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A queueing system with n-phases of service and (n-1) -types of retrial customers
European Journal of Operational Research 205 (2010) 638–649
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
A queueing system with n-phases of service and ðn 1Þ-types of retrial customers Christos Langaris, Ioannis Dimitriou * Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
a r t i c l e
i n f o
Article history: Received 18 June 2009 Accepted 21 January 2010 Available online 25 January 2010 Keywords: Poisson arrivals n-Phase service Retrial queues General services Single vacation
a b s t r a c t A queueing system with a single server providing n-phases of service in succession is considered. Every customer receives service in all phases. Arriving customers join a single ordinary queue, waiting to start the service procedure. When a customer completes his service in the ith phase he decides either to proceed to the next phase or to join the K i retrial box ði ¼ 1; 2; . . . ; n 1Þ, from where he repeats the demand for the ði þ 1Þth phase of service after a random amount of time and independently to the other customers in the system. Every customer can join during his service procedure a number of retrial boxes before departs from the system. When at the moment that a customer, either departs from the system or joins a retrial box and so releases the server, there are no other customers waiting in the ordinary queue, then the server departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service times are arbitrarily distributed. For such a system, the mean number of customers in the ordinary queue and in each retrial box separately are obtained, and used to investigate numerically system performance. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Queueing systems in which the server provides to each customer a number of phases of heterogeneous service in succession, can be proved very useful to model computer networks, production lines and telecommunication systems where messages are processed in n stages by a single server. Such kind of systems, with only two-phases of service, have firstly been discussed by Krishna and Lee [12] and Doshi [7], while more recently in a series of works (Madan [16], Choi and Kim [3], Choudhury and Madan [5], Katayama and Kobayashi [10], Ke [11]), the previous results, are extended to include systems allowing server vacations, Bernoulli feedback, N-policy, exhaustive or gated bulk service, start-up times etc., but again for models with only two-phases of service. Moreover, in all above mentioned papers one can find important applications to computer communication, production and manufacturing systems, central processor units and multimedia communications. Krishna Kumar et al. [14] and Choudhury [4] are the first who imposed the concept of ‘‘retrial customers” in the two-phase service models. Retrial queueing systems are characterized by the fact that an arriving customer who finds the server unavailable does not wait in a queue but instead he leaves the system, joining the so called retrial box, from where he repeats the demand for service later. Practical use of retrial queueing systems arises in telephone-switching systems and in telecommunication and computer networks. For complete surveys of past papers on such kind of models see Falin and Templeton [8], Artalejo and Gomez-Corral [2], Kulkarni and Liang [13] and Artalejo [1]. Krishna Kumar et al. [14] considered a two-phase service system where an arriving customer who finds the server unavailable joins the retrial box from where only the first customer can retry for service after an arbitrarily distributed time period while in the work of Choudhury [4] the investigated two-phase model includes Bernoulli server vacations and linear retrial policy. We have to observe here that in both papers the service procedure contains only two-phases of service, there is no any ordinary queue and all ‘‘waiting” customers are placed in the retrial box. Dimitriou and Langaris [6] considered a two-phase service model with both an ordinary queue and a retrial box receiving all customers who, upon finishing the first phase service, do not proceed to the second immediately but instead they decide to retry for the second phase later. In the present work, we generalize the work of Dimitriou and Langaris [6] by considering, for the first time in the literature, a model with n phases of service and n 1 retrial boxes, K 1 ; K 2 ; . . . ; K n1 say. All arriving customers are placed, upon arrival, in an ordinary queue (first phase) to receive service. When a customer completes his first phase service then, with probability 1 p1 , he proceeds to the second * Corresponding author. Tel.: +30 2651098242. E-mail addresses: [email protected] (C. Langaris), [email protected] (I. Dimitriou). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.01.034
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phase, while with probability p1 , he leaves the system and joins the K 1 retrial box. This procedure is repeated in each phase and so, when a customer completes his ith phase service, then either, with probability 1 pi , he proceeds to the ði þ 1Þth phase, or with probability pi he joins the K i retrial box in which case the server starts serving the next ordinary customer (if any) in the first phase. The customers in each retrial box retry, after a random amount of time and independently to each other, to find the server available and to proceed to the next phase of their service. Note here that every customer has to receive service in all phases and so he may join more than one retrial boxes during his service procedure. Obviously the customers in the K i retrial box are originated, either by ordinary customers, or by customers from K 1 ; K 2 ; . . . ; K i1 retrial boxes, that do not succeed to proceed directly to the ði þ 1Þth phase of service. When, at the moment of a customer’s departure from the system or his entrance in a retrial box, there are no more customers waiting in the ordinary queue, then the server departs for a single vacation (update devices, maintenance, etc.) of arbitrarily distributed length. In our model, and at any time, an ordinary and n 1 retrial queues must be taken into account. Due to the n-dimensional process that arise, the analysis of the present model cannot be handled by the methodology used to analyse the two-phase model in Dimitriou and Langaris [6], as here in case of a general n, the supplementary variable technique lead us to a ðn 1Þ-dimensional partial differential equation (Eq. (24) in Section 4) that hardly can be solved. Thus, our task now is the investigation of the mean number of customers in the ordinary queue and in each retrial box, by using a special methodology used firstly in Falin [9]. We have to point out here that this is not the first time in the literature the authors of the present work are engaged in the study of multiclass queueing systems. Thus, in Langaris and Katsaros [15], a system consisting of a single queue accepting n classes of ordinary customers, with nonpreemptive priorities between them and no retrial customers at all, is considered, while in Moutzoukis and Langaris [17], the above model is extended to contain a retrial box too, accepting a number of independent classes of customers, that retry to receive only a single service and to depart from the system. Although some theoretical results in [15,17], can also be used, after suitable modifications, in the present work, it is clear from the description above that these models, and the corresponding real life applications, are completely different from the present, new in the literature, model of n phases of heterogeneous service in succession, of n 1 retrial boxes, and of n 1 classes of retrial customers that change a number of times their class, jumping from one retrial box to another, after receiving a part of their service. As we mentioned before, the present model can only be considered as a thorough generalization of the model in Dimitriou and Langaris [6]. Our system can be used to model any situation with many stages of service, where in each stage a control and a separation of the serviced units must be taking place, and if a unit satisfies some quality standards then it proceeds immediately to the next phase of