Algorithm 2. We consider the case where c = K and νi = i (i = 0, 1, . . . , c − 1, c) in Figures 2 to 10. We first restrict ourself in the case % = λ/νK < 1, which is practical use in real world systems. Under this condition, we are able to determine the truncation point N0 by the single server retrial queue without nonpersistent customers presented in Section 4.1. For the case of % = λ/νK ≥ 1, we can choose a sufficiently large truncation point by try and errors (see Section 5.8).
5.3
Validation of the truncation point
First of all, we check the validity of our truncation point and our algorithms. Figure 2 shows the relative error of number of customers in the orbit for the cases µ = 0.01, 0.1, 1, 10, 100 while keeping % = 0.9 and r = 1. We observe that the relative error is around 10−12 which is almost the same as the order of . This suggests the validity of the truncation point N0 determined in Section 4.1 and of our algorithms. 1
µ=100 µ=10 µ=1 µ=0.1 µ=0.01
0.01
0.0001
Relative error
1e-006
1e-008
1e-010
1e-012
1e-014
1e-016 0
0.2
0.4 0.6 Retrial probabilities p = q
0.8
Figure 2: Relative error (% = 0.9, r = 1).
20
1
5.4
Effect of the number of servers
One of the most important questions for the manager of a call center is that how many operators do we need in order to satisfy a certain quality of service (QoS)? For example, how many operators do we need to have in order to achieve a blocking probability lower than 0.001? Numerical results in this section give the answer for this question under various parameter settings. Figures 3 and 4 present the blocking probability against the number of servers while keeping % = 0.9, µ = 10 and r = 1. In Figure 3, the three curves corresponding to three cases: p = q = 1, p = q = 0.7 and p = q = 0.1. In Figure 4, the curves correspond to the blocking probability with different values of µ and the blocking probability obtained by the Cohen-type equation. In all the curves in Figures 3 and 4, we observe that the blocking probability decreases with the number of servers as expected. This is because the offered load to each operator is smaller than one and thus operators can cooperate together. From the three curves in Figure 3, we observe that the blocking probability is sensitive to p = q. This means that when the retrial is fast (µ = 10 implies that the mean retrial interval is ten times shorter than the mean service time), we should carefully take into account p and q. Figure 4 shows the sensitivity of the retrial rate µ on the blocking probability where p = q = 0.9. We observe that the blocking blocking probability increases with µ when the number of servers is large enough while decreasing with µ when the number of servers is small enough. This phenomenon cannot be observed in retrial queues without nonpersistent customers, i.e., p = q = r = 1 [26], where the blocking probability is a monotonic function of µ. The reason is that the blocking probability is influenced by two factors. An increase in µ implies that a repeated call is likely blocked again, resulting in the increase of the blocking probability. At the same time, if the number of servers is small, the blocking probability is large. In our setting, a blocked customer may abandon with probability p = q. In this situation, an increase in µ implies that a repeated call is likely blocked and is lost. As a result, the blocking probability decreases with µ. Furthermore, an interesting insight is that all the curves are asymptotically lines when the number of servers is large. Finally, we observe from Figure 4 that the blocking probability obtained by the Cohen’s equation matches with those of µ = 0.1 and 0.01. This means that the Cohen’s equation is effective in approximating retrial system with slow retrial rate. 21
1 p=q=1, RQ p=q=0.7, RQ p=q=0.1, RQ
Blocking Probability
0.1
0.01
0.001
0.0001 0
20
40 60 The number of servers
80
100
Figure 3: Blocking probability against the number of servers (% = 0.9, r = 1).
1 p=q=0.9, Cohen’s eq. p=q=0.9, µ = 0.01 p=q=0.9, µ = 0.1 p=q=0.9, µ = 1 p=q=0.9, µ = 10 p=q=0.9, µ = 100
Blocking Probability
0.1
0.01
0.001
0.0001 50
100
150
200 250 300 The number of servers
350
400
450
500
Figure 4: Blocking probability against the number of servers (% = 0.9, p = q = 0.9, r = 1).
