Hindawi Publishing Corporation Advances in Operations Research Volume 2016, Article ID 3478709, 8 pages http://dx.doi.org/10.1155/2016/3478709
Research Article Selective Trunk with Multiserver Reservation Bacel Maddah1 and Muhammad El-Taha2 1
Department of Industrial Engineering and Management, Faculty of Engineering and Architecture, American University of Beirut, Beirut 1107 2020, Lebanon 2 Department of Mathematics and Statistics, University of Southern Maine, 96 Falmouth Street, Portland, ME 04104-9300, USA Correspondence should be addressed to Bacel Maddah;
[email protected] Received 3 November 2015; Revised 28 January 2016; Accepted 29 March 2016 Academic Editor: Yiqiang Zhao Copyright Β© 2016 B. Maddah and M. El-Taha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a queueing model that is primarily applicable to traffic control in communication networks that use the Selective Trunk Reservation technique. Specifically, consider two traffic streams competing for service at an π-server queueing system. Jobs from the protected stream, stream 1, are blocked only if all π servers are busy. Jobs from the best effort stream, stream 2, are blocked if π β π, π β₯ 1, servers are busy. Blocked jobs are diverted to a secondary group of π β π servers with, possibly, a different service rate. We extend the literature that studied this system for the special case of π = 1 and present an explicit computational scheme to calculate the joint probabilities of the number of primary and secondary busy servers and related performance measures. We also argue that the model can be useful for bed allocation in a hospital.
1. Introduction In this paper, we consider a simple and effective traffic control technique for communication networks with two priority traffic classes. The method, referred to as Selective Trunk Reservation (STR), assumes two traffic streams competing for service at an π-server queueing system. Jobs from stream 1, the protected stream, are blocked if all π servers are busy. Jobs from stream 2, the best effort stream, are blocked if π β π, π β₯ 1, servers are busy. Blocked jobs are sent to a secondary group of πβπ servers with, possibly, a different service rate from that of the primary servers. Jobs from both streams are lost when all π servers are busy. El-Taha and Heath [1] consider a similar problem but they restrict attention to the case π = 1 and describe a procedure to evaluate joint probabilities for π > 1. Their paper includes significant enhancements of earlier work (e.g., [1β7]). This paper is an extension of El-Taha and Heath [1] work and a response to inquiries about implementing the STR method of [1] for all π β₯ 1. In particular, we explicitly derive the joint probabilities for all π β₯ 1 and provide an efficient algorithm for computing them.
The block diagram of the model in this paper is shown in Figure 1. We derive the joint probabilities of the number of primary and secondary busy servers. From these probabilities, performance measures, including overflow probabilities of the server groups, are derived. We then apply our results to the specific case of two Poisson traffic streams and two heterogeneous groups of exponential servers. One alternative application of the model in this paper, which came to our attention recently, is for bed allocation in a hospital ward. The literature contains several examples of bed allocation with interesting similarities to the classic STR problem of communication. Examples of these works include [8β13]. In many of these works (e.g., [8, 10]), two types of patients are considered, serious and nonserious, which arrive according to a Poisson process and have an exponentially distributed duration of stay (service time), similar to the two streams of arrivals in the STR problem. More interestingly, this literature discusses the setting of a βcut-off occupancyβ in terms of a number of beds in the ward reserved for serious patients only, which is in the same spirit of STR. Esogbue and Singh [10] further consider βbuffer accommodationβ which mimics the secondary servers we
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Advances in Operations Research 1 π2
.. .
nβr
π1
.. .
n Primary servers
1 Overflow (stream 2) .. .
Overflow (stream 1)
cβn Loss (stream 2)
Loss (stream 1)
Secondary servers
Figure 1: Block diagram of the STR model.
consider in this paper. However, one distinctive difference between STR in communication and bed allocation in hospitals is that, in the latter, the service times depend on the type of arrival. This require state expansion, to track patient type, in Markovian STR models, such as the one in this paper. We believe that our efficient computational schemes can be tailored for such expanded-state bed allocation problems. The paper is organized as follows. In Section 2, we present an efficient iterative method to find the joint probability distribution of the number of busy primary and secondary servers. In Section 3, we provide an application that considers two Poisson streams, based on the analysis of Section 2, and we illustrate the application of the method by a numerical example. In Section 4, we draw some conclusions and discuss possible extensions. Finally, in the Appendix we provide an algorithm that is used for the development of a computer program of the present model.