service, while, if the quality of the unit is poor, then it is removed from the system and repeats its attempt to continue its service procedure later when the server is free from high quality units. For example, consider a manufacturing procedure consisting of n consecutive production levels. Any single item must be passed from and inspected in all these production levels to achieve high quality standards. Thus, if upon the end of the ith stage processing, the quality of the item is satisfactory, the item proceeds immediately to the ði þ 1Þth stage, while if it is not, the item is sent in the ith retrial box and it is characterized as an ði þ 1Þ-defective item. In the later case, the server starts processing a new item (if any) in the first production level. All ði þ 1Þ-defective items ‘‘seek” to continue their service procedure from the level of interruption (ði þ 1Þth stage) by receiving, in each of the following stages, special treatments that improve again their quality to the high standards. Thus, the processing of an ði þ 1Þ-defective item in levels ði þ 1Þ, ði þ 2Þ, etc. must be different (includes repairs for example) from the processing of an initial non-defective item in the same levels. Whenever an item, either completes the total service procedure or failed to pass an inspection and so it releases the server, and there are no new items waiting in the first stage, the server activates a maintenance of the system (vacation). Further applications of the presented model often arise in packet transmissions, in central processors, in multimedia communications etc. The article is organized as follows. A full description of the model is given in Section 2. Some, very useful for the analysis results, on the customer total service cycle and server busy and vacation periods, are given in Section 3, while a system states analysis is performed in Section 4. In Section 5 the mean number of customers in the ordinary queue and in each retrial box separately are obtained, and used to produce, in Section 6, numerical results and to compare numerically system performance under various changes of the parameters. 2. The model Consider a queueing system consisting of n phases of service and a single server who follows the customer in service when he passes from one phase to the next. Customers arrive to the system according to a Poisson process with parameter k, and are placed in a single queue (ordinary queue) waiting to start their service in first phase. When a customer finishes his service in the ith phase ði ¼ 1; 2; . . . ; n 1Þ then he either goes to the ði þ 1Þth phase with probability 1 pi , or he, with probability pi , joins the K th i retrial box from where he retries, independently to the other customers in the box, after an exponential time parameter liþ1 , to find the server idle and to proceed to the ði þ 1Þth phase of service. In case the customer chooses to join the retrial box, the server starts immediately to serve in the first phase the next customer (if any) in the queue. When a customer finishes the nth phase of service, he departs from the system. Every time the server becomes free (that is whenever the customer in service either finishes his nth phase service or joins a retrial box, and no other customers are waiting in the ordinary queue), he departs for a single vacation U 0 the length of which is arbitrarily distributed 0 and second moment about zero b ð2Þ . If with distribution function (D.F.) B0 ðxÞ, probability density function (p.d.f.) b0 ðxÞ, finite mean value b 0 the server, upon returning from a vacation, finds customers waiting for service in the first phase (ordinary queue) he starts serving them immediately, while if there are no customers waiting, he remains idle awaiting the first arrival, either from outside or from a retrial box, to start the service procedure again. From the above description, it is clear that the server’s idle period starts only when the server, upon returning from the vacation, finds no customers waiting in the ordinary queue. Moreover any retrial customer can find a position to continue his service procedure, only when the server is in the idle mode. Let us denote by P 1 , the customers who are waiting in the ordinary queue and by Pi ði ¼ 2; 3; . . . ; nÞ; those who joined the K th i1 retrial box. Any customer has to receive service in all phases before departs from the system and so he can join, during his service procedure, a number of retrial boxes. Therefore, any arriving customer may change type (class) several times during his service procedure. Thus, a Pi customer remains a Pi customer as far as he, upon emerging from the K th i1 retrial box, continues his service procedure, passing from one
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stage to the next without joining another retrial box, while if a Pi customer joins in the sequel the K th j1 retrial box ðj > iÞ then he becomes a P j customer. The service time of a P i customer in the jth phase, Bij say, must be different (contains special treatments) from the service time of a Pk ðk – iÞ customer in the same phase. Thus, it is assumed that Bij is arbitrarily distributed with D.F. Bij ðxÞ, p.d.f. bij ðxÞ , finite mean value and second moment about zero b ð2Þ (B ðxÞ; b ðxÞ; b ;b ð2Þ do not exist of course for j < i). Finally all random variables defined above b ij ij ij ij ij ij are assumed to be independent. Note finally that our model can also be described as a system of n-phases of service and a single retrial box that accepts now n 1 types Pi ; i ¼ 2; . . . ; n, say, of customers. The type of each customer in the retrial box characterizes the phase of service that he is due to receive next. Therefore, a Pi customer, upon emerging from the retrial box, will continue his service in the ith phase and if he, in the sequel, fails to proceed to the jth phase of service ðj > iÞ, then he returns into retrial box but now as a Pj customer. Thus, is the first time in the literature of the multiclass retrial queues, where every customer may join several times the retrial box, changing each time his type (class), before departs from the system. 3. General results 0 ¼ If a customer does not join any retrial box during his service procedure (with probability p P will be R0 ¼ nj¼1 B1j with Laplace-Stieltges Transform (LST) of its p.d.f.
r0 ðsÞ ¼
n Y
Qn1 i¼1
ð1 pi Þ) then his total service cycle
b1j ðsÞ;
j¼1
where bij ðÞ the LST of bij ðÞ. Let us suppose now that an arriving customer joins r retrial boxes ðr ¼ 1; 2; . . . ; n 1Þ during his service procedure, for example he joins the retrial boxes K m1 ; K m2 ; . . . ; K mr . Then it is clear that m1 ¼ 1; 2; . . . ; n r; m2 ¼ m1 þ 1; . . . ; n r þ 1; and so on, until mr ¼ mr1 þ 1; . . . ; n 1. Moreover the probability of this event is n1 Y
m1 m2 ...mr ¼ pm1 pm2 . . . pmr p
ð1 pi Þ;
i¼1 i–m1 ;...;mr
while the duration of the customer’s total service cycle in this case is (with m0 0)
Rm1 m2 ...mr ¼
!
r X
mi X
i¼1
j¼mi1 þ1
Bmi1 þ1j þ V
þ
n X
Bmr þ1j ;
j¼mr þ1
and the LST of its p.d.f.
rm1 m2 ...mr ðsÞ ¼ ðv ðsÞÞr
mi Y
rþ1 Y
bmi1 þ1j ðsÞ;
i¼1 j¼mi1 þ1
with mrþ1 n. Note here that V is the delay incurrent due to server absence (in vacations) that precedes the service of every customer emerging from a retrial box. This absence can be either of a single duration U 0 , if no customers arrive from outside during the vacation U 0 , or of a multiple duration, if at least one customer arrives during U 0 , in which case the server has to repeat the vacation as soon as he finishes the busy period of P1 customers, and before he becomes available to the customer emerging from the retrial box. Thus the p.d.f. v ðtÞ of V satisfies
v ðtÞ ¼ ekt b0 ðtÞ þ ð1 ekt Þb0 ðtÞ v ðtÞ; with LST
v ðsÞ ¼
1þ
b0 ðk þ sÞ b0 ðk þ sÞ
b0 ðsÞ
:
Thus the LST of the p.d.f. of the customer’s total service cycle R is given by
rðsÞ ¼ p 0r 0 ðsÞ þ
n1 X nr X
nrþ1 X
r¼1 m1 ¼1 m2 ¼m1 þ1
n1 X
...
m1 m2 ...mr r m1 m2 ...mr ðsÞ; p
mr ¼mr1 þ1
and if we take derivatives above, at s ¼ 0, we arrive after some algebra at
q k
n X d r ðsÞ ¼ kEðRÞ ¼ ðqj þ q0j Þ; ds s¼0 j¼1
ð1Þ
with q01 0,
0 kb p ; j ¼ 2; 3; . . . ; n; b0 ðkÞ j1 " # n k1 X Y qj ¼ kpj1 bjj þ bjk ð1 pm Þ ;
q0j ¼
k¼jþ1
ð2Þ j ¼ 1; 2; . . . ; n:
m¼j
and p0 1. Thus q must be considered as the mean number of new customers arriving during R and so, for an ergodic system, we have to assume q < 1.