22
5.5
Optimal number of servers
opt Figure 5 presents the optimal number servers C0.01 against the probability p = q while opt r = 1, % = 0.7 and µ = 0.01, 0.1, 1, 10, 100. We observe that C0.01 increases with p = q as
expected. The optimal number of servers is less sensitive to p = q for small value of µ, i.e., opt µ = 0.01, 0.1 while C0.01 is more sensitive to p = q when µ is large enough, i.e., µ = 10, 100.
These results suggest that we should carefully take into account the effect of p and q for systems with fast retrials. 50
µ = 0.01 µ = 0.1 µ=1 µ = 10 µ = 100
Optimal # of Servers
45
40
35
30 0.1
0.2
0.3
0.4
0.5 0.6 Probability p = q
0.7
0.8
0.9
1
Figure 5: Optimal number of servers (% = 0.7, r = 1).
5.6
Effect of retrial rate µ
In this section, we investigate the influence o the retrial rate µ on the performance measures. We fix K = c = 100 and % = λ/νK = 0.9. Figure 6 presents the relation between µ and the blocking probability. It should be noted that since the Cohen’s equation does not depend on µ, the blocking probability obtained by the equation is independent of µ. We observe that the blocking probability increases with relatively small µ while it decreases with relatively big µ. Especially, we observe two interesting asymptotic behaviors of the blocking probability when µ → 0 and µ → ∞. For the former, the blocking probability obtained by our retrial queueing model well matches that obtained from the Cohen-type 23
equation. This fact again suggests the effectiveness of the Cohen’s approximation for slow retrial model for which the algorithm is time consuming. In the latter, i.e., µ → ∞, the blocking probability converges to that of loss model without retrial which is consistent with the asymptotic result in [14]. Figure 7 shows the average number of servers against µ. It should be noted that the average number of busy servers is equal to the throughput from the servers under the current settings. We observe that the throughput decreases with the increase in µ in all three curves with different p, q. This implies that from a management point of view, a slow retrial is preferable for achieving a high throughput. We again find that the average number of busy servers converges to that of the loss system without retrial. 1
Blocking Probability
p=q=0.7, RQ p=q=0.7, Cohen’s eq. p=q=0.9, RQ p=q=0.9, Cohen’s eq. p=q=0.95, RQ p=q=0.95, Cohen’s eq.
0.1
0.01 0.01
0.1
1
10
100 Retrial rate µ
1000
10000
100000
1e+006
Figure 6: Blocking probability against µ (% = 0.9, r = 1).
5.7
Effect of the unsatisfactory probability r
In this section, keeping K = c = 100, p = q = 1 and % = 0.9, we investigate the influence of r on performance measures. Figure 8 illustrates the blocking probability again r for relatively small values of µ (µ = 0.01, 0.1, 1, 10). We observe that when µ is small (µ = 0.01, 0.1), the blocking probability is close to that obtained by the Cohen’s equation. We also observe that the blocking probability increases with r and µ. 24
p=q=0.7, RQ p=q=0.9, RQ p=q=0.95, RQ
Average # of Busy Servers
94
92
90
88
86
0.01
0.1
1
10
100 Retrial rate µ
1000
10000
100000
1e+006
Figure 7: Average number of busy servers (% = 0.9, r = 1).
In Figure 9, the curves for the blocking probability with relatively large value of µ (µ = 102 , 103 , · · · , 106 ) are plotted. We observe in all these curves that the blocking probability increases with r while it decreases with µ. We further observe that in the case of µ → ∞, i.e., µ = 106 , regardless the value of r, the blocking probability converges to that of the corresponding loss model without retrials.
5.8
Overload situation
Figure 10 presents the blocking probability against the number of servers under an overloaded situation where % = 1.5 and p = q = r = 0.7. After some try and errors, we choose the truncation point N0 = 1000, which is large enough. We plot three curves correspond to µ = 0.1, µ = 1 and µ = 10, respectively. Furthermore, the curve obtained by the Cohentype equation is also plotted. We observe from Figure 10 that the blocking probability also decreases with c = K and with µ. We also read from the Figure 10 that the curve by Cohen-type equation matches that with µ = 0.1. This again confirms the effectiveness of the Cohen-type approximation under an overloaded situation with slow retrial.