2. Joint Stationary Probabilities Formally, we describe the above model as a two-dimensional stochastic process X β‘ {(π1 (π‘), π2 (π‘)); π‘ β₯ 0} with finite state space S such that π1 (π‘) takes integer values in {0, . . . , π}, π β₯ 1; π2 (π‘) takes integer values in {0, . . . , π β π}, π β₯ π + 1. Here, π1 (π‘) represents the number of primary busy servers and π2 (π‘) represents the number of secondary busy servers.
Utilizing a similar notation to El-Taha and Heath [1], define π½1 (π, π) to be the state (π, π) stream 1 arrival rate and π½2 (π, π) to be the stream 2 arrival rate when the system state is (π, π). Let π½(π, π) = π½1 (π, π) + π½2 (π, π). Define πΎ1 (π) to be the primary group service rate when π primary servers are busy and πΎ2 (π) to be the secondary group service rate when π secondary servers are busy. Let ππ,π denote the state (π, π) joint stationary probability, π = 0, . . . , π, π = 0, . . . , π β π. The transition rate out of state (π, π) is denoted by πΌ(π, π) = π½(π, π)+πΎ1 (π)+πΎ2 (π). Then, the state probabilities for any π β₯ 1 satisfy the global balance equations: πΌ (π, π) ππ,π = π½ (π β 1, π) ππβ1,π + πΎ1 (π + 1) ππ+1,π + πΎ2 (π + 1) ππ,π+1 ,
(1)
π = 0, 1, . . . , π β π β 1; π = π β π, . . . , 0, πΌ (π β π, π) ππβπ,π = π½ (π β π β 1, π) ππβπβ1,π + π½2 (π β π, π β 1) ππβπ,πβ1 + πΎ1 (π β π + 1) ππβπ+1,π + πΎ2 (π + 1) ππβπ,π+1 ; π = π β π, . . . , 0,
(2)
Advances in Operations Research
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πΌ (π, π) ππ,π = π½1 (π β 1, π) ππβ1,π + π½2 (π, π β 1) ππ,πβ1 + πΎ1 (π + 1) ππ+1,π
(3)
+ πΎ2 (π + 1) ππ,π+1 ; π = π β π + 1, . . . , π β 1; π = π β π, . . . , 0,
The normalization equation is βππ=0 βπβπ π=0 ππ,π = 1. The relations in (1)β(4) are graphically illustrated in Figure 2. Equations (1)β(4) expand similar flow balance equations in El-Taha and Heath [1] by allowing for π > 1 in (3). Equations (1), (2), and (4) are the same as those in [1]. We present these equations here for completeness. The remainder of this section is devoted to solving (1)β (4) efficiently via an iterative scheme. The first step, in (5), expresses each ππ,π , π = 1, . . . , π β π, π = 0, . . . , π β π, in terms of the probabilities π0,π , π = π, . . . , π β π:
πΌ (π, π) ππ,π = π½1 (π β 1, π) ππβ1,π
min{π,πβπβπ}
ππ,π =
+ π½ (π, π β 1) ππ,πβ1 (4)
+ πΎ2 (π + 1) ππ,π+1 ;
β π=0
(β1)π πΊ (π, π, π) π0,π+π ,
(5)
π = 1, . . . , π β π, π = 0, . . . , π β π, π = π β π, . . . , 0.