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Let now Sj be the time interval from the epoch at which a P j customer starts his service in the jth phase until the epoch he either completes his service procedure and depart from the system or he joins another retrial box and releases the server. Let also N i ðSj Þ be the new Pi customers during Sj . Note here that during Sj we can have only new P1 customers (external arrivals) and/or one new Pjþ1 or P jþ2 or . . .or P n customer according to the retrial box that this specific Pj customer will join next (if any), while when j ¼ n we can have only new P1 customers during Sn . Define finally
aðjÞ ðt; k1 ; kjþ1 ; . . . ; kn Þdt ¼ P½Ni ðSj Þ ¼ ki i ¼ 1; j þ 1; . . . ; n; t < Sj 6 t þ dt;
aj ðz1 ; zjþ1 ; . . . ; zn Þ ¼
1 1 X X
...
k1 ¼0 kjþ1 ¼0
1 X
k
jþ1 zk11 zjþ1 . . . zknn
kn ¼0
Z
1
aðjÞ ðt; k1 ; kjþ1 ; . . . ; kn Þdt:
t¼0
Then it is easy to understand that, for any j ¼ 1; 2; . . . ; n
aj ðz1 ; zjþ1 ; . . . ; zn Þ ¼
n X
pm zmþ1
m¼j
m1 Y
ð1 pl Þ
l¼j
m Y
bjk ðk kz1 Þ;
ð3Þ
k¼j
with pn 1; znþ1 1 and in general for any Q l ;
Qi
l¼j Q l
¼ 1 if i < j. Moreover from relation (2)
d a1 ðz1 ; 1; 1; . . . ; 1Þjz1 ¼1 ¼ q1 : dz1 the ð1 nÞ In general and for any xj j ¼ 1; 2; . . . ; n; let as denote, for simplicity, by x the ð1 n 1Þ vector x ¼ ðx2 ; x3 ; . . . ; xn Þ and by x ¼ ðx1 ; x2 ; . . . ; xn Þ. To proceed further we need the following Lemma the proof of which is a simple application of the well known vector x result of Takacs [18] (Lemma 1, p. 47). Lemma 1. If (i) jzk j < 1 for at least one k ¼ 2; . . . ; n, and jzm j 6 1 for all other m ¼ 2; 3; . . . ; n with m – k, or (ii) jzm j 6 1; for all m ¼ 2; 3; . . . ; n and q1 > 1, then the relation
z1 a1 ðz1 ; z2 ; . . . ; zn Þ;
ð4Þ
has one and only one zero, z1 ¼ xðzÞ say, inside the region jz1 j < 1. Specifically for z ¼ 1; xð1Þ is the smallest positive real root of (4) with xð1Þ < 1 if q1 > 1 and xð1Þ ¼ 1 for q1 6 1. Proof. See Appendix A. h Let now T ðiÞ be the duration of a busy period of P 1 customers starting with i P 1 customers and N j ðT ðiÞ Þ be the number of new P j customers ðiÞ (joining the K th j1 retrial box) during T . Define ðiÞ
g k ðtÞdt ¼ P½N j ðT ðiÞ Þ ¼ kj j ¼ 2; 3; . . . ; n; t < T ðiÞ 6 t þ dt; Z 1 1 X ðiÞ GðiÞ ðs; zÞ ¼ zk est g k ðtÞdt: k¼0
t¼0
Now one can show, by following the analysis in Takacs [18, pp. 60–63], or in Langaris and Katsaros [15] (Theorem 4), that Lemma 2. For all jzm j 6 1; m ¼ 2; 3; . . . ; n;
GðiÞ ð0; zÞ ¼ xi ðzÞ; where xðzÞ the only zero of z1 a1 ðz1 ; z2 ; . . . ; zn Þ in jz1 j < 1. Proof. Let us suppose that the busy period of P 1 customers starts with only one P1 customer. By observing in the usual way (see Takacs [18, pp. 60–63]), that the duration of the busy period and the number of customers joining the retrial boxes during it, do not affected from the order in which the P 1 customers are selected for service, we can write easily ð1Þ
g k ðtÞ ¼
n X
ekt hi ðtÞdfk¼1i g þ
i¼1
n X 1 X i¼1
X
j¼1 m1 þm2 þ...mj ¼k1i
Z 0
t
h i eku ðkuÞj ð1Þ ð1Þ hi ðuÞ g ð1Þ m1 ðt uÞ g m2 ðt uÞ g mj ðt uÞ du j!
where 1i is the ð1 n 1Þ vector with 1 in the ith position and 0 in all other entries, 1n 0, and hi ðuÞ is the total duration of the consecutive service times of a P1 customer until the epoch he joins the ith retrial box, i ¼ 1; 2; . . . ; n; where i ¼ n means that the customer ends his service procedure and departs without joining any retrial box, i.e. for i ¼ 1; 2; . . . ; n
hi ðuÞ ¼ pi
i1 Y
ð1 pl Þb11 ðuÞ b12 ðuÞ b1i ðuÞ
l¼1
with pn 1 and LST
hi ðsÞ ¼ pi
i1 Y l¼1
ð1 pl Þ
i Y k¼1
b1k ðsÞ:
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Now it is clear that
Gð1Þ ðs; zÞ ¼
1 Z X 0
j¼0
¼
Z 1 n n X ð1Þ eðkþsÞu ðkuÞj X ziþ1 hi ðuÞ½Gð1Þ ðs; zÞj du ¼ eðsþkkG ðs;zÞÞu ziþ1 hi ðuÞdu j! 0 i¼1 i¼1
1
n X
pi ziþ1
i¼1
i1 Y
ð1 pl Þ
l¼1
i Y
b1k ðs þ k kGð1Þ ðs; zÞÞ
k¼1
and using relation (3) with s ¼ 0 we arrive at
Gð1Þ ð0; zÞ ¼ a1 ðGð1Þ ð0; zÞ; z2 ; . . . ; zn Þ and from Lemma 1 Gð1Þ ð0; zÞ ¼ xðzÞ. Now it is easy to understand that if the busy period starts with i P1 customers, then it can be considered as the sum of i busy periods each of which starts with only one P 1 customer, and the lemma has been proved. Let V be the time interval from the epoch the server departs for a single vacation until the epoch he becomes idle for the first time. Denote also N j ðVÞ the number of new Pj customers during V. If we define