25
0.1
Blocking Probability
µ = 10, RQ µ = 1, RQ µ = 0.1, RQ µ = 0.01, RQ Cohen’s eq.
0.01 0
0.2
0.4 0.6 Retrial probability (r)
0.8
1
Figure 8: Blocking probability against r with small µ (% = 0.9).
0.1
Blocking Probability
µ = 1000000, RQ µ = 10000, RQ µ = 1000, RQ µ = 100, RQ Erlang B without retrial
0.01 0
0.2
0.4 0.6 Retrial probability (r)
0.8
1
Figure 9: Blocking probability against r with big µ (% = 0.9).
26
0.65
p=q=r=0.7, µ = 0.1 p=q=r=0.7, µ = 1 p=q=r=0.7, µ = 10 Cohen’s equation
Blocking Probability
0.6
0.55
0.5
0.45 5
10
15
20 25 30 The number of servers
35
40
45
50
Figure 10: Blocking probability against the number of servers (% = 1.5).
5.9
Average number of waiting customers
Figure 11 presents the average number of waiting customers in the buffer against the buffer size K − c for c = 10, p = q = 0.7, r = 1 and µ = 0.01, 1, 100, % = 0.7, 0.9. It should be noted that νi = i (i = 0, 1, . . . , c) and νi = c (i = c + 1, c + 2, . . . , K) in this scenario. We observe that the average number of waiting customers E[Q] increases with K and converges as K → ∞. We further observe that E[Q] increases with µ. The reason is that when the retrial rate µ is high, a customer has a higher chance of occupying a waiting position rather than in the orbit.
6
Conclusion and Future Work
In this paper, we have presented an extensive analysis of a multiserver retrial queue with two types of nonpersistent customers which is motivated from modelling of a call center operating in a group of cooperative call centers. We have derived an efficient algorithm for analyzing the stationary behavior of the system. For the case where the algorithm is timeconsuming, i.e., slow retrial, we have proposed a Cohen-type fixed point equation which could be solved fast. Various insights into the behavior of the model have been observed.
27
Average # of Waiting Customers
10
1
0.1
ρ = 0.7, µ = 1 ρ = 0.7, µ = 100 ρ = 0.7, µ = 0.01 ρ = 0.9, µ = 1 ρ = 0.9, µ = 100 ρ = 0.9, µ = 0.01
0.01 0
10
20
30
40
50
Buffer size (K-c)
Figure 11: Average number of waiting customers against K − c (% = 0.7, 0.9).
We also have derived explicit solution in terms of hypergeometric functions for the case of single server. For the future work, we would like to investigate some asymptotic behavior of the system when the number of servers is very large. In particular, we plan to analyze the slop of the curves in Figures 3 and 4. A closely related work in this direction has been carried out by Avram et al. [8]. An appropriate choice of the truncation point for the overloaded situation should be also taken into consideration.
A
Proof of Theorem 3.1
Proof. Let U (n) denote (n+1)
(n)
U (n) = Q1 + R(n+1) Q2
,
n ∈ N.
which is the defective infinitesimal generator of the restricted process of {X(t)} on level n, (n+1)
under the taboo of levels n − 1. Due to the special structure of R(n+1) and of Q2 (
have (n)
U (n) = Q1 +
O e(n+1) r 28
, we
) ,
n ∈ N,
(22)
where e r
(
(n)
=
(n) (n) (n) re0 , re1 , . . . , reK
)
n ∈ N,
,
and (n)
= nµ¯ r r0 ,
(n)
(n)
= nµrri−1 + nµ¯ r ri ,
(n)
= nµrrK−1 + nµ(¯ r + r¯ q )rK ,
re0
i = 0,
(n)
rei
(23)
(n)
(n)
reK We also have
R
(n)
(n−1) Q0
=
(
−U
(n)
i = 1, 2, . . . , K − 1,
(24)
(n)
(25)
)−1
i = K.
n ∈ N,
,
which is equivalent to (n−1)
R(n) U (n) = −Q0
n ∈ N.