where, for each π = π β π, . . . , 0,
πΊ (π, π, π) 1 { { { { = {πΎ1 (π)β1 [πΌ (π β 1, π) πΊ (π β 1, π, π) β π½ (π β 2, π) πΊ (π β 2, π, π) + πΎ2 (π + 1) πΊ (π β 1, π + 1, π β 1)] , { { { {0
Equation (5) can be proved in a similar manner to Lemma 2.1 of El-Taha and Heath [1]. In our main result, Theorem 1, (5) is utilized in obtaining {ππ,π , π = 0, . . . , π β π; π = 0, . . . , π β π}, in terms of ππβπ,πβπ , π = 0, 1, . . . , π. The ππβπ,πβπ probabilities are obtained by solving a system of π + 1 linear equations formulated by applying Theorem 1 in combination with the limiting state probabilities of the number of busy primary servers. Finally, all joint probabilities are computed by appealing again to Theorem 1 and by replacing ππβπ,πβπ , π = 0, 1, . . . , π, by their computed values. Theorem 1 is our main contribution with respect to ElTaha and Heath [1], as it gives explicit expressions of the limiting probabilities ππ,π in terms of a smaller set of these probabilities, ππβπ,πβπ , π = 0, 1, . . . , π. The main result in [1] is a special case of Theorem 1 with π = 1.
min(πβπ,πβπβπ)
β
β π=1
π = 0, π = 0; π = 0, . . . , min {π, π β π β π} , π = 1, . . . , π β π; otherwise.
(β1)π πΊ (π β π, π, π) π΅π (0, π + π)] , π = 0, . . . , π; π = π β π β 1, . . . , 0,
(9) π΅π (π β π, π) = π½2 (π β π, π)
β π½ (π β π β 1, π + 1) π΅π (π β π β 1, π + 1) β πΎ1 (π β π + 1) π΅π (π β π + 1, π + 1)
(10)
β πΎ2 (π + 2) π΅π (π β π, π + 2)] , π = 0, . . . , π; π = π β π β 1, . . . , 0,
π
β1
(7)
π=0
π΅π (π, π) = π½2 (π, π) [πΌ (π, π + 1) π΅π (π, π + 1) β π½1 (π β 1, π + 1) π΅π (π β 1, π + 1)
where
(11)
β πΎ1 (π + 1) π΅π (π + 1, π + 1) β πΎ2 (π + 2) π΅π (π, π + 2)] , min(π,πβπβπ)
π΅π (π, π) =
β1
β
[πΌ (π β π, π + 1) π΅π (π β π, π + 1)
Theorem 1. For all (π, π), π = 0, . . . , π and π = 0, . . . , π β π, ππ,π = β π΅π (π, π) ππβπ+π,πβπ ,
(6)
β π=0
(β1)π πΊ (π, π, π) π΅π (0, π + π) ,
π = 0, . . . , π; π = 1, . . . , π β π β 1; π β€ π β 2; π = π β π β 1, . . . , 0;
and the βboundaryβ conditions are β1
π΅π (0, π) = πΊ (π β π, π, 0) [π΅π (π β π, π)
(8)
π = 0, . . . , π; π = π β π + 1, . . . , π β 1; π = π β π β 1, . . . , 0, β1
π΅π (π, π) = π½ (π, π) [πΌ (π, π + 1) π΅π (π, π + 1) β π½1 (π β 1, π + 1) π΅π (π β 1, π + 1) β πΎ2 (π + 2) π΅π (π, π + 2)] , π = 0, . . . , π; π = π β π β 1, . . . , 0,
(12)
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Advances in Operations Research
πΎ2 (c β n)
πΎ2 (2)
πΎ2 (1)
0, 0
0, 1
0, j
0, 2
0, c β n β 1
0, c β n
i β 1, j πΎ1 (i)
π½(i β 1, j) πΎ2 (j) i, 0
πΎ2 (j + 1)
i, j β 1
i, j
i, j + 1
i, c β n
πΎ1 (i + 1)
π½(i, j)
i + 1, j
n β r β 1, j πΎ1 (n β r)
π½(n β r β 1, j) πΎ2 (j) n β r, 0
πΎ2 (j + 1) n β r, j
n β r, j β 1
n β r, j + 1
π½2 (n β r, j β 1)
π½2 (n β r, j)
π½1 (n β r, j)
πΎ1 (n β r + 1)
n β r, c β n
n β r + 1, j
iσ³° β 1, j π½1 (iσ³° β 1, j)
πΎ1 (iσ³° ) πΎ2 (j + 1)
πΎ2 (j) iσ³° , 0
iσ³° , j
iσ³° , j β 1
iσ³° , j + 1
iσ³° , c β n
n, j + 1
n, c β n
σ³°
σ³°
π½2 (i , j)
π½2 (i , j β 1) σ³°
π½1 (i , j)
πΎ1 (iσ³° + 1) iσ³° + 1, j
n β 1, j π½1 (n β 1, j)
πΎ1 (n) πΎ2 (j + 1)
πΎ2 (j) n, 0
n, j β 1
n, j π½(n, j β 1)
π½(n, j)
Figure 2: Transition diagram of the model.