v k ðtÞdt ¼ P½Nj ðVÞ ¼ kj v ðs; zÞ ¼
1 X
zk
Z
1
j ¼ 2; 3; . . . ; n; t < V 6 t þ dt;
est v k ðtÞdt;
t¼0
k¼0
then
v k ðtÞ ¼ ekt b0 ðtÞdfk¼0g þ
k X 1 X ðktÞi kt e b0 ðtÞ g ðiÞ m ðtÞ v km ðtÞ; i! m¼0 i¼1
where
dfAg ¼
1 if A holds; 0 otherwise;
and so
v ð0; zÞ ¼
1þ
b0 ðkÞ
b0 ðkÞ : b0 ðk kxðzÞÞ
ð5Þ
Let DðiÞ be the time interval from the epoch a P i i ¼ 2; 3; . . . ; n retrial customer finds a position for service, until the epoch the server departs for a vacation, and denote by N j ðDðiÞ Þ, the number of the new P j customers during DðiÞ . Define finally ðiÞ
dk ðtÞdt ¼ P½Nj ðDðiÞ Þ ¼ kj j ¼ 2; 3; . . . ; n; t < DðiÞ 6 t þ dt; Z 1 1 X ðiÞ zk est dk ðtÞdt; DðiÞ ðs; zÞ ¼ k¼0
t¼0
then ðiÞ
dk ðtÞ ¼ ekt
n X
dfk¼1rþ1 g sir ðtÞ þ
r¼i
1 n X ðktÞm kt X ðmÞ sir ðtÞ g k1rþ1 ðtÞ; e m! m¼1 r¼i
ð6Þ
where 1j ¼ ð0; . . . ; 0; 1; 0; . . . ; 0Þ with the 1 in the jth position, 1nþ1 0; and
sir ðtÞ ¼ ð1 pi Þbii ðtÞ ð1 pr1 Þbir1 ðtÞ pr bir ðtÞ;
r ¼ i; i þ 1; . . . ; n;
is in fact the total time the P i retrial customer holds the server from the epoch he finds a position for service until the epoch he joins the K th r retrial box and becomes a Prþ1 customer or departs from the system (case r ¼ n). By taking LST in (6) above we arrive at
DðiÞ ð0; zÞ ¼ ai ðxðzÞ; ziþ1 ; . . . ; zn Þ;
ð7Þ
where the function ai ðxðzÞ; ziþ1 ; . . . ; zn Þ has been defined in (3). Define finally C ðiÞ as the time interval from the epoch at which a P i customer finds a position for service until the epoch the server becomes for the first time idle and ready to accept the next customer from outside or from a retrial box. If N j ðC ðiÞ Þ is the number of new P j customers during C ðiÞ and define ðiÞ
ck ðtÞdt ¼ P½Nj ðC ðiÞ Þ ¼ kj j ¼ 2; 3; . . . ; n; t < C ðiÞ 6 t þ dt; Z 1 1 X ðiÞ C ðiÞ ðs; zÞ ¼ zk est ck ðtÞdt; k¼0
t¼0
then it is easy to realize that C ðiÞ ¼ DðiÞ þ V and from (7)
C ðiÞ ð0; zÞ ¼ ai ðxðzÞ; ziþ1 ; . . . ; zn Þv ð0; zÞ:
ð8Þ
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We have to state here the following theorem. The proof is similar to the proof of Theorem 3.2 in Moutzoukis and Langaris [17]. h Theorem 3. For any permutation ði2 ; i3 ; . . . ; in Þ of the set ð2; 3; . . . ; nÞ and for j ¼ 2; 3; . . . ; n; if (a) jzim j < 1 for at least one m ¼ j þ 1; . . . ; n, and ij1 > 1ðq i1 q1 Þ, or (c) jzir j 6 1, for all jzir j 6 1, for all other r ¼ j þ 1; . . . ; n with r – m, or (b) jzir j 6 1, for all r ¼ j þ 1; . . . ; n, and q ij1 , the equation ij > 1 P q r ¼ j þ 1; . . . ; n, and q
zij C ðij Þ ð0; wij1 ðzij ; zijþ1 ; . . . ; zin ÞÞ ¼ 0;
ð9Þ
has one and only one root, zij ¼ xij ðzijþ1 ; . . . ; zin Þ; j – n; zin ¼ xin say, inside the region jzij j < 1, where the vector wij ðzijþ1 ; zijþ2 ; . . . ; zin Þ is defined by
wi1 ðzi2 ; zi3 ; . . . ; zin Þ ¼ ðz2 ; z3 ; . . . ; zn Þ; wi2 ðzi3 ; zi4 ; . . . ; zin Þ ¼ wi1 ðxi2 ðzi3 ; . . . ; zin Þ; zi3 ; . . . ; zin Þ; wik ðzikþ1 ; zikþ2 ; . . . ; zin Þ ¼ wik1 ðxik ðzikþ1 ; . . . ; zin Þ; zikþ1 ; . . . ; zin Þ;
k ¼ 2; . . . ; n 1;
while for j ¼ 2; 3; . . . ; n
q ij ¼
@ ðij Þ C ð0; wij1 ðzij ; zijþ1 ; . . . ; zin ÞÞjzi ¼zi ¼¼zi ¼1 : n j jþ1 @zij
ð10Þ
ij1 6 1 the root xij ð1; . . . ; 1Þ is the smallest positive real root of (9) with xij ð1; . . . ; 1Þ < 1 if Moreover for real zir ¼ 1; r ¼ j þ 1; . . . ; n, and q
q ij > 1 and xij ð1; . . . ; 1Þ ¼ 1 for q ij 6 1. Proof. See Appendix A. h in in (10) is given by, One can show here that, for any permutation ði2 ; i3 ; . . . ; in Þ of the set ð2; 3; . . . ; nÞ, the final term q
P
q in ¼
qin þ q0in þ dfin
ð11Þ
in (and all other q ij ) is always less than one. and so it is clear, by comparing relations (1) and (11) that, for q < 1; q 4. Steady states analysis Let us assume that a state of statistical equilibrium exists and let N 1 ; N i ; i ¼ 2; . . . ; n denote the number of P1 ; P i , customers in the ordinary queue (not in service) and in the K th i1 retrial box, respectively. Let also
8 if server on vacation; > <0 n ¼ ði; jÞ if server busy on jth phase with P i customer; > : id if server idle;
and
qðkÞ ¼ Pðn ¼ id; N 1 ¼ 0; Nm ¼ km ; m ¼ 2; 3; . . . ; nÞ; xÞdx ¼ Pðn ¼ 0; N m ¼ km ; m ¼ 1; 2; . . . ; n; x < U 0 6 x þ dxÞ; p0 ðk; xÞdx ¼ Pðn ¼ ði; jÞ; N m ¼ km ; m ¼ 1; 2; . . . ; n; x < U ij 6 x þ dxÞ; p ðk; ij
¼ ðk1 ; k2 ; . . . ; kn Þ ðk1 ; kÞ, and U ; U 0 the elapsed service or vacation time, respectively. If where, as it is stated before, k ¼ ðk2 ; . . . ; kn Þ; k ij finally
Q ðzÞ ¼
X
qðkÞzk
kP0
P0 ðz; xÞ ¼
X
X
...