,
(26)
Because the first K rows of both sides of (26) are zero vectors, (26) is equivalent to (n)
(n)
(n)
(n)
(r0 , r1 , . . . , rK−1 , rK )U (n) = (0, 0, . . . , 0, −λp).
(27)
From (22), we can solve (27) efficiently. Indeed, we rewrite (27) as the following system of equations. (n) (n)
(n)
(n+1) (n) rK
(n)
(n+1) (n)
b0 r0 + ν1 r1 + re0 (n)
(n) (n)
= 0,
i = 0,
(28)
+ νi+1 ri+1 + rei rK = 0, i = 1, 2, . . . , K − 1, ( ) (n) (n) (n+1) (n) λrK−1 + bK + reK rK = −pλ, i = K.
λri−1 + bi ri
(n)
We assume that ri
(29) (30)
(n)
(i = 0, 1, . . . , K) can be expressed in terms of rK as (n)
ri
(n)
= αi
(n) (n)
+ βi rK ,
i = 0, 1, . . . , K.
Substituting (31) into (29) and (30) yields (n)
(n)
(n)
λαi−1 + bi αi (n)
(n) (n)
λβi−1 + bi βi
(n)
+ νi+1 αi+1 = 0,
(n)
(n+1)
+ νi+1 βi+1 + rei 29
= 0,
(31)
for i = K − 1, K − 2, . . . , 1. Substituting (6) into (28) yields (n)
(n)
(n) (n)
(n) (n)
(n)
(n+1) (n) rK
b0 (α0 + β0 rK ) + ν1 (α1 + β1 rK ) + re0
= 0.
(32)
Thus, we have (n)
(n) rK
=−
(n)
(n)
b0 α0 + ν1 α1 (n) (n)
(n)
(n+1)
b0 β0 + ν1 β1 + re0
.
Theorem 3.1 follows from (6) and (32).
B
Proof of Corollary 1
Proof. We observe that the system of linear equations: (0)
(1)
x0 (Q1 + R(1) Q2 ) = 0,
(33)
expresses a special case of (27) where n = 0 and the λ in the right hand side is equal to (0)
(0)
0. Note that for this case, αK−1 = 0 and thus αi
= 0 (i = 0, 1, . . . , K). Therefore, from
(0)
xi,0 = βi xK,0 and x0 e = 1, Corollary 1 is proved.
C
Proof of Theorem 3.2
Proof. Equation (8) is easily proved using mathematical induction. We show (10) also by mathematical induction. We confirm that (10) is true for i = 0. Assuming that (10) is true for all i = 0, 1, . . . , m, where m = 0, 1, . . . , K − 2, we prove that (10) is also true for i = m + 1. Indeed, it follows from (9) that (n)
Bm+1 = λ + νm+1 + nµ −
λνm+1 (n)
Bm λνm+1 > λ + νm+1 + nµ − λ = λ + nµ > λ,
(n)
where Bm > λ is used in the first inequality. It follows from (9) and (n+1)
rei
> 0,
(n)
rK > 0,
30
(n)
Bi
> 0,
(n)
that Di
≥ 0 (i = 0, 1, . . . , K − 1).
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[23] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi (2009). M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial and Management Optimization, 5, 431– 451. [24] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi (2010). State-dependent M/M/c/c + r retrial queues with Bernoulli abandonment. Journal of Industrial and Management Optimization, 6, 517–540. [25] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi (2010). A simple algorithm for the rate matrices of level-dependent QBD processes. In Proceedings of the 5th International Conference on Queueing Theory and Network Applications. [26] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi (2013). A matrix continued fraction approach to multiserver retrial queues. Annals of Operations Research, 202, 161-183. [27] P. Tran-Gia and M. Mandjes (1997). Modeling of customer retrial phenomenon in cellular mobile networks. IEEE Journal on Selected Areas in Communications, 15, 1406–1414. Tuan Phung-Duc is an Assistant Professor in Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. He received a Ph.D. in Informatics from Kyoto University in 2011. He is currently in the Editorial Board of the KSII Transactions on Internet and Information Systems. His research interests include Queueing Theory and Performance Analysis of Telecommunication and Service Systems.
33