Advances in Operations Research
5 β πΎ1 (π + 1) π΅π (π + 1, π + 1)
π΅π (π, π β π) πΊ (π, π β π, 0) { ; π = 0; π = 0, . . . , π β π { { { πΊ (π β π, π β π, 0) = {1; π = π β π + π; π = 1, . . . , π; { { { ππ‘βπππ€ππ π. {0;
π
(13)
(14)
which implies ππβπ,πβπ = πΊ (π β π, π β π, 0) π0,πβπ ,
(15)
π0,πβπ = πΊ (π β π, π β π, 0)β1 ππβπ,πβπ .
(16)
and then
(21) where π΅π (π, π) is given by (11). Similarly, by manipulating (2) and (4) one can show that ππβπ,π = βππ=0 π΅π (π β π, π)ππβπ+π,πβπ , π = 0, . . . , π β π, where π΅π (π β π, π) is given by (10) and that ππ,π = βππ=0 π΅π (π, π)ππβπ+π,πβπ , π = 0, . . . , π β π, where π΅π (π, π) is given by (12). Next, we evaluate π0,π , π = π β π β 1, . . . , 0. It follows from the above equations and (5) that π
It follows that ππ,πβπ =
π=0
π) ππβπ+π,πβπ ,
Proof. The joint probabilities are derived iteratively. First consider π = π β π. By (5), ππ,πβπ = πΊ (π, π β π, 0) π0,πβπ ; π = 0, 1, . . . , π β π,
β πΎ2 (π + 2) π΅π (π, π + 2)] ππβπ+π,πβπ } = β π΅π (π,
πΊ (π, π β π, 0) π πΊ (π β π, π β π, 0) πβπ,πβπ
ππβπ,π = β π΅π (π β π, π) ππβπ+π,πβπ π=0
(17)
min(πβπ,πβπβπ)
=
= π΅0 (π, π β π) ππβπ,πβπ , π = 0, . . . , π β π,
π=0
where π΅0 (π, π β π) is given by (13). Note that, for π = π β π + 1, . . . , π and all π = 0, . . . , π, π΅π (π, π β π) = 0 except π΅π (π β π + π, π β π) = 1. Therefore, the theorem is valid for π = π β π. Next, we consider the general case. Rewrite (3) as
β πΎ1 (π + 1) ππ+1,π β πΎ2 (π + 1) ππ,π+1 ] .
(18)
(β1)π πΊ (π β π, π, π) π0,π+π
min(πβπ,πβπβπ)
+
β
(β1)π πΊ (π β π, π, π) π0,π+π .