k2 P0
X
qðk2 ; . . . ; kn Þzk22 zk33 . . . zknn ;
kn P0
xÞzk ; p0 ðk;
0 kP
Pij ðz; xÞ ¼
X
xÞzk ; pij ðk;
i; j ¼ 1; 2;
0 kP
then we arrive easily, for x > 0, at
P0 ðz; xÞ ¼ P0 ðz; 0Þð1 B0 ðxÞÞ exp½ðk kz1 Þx; Pij ðz; xÞ ¼ Pij ðz; 0Þð1 Bij ðxÞÞ exp½ðk kz1 Þx;
ð12Þ
and n X j¼2
lj zj
@ Q ðzÞ þ kQ ðzÞ ¼ P0 ðð0; zÞ; 0Þb0 ðkÞ; @zj
with b0 ð:Þ the LST of b0 ð:Þ. For the boundary conditions ðx ¼ 0Þ we obtain in a similar way
ð13Þ
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C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
P0 ðð0; zÞ; 0Þ ¼
n1 X
pm zmþ1
m¼1
m X
Pim ðð0; zÞ; 0Þbim ðkÞ þ
i¼1
n X
Pin ðð0; zÞ; 0Þbin ðkÞ;
ð14Þ
i¼1
@ Q ðzÞ; j ¼ 2; . . . ; n; @zj j1 Y ð1 pm Þb1m ðk kz1 Þ; P1j ðz; 0Þ ¼ P11 ðz; 0Þ
Pjj ðð0; zÞ; 0Þ ¼ lj
j ¼ 2; . . . ; n;
m¼1
Pij ðz; 0Þ ¼ li
j1 Y @ Q ðzÞ ð1 pm Þbim ðk kz1 Þ; @zi m¼i
i ¼ 2; . . . ; n j ¼ i þ 1; . . . ; n;
ð15Þ
while for the P11 ðz; 0Þ we obtain using relations (14) and (15)
( P11 ðz; 0Þ ¼
kz1 QðzÞ þ
n X
), aj ðz1 ; zjþ1 ; . . . ; zn ÞP jj ðð0; zÞ; 0Þ P0 ðð0; zÞ; 0Þ½1 þ b0 ðkÞ b0 ðk kz1 Þ
½z1 a1 ðzÞ:
ð16Þ
j¼2
Replacing now in the numerator of (16) the zero xðzÞ of the denominator (see Lemma 1), we arrive at
P0 ðð0; zÞ; 0Þ ¼
P kxðzÞQðzÞ þ nj¼2 aj ðxðzÞ; zjþ1 ; . . . ; zn ÞPjj ðð0; zÞ; 0Þ ; 1 þ b0 ðkÞ b0 ðk kxðzÞÞ
ð17Þ
and substituting back from (14)–(17) in (12) and (13) and integrating with respect to x we obtain for j ¼ 2; . . . ; n
e1j ðz1 ÞP1j ðzÞ ¼ e11 ðz1 ÞP11 ðzÞ
j1 Y
ð1 pm Þb1m ðk kz1 Þ;
ð18Þ
m¼1
ejj ðz1 ÞPjj ðzÞ ¼ lj
@ Q ðzÞ; @zj
ð19Þ
j1 Y i ¼ 2; . . . ; n @ Q ðzÞ ð1 pm Þbim ðk kz1 Þ; @zi j ¼ i þ 1; . . . ; n; m¼i Pn kxðz2 ; . . . ; zn ÞQ ðzÞ þ j¼2 ejj ðz1 Þaj ðxðz2 ; . . . ; zn Þ; zjþ1 ; . . . ; zn ÞPjj ðzÞ ; e0 ðz1 ÞP0 ðzÞ ¼ 1 þ b0 ðkÞ b0 ðk kxðzÞÞ Xn e ðz1 Þ½aj ðz1 ; zjþ1 ; . . . ; zn Þ aj ðxðzÞ; zjþ1 ; . . . ; zn ÞPjj ðzÞ e11 ðz1 ÞP11 ðzÞ ¼ fk½z1 xðzÞQ ðzÞ þ j¼2 jj
eij ðz1 ÞPij ðzÞ ¼ li
þ e0 ðz1 Þ½b0 ðk kz1 Þ b0 ðk kxðzÞÞP0 ðzÞg=½z1 a1 ðzÞ; n X
lj zj
j¼2
@ Q ðzÞ þ kQ ðzÞ ¼ e0 ðz1 ÞP0 ðzÞb0 ðkÞ; @zj
where in general P ðzÞ ¼
eij ðz1 Þ ¼
R
ð20Þ ð21Þ
ð22Þ ð23Þ
P ðz; xÞdx and
k kz1 ; 1 bij ðk kz1 Þ
1 eij ð1Þ ¼ : bij
By substituting finally from (21) to (23) and using relations (5), (8), (19), we arrive easily at the ðn 1Þ-dimensional partial differential equation n X
lj ½zj C ðjÞ ð0; zÞ
j¼2
@ Q ðzÞ þ k½1 xðzÞv ð0; zÞQ ðzÞ ¼ 0 @zj
ð24Þ
Now to obtain, from (18)–(22) the generating functions, one has to solve first Eq. (24) to obtain Q ðzÞ, and it is easy to understand that Eq. (24) hardly can be solved. In the sequel we will use the relations obtained so far to investigate, using the methodology in Falin [9], the mean number of customers in the ordinary queue and in each retrial box separately. 5. Mean number of ordinary and retrial customers To start the analysis we need to calculate the generating functions at the point z ¼ 1: are given by z¼1 Theorem 4. For q < 1 the generating functions Pij ð:Þ; P 0 ð:Þ; Q ð:Þ at the point
11 ; ¼ kb P11 ð1Þ ¼ kp b Pij ð1Þ i1 ij
¼ kp b Pjj ð1Þ j1 jj ; j1 Y m¼i
ð1 pm Þ;
i ¼ 1; . . . ; n j ¼ i þ 1; . . . ; n
1q 0 =b ðkÞ ; 1 þ kb " 0 # n 0 X k b ¼ p þ Qð1Þ : P0 ð1Þ b0 ðkÞ j¼2 j1
Q ð1Þ ¼
j ¼ 2; . . . ; n
ð25Þ
C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
645
Proof. Let us define
N ðzÞ ¼ Q ðzÞ þ P 0 ðzÞ þ
n X n X i¼1
Pij ðzÞ;
ð26Þ
j¼i
then from relations (18)–(20)
Pij ðzÞ ¼
j1 Y i ¼ 1; 2; . . . ; n eii ðz1 Þ Pii ðzÞ ð1 pm Þbim ðk kz1 Þ; eij ðz1 Þ j ¼ i þ 1; . . . ; n m¼i
i.e. j1 Y ¼ P ii ð1Þ bij Pij ð1Þ ð1 pm Þ; b ii m¼i
i ¼ 1; 2; . . . ; n
ð27Þ
j ¼ i þ 1; . . . ; n
¼ 1 we obtain after manipulations and substituting back in (26), using (21) and observing that N ð1Þ
! " # n n k1 0 0 X Y X b kb Pjj ð1Þ Q ð1Þ þ ¼ 1: b b P ð 1Þ þ 1 þ þ ð1 p Þ þ jj jk m jj 11 11 b0 ðkÞ b0 ðkÞ b kb j¼2 m¼j k¼jþ1
q1
ð28Þ
we arrive at Using now (21) in (22) and putting z¼1
P11 ð1Þ k ¼ 1 q1 b11
( 1þ
! " #) n n k1 0 0 X X Y kb Pjj ð1Þ b þ Q ð1Þ þ ; ð1 p Þ þ b b jj jk m b0 ðkÞ b0 ðkÞ b jj j¼2 m¼j k¼jþ1
ð29Þ
where q1 has been defined in (2), and so from (27)–(29) above we conclude
; ¼ kb P11 ð1Þ 11
¼ kb P1j ð1Þ 1j
j1 Y
ð1 pm Þ;
j ¼ 2; 3; . . . ; n:
ð30Þ
m¼1
Multiplying relation (19) by zj , adding for all j, subtracting from (23) and using (21) we arrive at n X ejj ðz1 ÞPjj ðzÞ ½Dj ðzÞ zj ¼ k½1 xðzÞf ðxðzÞÞ; Q ðzÞ j¼2
ð31Þ
with
f ðxðzÞÞ ¼
b0 ðkÞ ; 1 þ b0 ðkÞ b0 ðk kxðzÞÞ
ð32Þ
Dj ðzÞ ¼ f ðxðzÞÞaj ðxðzÞ; zjþ1 ; . . . ; zn Þ:
ð33Þ
Here the great importance of the analysis and the results obtained in Section 2 is apparent. By comparing relations (32) and (33) above with relations (5) and (8) in Section 2 it is clear that
Di ðzÞ C ðiÞ ð0; zÞ; and so for q < 1, the main Theorem 3 can be used directly in (31). Thus by putting in (31) z1 ¼ 1, and, for any permutation ði2 ; i3 ; . . . ; in Þ of ð2; 3; . . . ; nÞ; replacing in succession zi2 ; zi3 ; . . . ; zin1 by the corresponding zeros xi2 ðÞ; xi3 ðÞ; . . . ; xin1 ðÞ we succeed to eliminate all except the final term of the sum in the left hand side of (31). Thus the only unknown remaining in (31) is P in in ðÞ and this for all in ¼ 2; 3; . . . ; n; and so finally
¼ Pjj ð1Þ
! 0 kpj1 b kb jj Q ð1Þ; 1 þ 1 q b0 ðkÞ
j ¼ 2; 3; . . . ; n;
ð34Þ
and replacing in (28) we obtain after manipulations
Q ð1Þ ¼
1 q
0 1 þ bkbðkÞ
ð35Þ
;
0
which is the third of (25). in (27) and (21) we arrive at the second and forth of (25), From (35) and (34) we obtain the first of (25) and putting back Pjj ð1Þ respectively and the theorem has been proved. h For p0 1 and j ¼ 1; . . . ; n, let us define now
(
pj ¼ pj1
ð2Þ þ b jj
n X k¼jþ1
ð2Þ b jk
k1 Y m¼j
ð1 pm Þ þ 2
n1 Y r X r¼j m¼j
then, for the mean length of the ordinary queue,
" #) n k1 X Y ð1 pm Þbjr bjrþ1 þ ð1 pm Þ ; bjk k¼rþ2
m¼rþ1
ð36Þ
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C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
Theorem 5. The mean number of P1 customers, in the ordinary queue is given by,
( ) " # n n 0 X X k2 b EðN1 Þ ¼ p þ Qð1Þ þ pk ; 2ð1 q1 Þ b0 ðkÞ k¼2 k1 k¼1
where Q ð1Þ and
ð37Þ
pk are given by (35) and (36), respectively.