Solving for π0,π , using the induction argument, and simplifying, min(πβπ,πβπβπ)
β1
π0,π = πΊ (π β π, π, 0) [ππβπ,π β
Using a change of variable (π β 1 β π), we obtain
β
β1
β1
(19)
β πΎ2 (π + 2) ππ,π+2 ] ,
β
πΊ (π β π, π, π) π0,π+π ] = πΊ (π β π, π, 0) π
where π = π β π + 1, . . . , π β 1; and π = π β π β 1, . . . , 0. Now, suppose that the theorem holds for π+1, π+2, . . . , πβ π. Then for π = π β π + 1, . . . , π β 1; and π = π β π β 1, . . . , 0 β1
β
[ β π΅π (π β π, π) ππβπ+π,πβπ π=0
π min(πβπ,πβπβπ)
ββ
β
π=0
π=1
ππ,π = π½2 (π, π)
π
(β1)π πΊ (π β π, π, π) π΅π (0, π + π) (23) β1
β
ππβπ+π,πβπ ] = πΊ (π β π, π, 0)
β
{πΌ (π, π + 1) β π΅π (π, π + 1) ππβπ+π,πβπ π=0
min(πβπ,πβπβπ)
π
β π½1 (π β 1, π + 1) β π΅π (π β 1, π + 1) ππβπ+π,πβπ π=0
(20)
β
β π=1
π
β [π΅π (π β π, π)
π=0
(β1)π πΊ (π β π, π, π) π΅π (0, π + π)] π
π
β πΎ1 (π + 1) β π΅π (π + 1, π + 1) ππβπ+π,πβπ
β
ππβπ+π,πβπ = β π΅π (0, π) ππβπ+π,πβπ , π=0
π=0 π
β πΎ2 (π + 2) β π΅π (π, π + 2) ππβπ+π,πβπ } ,
where π΅π (0, π) is given by (9). Now, for π = 1, 2, . . . , π β π β 1, using (5),
π=0
min(π,πβπβπ)
which leads to β1
π
ππ,π = π½2 (π, π) { β [πΌ (π, π + 1) π΅π (π, π + 1) π=0
β π½1 (π β 1, π + 1) π΅π (π β 1, π + 1)
(β1)π
π=1
ππ,π = π½2 (π, π) [πΌ (π, π + 1) ππ,π+1 β π½1 (π β 1, π + 1) ππβ1,π+1 β πΎ1 (π + 1) ππ+1,π+1
(22)
= πΊ (π β π, π, 0) π0,π
π=1
β1
ππ,πβ1 = π½2 (π, π β 1) [πΌ (π, π) ππ,π β π½1 (π β 1, π) ππβ1,π
β
ππ,π =
β
=
β
π=0 min(π,πβπβπ) π=0
(β1)π πΊ (π, π, π) π0,π+π
(β1)π πΊ (π, π, π)
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Advances in Operations Research π
a best effort stream with arrival rate π 2 . Both primary and secondary groups of servers have exponentially distributed service times with constant service rates π1 and π2 , respectively. The transition rates are then obtained from the general model as
β
β π΅π (0, π + π) ππβπ+π,πβπ π=0 π
min(π,πβπβπ)
= β[
β
π=0
π=0
(β1)π πΊ (π, π, π) π΅π (0, π + π)]
π
π½ (π, π) = π 1 + π 2 , π = 0, . . . , π β π β 1, π = π; π = 0, . . . , π β π,
β
ππβπ+π,πβπ = β π΅π (π, π) ππβπ+π,πβπ , π=0
(24) where π΅π (π, π) is given by (8). The induction argument completes the proof of the theorem.
π½1 (π, π) = π 1 , π = π β π, . . . , π β 1, π = 0, . . . , π β π, π½2 (π, π) = π 2 , π = π β π, . . . , π β 1 , π = 0, . . . , π β π β 1, (27) πΎ1 (π) = ππ1 , π = 0, . . . , π,
Now, we proceed to evaluate ππβπ+π,πβπ , π = 0, . . . , π. In the following theorem, these are obtained based on the marginal limiting probabilities of the number of busy primary servers, ππ(π) , π = 0, 1, . . . , π, which have a simple closed form. Theorem 2. The probabilities ππβπ+π,πβπ , π = 0, . . . , π, are the solution to the following system of linear equations: π
β πΆ (π, π) ππβπ+π,πβπ = ππ(π) ,
(25)
π=0
where ππ(π) = (βππ=1 π½(π β 1)πΎ1 (π)β1 )/(1 + βππ=1 βππ=1 π½(π β 1)πΎ1 (π)β1 ), πΆ(π, π) = βπβπ π=0 π΅π (π, π), and π΅π (π, π), π = 0, 1, . . . , π, π = 0, 1, . . . , π β π are as given in Theorem 1. Proof. The proof follows by noting that ππ(π) = βπβπ π=0 ππ,π , π =
0, 1, . . . , π. In addition, the probabilities ππ(π) satisfy the flow balance equations: (π) π½ (π) ππ(π) = πΎ1 (π + 1) ππ+1 ,
π = 0, 1, . . . , π.
(26)
Utilizing π0(π) as a normalizing constant and utilizing the expression in Theorem 1 of ππ,π , π = 0, 1, . . . , π, π = 0, . . . , πβ π, complete the proof.
3. Application Similar to El-Taha and Heath [1], we present an application for the special case of two, state-independent, Poisson arrival streams, a real time (protected) stream with arrival rate π 1 and 0.000354 0.002208 0.006793 0.013680 0.020103 0.022623 0.019626 0.012305 0.003575 0.000944 0.000201
0.000457 0.002936 0.009354 0.019626 0.030334 0.036486 0.034914 0.026207 0.008741 0.002585 0.000630
0.000392 0.002604 0.008607 0.018835 0.030609 0.039187 0.040776 0.034800 0.012867 0.004160 0.001130
πΎ2 (π) = ππ2 , π = 0, . . . , π β π, π 1 + π 2 , π = 0, . . . , π β π β 1, π½ (π) = { π = π β π, . . . , π β 1. π1, The joint probabilities, ππ,π , are obtained from Theorem 1 and subsequent results of Section 2. The overflow probabilities, the loss probabilities of the protected and the best effort streams, and the mean number of busy primary and secondary servers are then obtained from the joint probabilities, ππ,π . For simplicity, one may use the distribution of the number of busy primary servers, ππ(π) , as defined in Theorem 2, in obtaining some of these measures of performance. We have developed a C++ program for obtaining the joint probabilities and other measures of performance. A summary of the program algorithm is presented in the Appendix. To solve the system of linear equations in (25), we use the LU decomposition technique with partial pivoting as presented by Press et al. [14] (pp. 43β48). Press et al. [14] report that this method is three times faster than the classical GaussJordan elimination method. In addition, the method is stable and yield results with high precision for systems of linear equations of dimension in the order of couple of hundreds. Therefore, our computational scheme is expected to be fast and robust. Example. To illustrate the application of our method, consider an example with (π, π, π) = (16, 10, 3) and arrival and service rates (π 1 , π 2 , π1 , π2 ) = (10.0, 10.0, 3.0, 1.0). Joint probabilities, generated by our C++ program, are presented as follows: ππ,π ; π = 0, 1, . . . , 10; π = 0, . . . , 6
0.000272 0.001875 0.006461 0.014818 0.025395 0.034594 0.038837 0.036657 0.014727 0.005131 0.001528
0.000156 0.001134 0.004130 0.010054 0.018390 0.026899 0.032662 0.033678 0.014597 0.005459 0.001762
0.000070 0.000544 0.002145 0.005703 0.011492 0.018671 0.025321 0.029067 0.013846 0.005659 0.001999
0.000019 0.000161 0.000715 0.002184 0.005178 0.010206 0.017494 0.026934 0.014834 0.006872 0.003019
(28)
Advances in Operations Research