Proof. Differentiating relations (19)–(21), with respect to z1 , and setting zl ¼ 1; l ¼ 1; 2; . . . ; n, we arrive easily at,
ð2Þ k2 pi1 b @Pii ðzÞ ii ¼ ; @z1 z¼1 2 ð2Þ Y j1 j1 j1 X Y k2 pi1 b @Pij ðzÞ ij ; ¼ ð1 pm Þ þ k2 pi1 b ð1 pm Þ b ij im @z1 z¼1 2 m¼i m¼i m¼i " # n ð2Þ X @P0 ðzÞ k2 b ¼ 0 p þ Q ð1Þ : @z1 z¼1 2b0 ðkÞ k¼2 k1
i ¼ 2; 3; . . . ; n; j ¼ i þ 1; . . . ; n;
ð38Þ
In a similar way, from (22) and (18),
" # " # n n ð2Þ X ð2Þ X @P11 ðzÞ k3 b k2 b b 11 0 ¼ pk1 þ Q ð1Þ þ pk þ 11 ; @z1 z¼1 2ð1 q1 Þ b0 ðkÞ k¼2 2 k¼1 " # " # j1 ð2Þ n n 3 X Y X b @P1j ðzÞ k b 1j 0 ¼ ð1 pm Þ p þ Q ð1Þ þ pk @z1 z¼1 2ð1 q1 Þ m¼1 b0 ðkÞ k¼2 k1 k¼1 " # j1 j1 ð2Þ Y X þ b1j ; j ¼ 2; 3; . . . ; n: ð1 pm Þ b b þ k2 1j 1m 2 m¼1 m¼1
ð39Þ
Observing now that
EðN1 Þ ¼
n X n X @N ðzÞ @Pij ðzÞ @P0 ðzÞ ¼ þ ; @z1 z¼1 @z1 z¼1 @z1 z¼1 i¼1 j¼i
and replacing from (38) and (39) we obtain relation (37) and the theorem has been proved. h Before giving the mean queue lengths for the retrial customers we have to state some preliminary results. Let, for k; j ¼ 2; . . . ; n; m ¼ 1; 2, ðmÞ
hk
¼
m d xðzÞ ; m dzk z¼1
q^ ðmÞ jk ¼
m d aj ðxðzÞ; . . . ; zk ; . . . 1Þ m ; dzk z¼1
^ ðuÞ ^ ðuÞ where xðzÞ is defined in Lemma 1 (note that q jk – qkj ; k – j). Then after some algebra we obtain, ð1Þ
hk ¼ ð2Þ
k2 pk1 Y ð1 pm Þ; ð1 q1 Þ m¼1
ð1Þ k1 X k2 ðhk Þ2 p1 þ 2kðhkð1Þ Þ2 b1m ; ð1 q1 Þ m¼1 ! n r1 X Y ð1Þ ¼ kh ð1 p Þ þ d b þ b
hk ¼
q^ jkð1Þ
k
jj
jr
þ q^ jkð2Þ ¼ khð2Þ b jj k
m¼j
n X
r1 Y
r¼jþ1
b jr
fk>jg
m
r¼jþ1
"
! ð1 pm Þ þ
m¼j
# k2 Y pk1 ð1 pm Þ ; m¼j
ð1Þ ðkhk Þ2
pj1
pj þ dfk>jg 2khð1Þ k pk1
k2 Y m¼j
ð1 pm Þ
k1 X
b jr
k; j ¼ 2; . . . ; n:
r¼j
Moreover define
sk ¼
ð2Þ n h i X kQ ð1; . . . ; 1Þhk k2 pk1 P0 ð1; . . . ; 1Þ k ð2Þ þ 0 hð2Þ þ ðkhð1Þ Þ2 b ^ ð2Þ þ pj1 q þ kb Qk2 Qk2 Qk2 0 k k jk l ð1 q Þ 2pk1 m¼1 ð1 pm Þ 2pk1 b0 m¼1 ð1 pm Þ 2pk1 m¼1 ð1 pm Þ j¼2 1 k ! ð1Þ n 0 X khk b k2 b kp k 0 ð1Þ ; k ¼ 2; . . . ; n: þ q0j q^ ð1Þ ð1 q Þhk þ k1 þ jk þ ð1 q1 Þ j¼2 ð1 q1 Þb0 ðkÞ lk b0 ðkÞ
and denote
@2 Dkl ¼ Q ðzÞ @zk @zl
;
k; l ¼ 2; 3; . . . ; n:
z¼1
Note here that Dkl ¼ Dlk 8 k; l. Then we state the following theorem. Theorem 6. The quantities Dkl ; k; l ¼ 2; . . . ; n can be found as the solution of the system of linear equations,
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C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
ðlk þ ll ÞDkl ¼
n1 X
!
lj Dkj pl1
j¼2
þ pl1
l2 n1 k2 X Y Y ð1 pm Þ dfl>jg þ lj Dlj dfk>jg pk1 ð1 pm Þ m¼j
l2 Y
ð1 pm Þ
m¼1
" n X j¼2
j¼2
m¼j
#
!