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INITIALIZE: input π, π, π, πΌ(π, π), π½1 (π, π), π½2 (π, π), π½(π, π), πΎ1 (π), πΎ2 (π) β0 β€ π β€ π, β0 β€ π β€ π β π; STEP 1. CASE OF (π = π β π) πΊ(0, π β π, 0) = 1; for (π = 1; π β€ π β π; π + +) πΊ(π, π β π, 0) = πΎ1 (π)β1 [πΌ(π β 1, π β π)πΊ(π β 1, π β π, 0) βπ½(π β 2, π β π)πΊ(π β 2, π β π, 0)]; for (π = 0; π β€ π β π; π + +){ π΅0 (π, π β π) = πΊ(π, π β π, 0)/πΊ(π β π, π β π, 0); for (π = 1; π β€ π; π + +){ π΅π (π β π + π, π β π) = 1; } } STEP 2. CASE OF (π β€ π β π β 1) for (π = π β π β 1, π β₯ 0; π β β){ πΊ(0, π, 0) = 1; for (π = 1; π β€ π β π; π + +){ for (π = 0; π β€ min(π, π β π β π); π + +) πΊ(π, π, π) = πΎ1 (π)β1 [πΌ(π β 1, π)πΊ(π β 1, π, π) βπ½(π β 2, π)πΊ(π β 2, π, π) + πΎ2 (π + 1)πΊ(π β 1, π + 1, π β 1)] } for (π = 0; π β€ π; π + +){ π΅π (π, π) = π½(π, π)β1 [πΌ(π, π + 1)π΅π (π, π + 1) βπ½1 (π β 1, π + 1)π΅π (π β 1, π + 1) β πΎ2 (π + 2)π΅π (π, π + 2)]; π΅π (π β π, π) = π½2 (π β π, π)β1 [πΌ(π β π, π + 1)π΅π (π β π, π + 1) βπ½(π β π β 1, π + 1)π΅π (π β π β 1, π + 1) β πΎ1 (π β π + 1)π΅π (π β π + 1, π + 1) βπΎ2 (π + 2)π΅π (π β π, π + 2)]; for (π = π β π + 1; π β€ π β 1; π + +){ π΅π (π, π) = π½2 (π, π)β1 [πΌ(π, π + 1)π΅π (π, π + 1) βπ½1 (π β 1, π + 1)π΅π (π β 1, π + 1) β πΎ1 (π + 1)π΅π (π + 1, π + 1) βπΎ2 (π + 2)π΅π (π, π + 2)]; } π΅π (0, π) = πΊ(π β π, π, 0)β1 [π΅π (π β π, π) min(πβπ,πβπβπ) β βπ=1 (β1)π πΊ(π β π, π, π)π΅π (0, π + π)]; for (π = 1; π β€ π β π β 1; π + +) min(π,πβπβπ) (β1)π πΊ(π, π, π)π΅π (0, π + π) π΅π (π, π) = βπ=0 } } } COMPUTE JOINT PROBABILITIES: input π½(π), πΎ1 (π + 1) βπ < π; for (π = π β π; π β€ π; π + +) π π π ππ(π) = βπ=1 π½(π β 1)/πΎ1 (π)[1 + βπ=1 βπ=1 π½(π β 1)/πΎ1 (π)]β1 ; } πβπ Compute: πΆ(π, π) = βπ=0 π΅π (π, π) Solve the system of equations π βπ=0 πΆ(π, π)ππβπ+π,πβπ = ππ(π) (Note; the unknowns are ππβπ+π,πβπ , π = 0, . . . , π, π = π β π, . . . , π.) for (π = π β π; π β₯ 0; π β β) for (π = 0; π β€ π; π + +) π ππ,π = βπ=0 Bπ (π, π)ππβπ+π,πβπ END Algorithm 1
Measures of performance are easily obtained from the above joint probabilities or the distribution of the number of busy primary servers, ππ(π) . These measures of performance are computed as follows: protected stream overflow probability, π1 = ππ(π) = βπβπ π=0 ππ,π = 0.010; best effort stream overflow
probability, π2 = βππ=πβπ ππ(π) = βππ=πβπ βπβπ π=0 ππ,π = 0.324; protected stream loss probability, π1 = ππ,πβπ = 0.003; best effort stream loss probability, π2 = βππ=πβπ ππ,πβπ = 0.052; mean number of busy primary servers, πΏ 1 = βππ=1 πππ(π) = βππ=1 βπβπ π=0 πππ,π = 5.553; and mean number of busy secondary
8 servers, πΏ 2 = βππ=0 βπβπ π=1 πππ,π = 2.795. These results have been validated via discrete-event simulation. Note finally that the example here is selected for illustrative purposes only. Our computational scheme can be further utilized in determining an optimal π value that meets a specific criterion such as a specified overflow probability for protected traffic and studying the effect of increasing the best effort traffic on the protected traffic quality of service. For details, see Examples 2 and 3 in El-Taha and Heath [1].