"
#
k2 n X Y lj ðqj þ q0j Þ lj ðqj þ q0j Þ ð1 pm Þ Dkj þ sk þ pk1 Dlj þ sl : pj1 ð1 q1 Þ pj1 ð1 q1 Þ m¼1 j¼2
ð40Þ
Proof. Replacing (21) to (22) we arrive at
e0 ðz1 ÞP0 ðzÞ ¼
kz1 Q ðzÞ þ
Pn
z1 Þ
j¼2 ejj ðz1 Þaj ðz1 ; zjþ1 ; . . . ; zn ÞP jj ðz Þ þ e11 ðz1 ÞP 11 ðz Þða1 ðz Þ 1 þ b0 ðkÞ b0 ðk kz1 Þ
;
and replacing to (23) and setting z1 ¼ 1 we obtain n X
lj zj
j¼2
n X @ Pjj ð1; zÞ P11 ð1; zÞ Q ðzÞ ¼ jj aj ð1; zjþ1 ; . . . ; zn Þ þ b 11 ða1 ð1; zÞ 1Þ: @zj b j¼2
ð41Þ
Now adding relation (19) for all j ¼ 2; . . . ; n, putting z1 ¼ 1 and subtracting from (41) we arrive at our basic equation, n X
lj ðzj 1Þ
j¼2
n1 X @ Pjj ð1; zÞ QðzÞ ¼ ½aj ð1; zjþ1 ; . . . ; zn Þ 1: @zj b jj j¼1
ð42Þ
Taking finally derivatives above with respect to zk ; zl , and using the fact that from (19) after some algebra at relation (40). h
@P jj ð zÞ jz¼1 @zk
; 8 k; j ¼ 2; 3; . . . ; n; we arrive ¼ lj Djk b jj
Now we are ready to give the mean number of customers in the retrial boxes. Lemma 7. The mean number of Pk ; k ¼ 2; 3; . . . ; n customers in K th k1 retrial box is given by
EðNk Þ ¼
ð1Þ n 0 khk b lm ðqm þ q0m Þ sk q1 X kb kp kp 0 ð1Þ q0m ðq^ ð1Þ Dmk þ þ Þþ ð1 q Þhk þ k1 þ k1 : mk þ kp ð1 q Þ k b ðkÞ b ðkÞ l lk m1 1 k 0 0 m¼2 m¼2 n X
ð43Þ
where Dkl are found in Theorem 6. Proof. Differentiating (22) and (18), with respect to zk , and setting zl ¼ 1; l ¼ 1; 2; . . . ; n, we obtain after manipulations,
" # n X @P 11 ðzÞ lm ðqm þ q0m Þ 11 ¼ b þ s D mk k ; @zk z¼1 pm1 ð1 q1 Þ m¼2 j1 Q ð1 p Þ b 1j r @P 1j ðzÞ @P11 ðzÞ r¼1 ¼ ; j ¼ 2; . . . ; n: 11 @zk z¼1 @zk b
ð44Þ
z¼1
In a similar way from (19)–(21),
@Pii ðzÞ @zk z ¼1
@Pij ðzÞ @zk
z¼1
@P0 ðzÞ @zk
z¼1
; ¼ li Dik b ii Q j1 b ð1p ij r Þ @P ii ð z Þ r¼i ¼ @z b k
ii
¼
n P lm q0m D m¼2
kpm1
mk
z¼1
i ¼ 2; . . . ; n;
; h
j ¼ i þ 1; . . . ; n; ð1Þ
ð45Þ i
0 þ bkbðkÞ ð1 q Þhk þ kplk1 þ 0
k
n P m¼2
q0m
ð1Þ khk b 0 q^ ð1Þ : mk þ b ðkÞ 0
Observing now that, for any k ¼ 2; 3; . . . ; n;
EðNk Þ ¼
n X n X @N ðzÞ @Q ðzÞ @Pij ðzÞ @P0 ðzÞ ¼ þ þ ; @zk z¼1 @zk z¼1 i¼1 j¼i @zk z¼1 @zk z¼1
ð46Þ
and replacing from (44), (45) to (46) we arrive easily at (43) and the theorem has been proved. h
6. Numerical results In this section we consider a system of n ¼ 4 phases of service and so three retrial boxes and use the formulae derived previously to obtain numerical results and to investigate the way the mean number of customers in the retrial boxes EðN i Þ i ¼ 2; 3; 4 are affected when 11 , and the mean retrial interval in the first box 0 , the mean service time of the ordinary customers b we vary the mean vacation time b Eðretrial K 1 Þ ¼ 1=l2 , always for increasing values of the mean arrival rate k. To construct the tables we assume that the vacation time U 0 and the service times follow exponential distributions with p.d.f.’s respectively,
1 b0 ðxÞ ¼ eð1=b0 Þx ; b0
1 bij ðxÞ ¼ eð1=bij Þx ; bij
i ¼ 1; . . . ; 4 j ¼ i; . . . ; 4:
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C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
14 ¼ 0:2, b 12 ¼ b 22 ¼ b 23 ¼ b 33 ¼ 0:33, b 13 ¼ b 24 ¼ b 34 ¼ b 44 ¼ 0:25; p ¼ 0:7, Moreover we assume that in all tables below b 1 p2 ¼ 0:5; p3 ¼ 0:1. Finally l3 ¼ 0:5; l4 ¼ 2. Table 1 shows the way EðN i Þ i ¼ 2; 3; 4 changes when we vary the mean vacation time, for increasing values of the mean arrival rate k. Here one can observe the crucial role that the vacation plays on the number of retrial customers. Thus, even for a small value of k; k ¼ 0:15 0 ¼ 0Þ to the system with for example, EðN2 Þ increases from 0.1488 to 46.026 when we pass from a system without vacation period (b
Table 1 Values of EðN i Þ; i ¼ 2; 3; 4; for
l2 ¼ 1; b11 ¼ 0:5.
knb 0
0
0.2
0.6
1.3
2.7
0.1488 0.2029 0.0112
0.1926 0.2451 0.0161
0.3084 0.3602 0.0282
0.6816 0.7605 0.0623
46.026 64.83 3.316
0.27
0.3798 0.4958 0.0287
0.5273 0.6693 0.0435
1.1548 1.3906 0.0954
120.04 170.57 8.599
0.42
1.045 1.3022 0.0768
1.8373 2.2913 0.1395
267.02 380.45 19.077
0.59
4.1926 5.2556 0.2956
221.4 314.4 15.814
0.71
126.8 179.3 9.0327
0.15
EðN 2 Þ EðN3 Þ EðN4 Þ
Table 2 Values of EðN i Þ; i ¼ 2; 3; 4; for
l2 ¼ 1; b0 ¼ 0:2.