4. Conclusion We have extended the results of El-Taha and Heath [1] to explicitly allow multiple reserved channels, π > 1. As in [1], we present an efficient iterative technique to evaluate the joint probabilities of busy primary and secondary servers. A new component of the present model is the system of π + 1 linear equations given in (25). By using effective numerical methods and with the availability of computers with high processing speed and storage capacity, this part of the computational scheme will not be of concern even for large values of π. We have illustrated the application of the method for the case of two Poisson arrival streams competing for service on two sets of heterogeneous servers with constant service rates. It is important to note that the present model may be applicable for more general situations such as arrivals from a finite source and state dependent service rates. This may be an area for further extensions and applications of the results of this paper. Another area for future work is to tailor the current model to the problem of bed allocation in a hospital ward, with service rates depending on the type of arrival, which requires state expansion of the Markov chain.
Appendix For a π-server system with π primary servers and π = π β π secondary servers, Algorithm 1 computes the joint stationary probabilities ππ,π .
Competing Interests The authors declare that they have no competing interests.
References [1] M. El-Taha and J. R. Heath, βTraffic overflow in loss systems with selective trunk reservation,β Performance Evaluation, vol. 41, no. 4, pp. 295β306, 2000. [2] J. M. Akinpelu, βThe overload performance of engineered networks with nonhierarchical and hierarchical routing,β The Bell System Technical Journal, vol. 63, no. 7, 1984. [3] A. Girard, βBlocking probability of noninteger groups with trunk reservation,β IEEE Transactions on Communications, vol. 33, no. 2, pp. 113β120, 1985. [4] F. Le Gall and J. Bernussou, βBlocking probabilities for trunk reservation policy,β IEEE Transactions on Communications, vol. 35, no. 3, pp. 313β318, 1987. [5] L. Paratz, βTraffic theory of circuit reservation,β Australian Telecommunication Research, vol. 20, no. 2, pp. 51β60, 1986.
Advances in Operations Research [6] E. A. Van Doorn, βOn the overflow process from a finite Markovian queue,β Performance Evaluation, vol. 4, no. 4, pp. 233β240, 1984. [7] J. F. E. Wallin and B. Sanders, βThe calculation of overflow moments in loss systems with selective trunk reservation,β Performance Evaluation, vol. 15, no. 3, pp. 195β202, 1992. [8] J. P. Young, A queueing theory approach to the control of hospital in-patient census [Ph.D. dissertation], School of Engineering, The Johns Hopkins University, Baltimore, Md, USA, 1962. [9] P. Kolesar, βA markovian model for hospital admission scheduling,β Management Science, vol. 16, no. 6, pp. 384β396, 1970. [10] A. O. Esogbue and A. J. Singh, βA stochastic model for an optimal priority bed distribution problem in a hospital ward,β Operations Research, vol. 24, no. 5, pp. 884β898, 1976. [11] E. P. C. Kao and G. G. Tung, βBed allocation in a public health care delivery system,β Management Science, vol. 27, no. 5, pp. 507β520, 1981. [12] L. V. Green and V. Nguyen, βStrategies for cutting hospital beds: the impact on patient service,β Health Services Research, vol. 36, no. 2, pp. 421β442, 2001. [13] J. K. Cochran and K. T. Roche, βA multi-class queuing network analysis methodology for improving hospital emergency department performance,β Computers & Operations Research, vol. 36, no. 5, pp. 1497β1512, 2009. [14] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 1992.
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