knb 11
0.2
0.8
1.3
2.1
2.8
5.5
0.1711 0.2257 0.0145
0.2233 0.271 0.0181
0.3014 0.3291 0.0226
0.5481 0.4882 0.0346
1.0297 0.7737 0.0595
18522.3 26410.6 1321.15
0.25
0.3646 0.4783 0.0306
0.6088 0.7108 0.047
1.1043 1.1131 0.0755
4.3585 3.97 0.25
193.69 263.19 13.378
0.3
0.5051 0.662 0.042
0.9877 1.1428 0.0742
2.267 2.354 0.1496
64.9 85.09 4.423
0.4
0.9447 1.2382 0.0765
3.034 3.6085 0.2161
112.03 156.15 7.9094
0.5
1.8448 2.4349 0.1444
108.39 152.81 7.717
0.71
92.598 131.62 6.632
0.15
EðN 2 Þ EðN 3 Þ EðN 4 Þ
Table 3 ¼ 0:2; b ¼ 0:5. Values of EðN i Þ; i ¼ 2; 3; 4 for b 0 11 0.02
0.2
1
2
10
0.0452 0.2539 0.0177
0.0727 0.2515 0.0171
0.1926 0.2454 0.0161
0.3397 0.2409 0.0157
1.4994 0.2329 0.0151
0.27
0.1423 0.7229 0.0521
0.2185 0.7087 0.0488
0.5373 0.6693 0.0435
0.9183 0.6443 0.413
3.862 0.5987 0.0382
0.42
0.4837 2.7032 0.1904
0.761 2.5895 0.1699
1.8373 2.2913 0.1395
3.045 2.1153 0.1271
11.953 1.8179 0.1093
0.59
8.33 464.3 23.5
60.25 419.03 21.12
221.4 314.4 15.814
371.7 265.23 13.31
1347.5 192.75 9.6763
k n Eðretrial K 1 Þ 0.15
EðN 2 Þ EðN 3 Þ EðN 4 Þ
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0 ¼ 2:7; while the corresponding value for EðN 3 Þ increases from 0.2029 to 64.83. When now the arrival rate k becomes k ¼ 0:42 then even a b 0 ¼ 0:6 increases dramatically the mean number of retrial customers to 267.02 in retrial box K 1 and to 380.45 0 ¼ 0 to b small change from b in retrial box K 2 , respectively. Thus we must be very careful on the vacation period that we must allow, to avoid overcrowded retrial boxes. The behavior of the third retrial box K 3 ðEðN 4 ÞÞ is smoother, and it shows us a way to reduce this dramatic effect of the vacation period by allowing faster retrials (Eðretrial K 3 Þ ¼ 1=l4 ¼ 0:5 here) and/or less preference of the box ðp3 ¼ 0:1Þ. 11 . Similar observations can be deduced from Table 2 that contains values of EðNi Þ i ¼ 2; 3; 4 when we vary the mean first stage service b One can observe again the way the mean number of retrial customers in each box increases when b11 increases. An increase that depends on how fast or slow the mean retrial Eðretrial K i Þ is and/or on the preference that customers show to the corresponding box pi . Table 3 finally depicts the way the EðN i Þ i ¼ 2; 3; 4 are affected when we vary the mean retrial rate in the first box Eðretrial K 1 Þ ¼ 1=l2 . One can observe here not only the increase of EðN 2 Þ, but mainly the reduction of EðN 3 Þ and EðN 4 Þ in the other two boxes when we increase 1=l2 , a reduction which is more apparent when k increases. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.ejor.2010.01.034. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
J.R. Artalejo, A classified bibliography of research on retrial queues: progress in 1990–1999, Top 7 (2) (1999) 187–211. J.R. Artalejo, A. Gomez-Corral, Retrial queueing systems, A Computational Approach, Springer-Verlag, Berlin, Heidelberg, 2008. D.I. Choi, T. Kim, Analysis of a two-phase queueing system with vacations and Bernoulli feedback, Stochastic Analysis and Applications 21 (5) (2003) 1009–1019. G. Choudhury, Steady state analysis of a M/G/1 queue with linear retrial policy and two-phase service under Bernoulli vacation schedule, Applied Mathematical Modelling 32 (12) (2008) 2480–2489. G. Choudhury, K.C. Madan, A two stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy, Mathematical and Computer Modelling 42 (2005) 71–85. I. Dimitriou, C. Langaris, Analysis of a retrial queue with two-phase service and server vacations, Queueing Systems 60 (1–2) (2008) 111–129. B.T. Doshi, Analysis of a two-phase queueing system with general service times, Operation Research Letters 10 (1991) 265–272. G.I. Falin, J.G.C. Templeton, Retrial Queues, Chapman and Hall, London, 1997. G.I. Falin, On a multiclass batch arrival retrial queue, Advances in Applied Probability 20 (1988) 483–487. T. Katayama, K. Kobayashi, Sojourn time analysis of a queueing system with two-phase service and server vacations, Naval Research Logistics 54 (1) (2006) 59–65. J.-C. Ke, An M[x]/G/1 system with startup server and J additional options for service, Applied Mathematical Modelling 32 (4) (2008) 443–458. C.M. Krishna, Y.H. Lee, A study of a two-phase service, Operation Research Letters 9 (1990) 91–97. V.G. Kulkarni, H.M. Liang, Retrial queues revisited, in: J.H. Dshalalow (Ed.), Frontiers in Queueing, CRP Press, Boca Raton, 1997, pp. 19–34. A. Krishna Kumar, A. Vijayakumar, D. Arivudainambi, An M/G/1 retrial queueing system with two-phases of service and preemptive resume, Annals of Operation Research 113 (2002) 61–79. C. Langaris, A. Katsaros, Time depended analysis of a queue with batch arrivals and N levels of non preemptive priority, Queueing Systems 19 (1995) 269–288. K.C. Madan, On a single server queue with two stage heterogeneous service and deterministic server vacations, International Journal of Systems Science 32 (7) (2001) 837–844. E. Moutzoukis, C. Langaris, Non-preemptive priorities and vacations in a multiclass retrial queueing system, Stochastic Models 12 (3) (1996) 455–472. L. Takacs, Introduction to the Theory of Queues, Oxford University Press, New York, 1962.
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
A queueing system with n-phases of service and ðn 1Þ-types of retrial customers Christos Langaris, Ioannis Dimitriou * Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
a r t i c l e
i n f o
Article history: Received 18 June 2009 Accepted 21 January 2010 Available online 25 January 2010 Keywords: Poisson arrivals n-Phase service Retrial queues General services Single vacation
a b s t r a c t A queueing system with a single server providing n-phases of service in succession is considered. Every customer receives service in all phases. Arriving customers join a single ordinary queue, waiting to start the service procedure. When a customer completes his service in the ith phase he decides either to proceed to the next phase or to join the K i retrial box ði ¼ 1; 2; . . . ; n 1Þ, from where he repeats the demand for the ði þ 1Þth phase of service after a random amount of time and independently to the other customers in the system. Every customer can join during his service procedure a number of retrial boxes before departs from the system. When at the moment that a customer, either departs from the system or joins a retrial box and so releases the server, there are no other customers waiting in the ordinary queue, then the server departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service times are arbitrarily distributed. For such a system, the mean number of customers in the ordinary queue and in each retrial box separately are obtained, and used to investigate numerically system performance. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Queueing systems in which the server provides to each customer a number of phases of heterogeneous service in succession, can be proved very useful to model computer networks, production lines and telecommunication systems where messages are processed in n stages by a single server. Such kind of systems, with only two-phases of service, have firstly been discussed by Krishna and Lee [12] and Doshi [7], while more recently in a series of works (Madan [16], Choi and Kim [3], Choudhury and Madan [5], Katayama and Kobayashi [10], Ke [11]), the previous results, are extended to include systems allowing server vacations, Bernoulli feedback, N-policy, exhaustive or gated bulk service, start-up times etc., but again for models with only two-phases of service. Moreover, in all above mentioned papers one can find important applications to computer communication, production and manufacturing systems, central processor units and multimedia communications. Krishna Kumar et al. [14] and Choudhury [4] are the first who imposed the concept of ‘‘retrial customers” in the two-phase service models. Retrial queueing systems are characterized by the fact that an arriving customer who finds the server unavailable does not wait in a queue but instead he leaves the system, joining the so called retrial box, from where he repeats the demand for service later. Practical use of retrial queueing systems arises in telephone-switching systems and in telecommunication and computer networks. For complete surveys of past papers on such kind of models see Falin and Templeton [8], Artalejo and Gomez-Corral [2], Kulkarni and Liang [13] and Artalejo [1]. Krishna Kumar et al. [14] considered a two-phase service system where an arriving customer who finds the server unavailable joins the retrial box from where only the first customer can retry for service after an arbitrarily distributed time period while in the work of Choudhury [4] the investigated two-phase model includes Bernoulli server vacations and linear retrial policy. We have to observe here that in both papers the service procedure contains only two-phases of service, there is no any ordinary queue and all ‘‘waiting” customers are placed in the retrial box. Dimitriou and Langaris [6] considered a two-phase service model with both an ordinary queue and a retrial box receiving all customers who, upon finishing the first phase service, do not proceed to the second immediately but instead they decide to retry for the second phase later. In the present work, we generalize the work of Dimitriou and Langaris [6] by considering, for the first time in the literature, a model with n phases of service and n 1 retrial boxes, K 1 ; K 2 ; . . . ; K n1 say. All arriving customers are placed, upon arrival, in an ordinary queue (first phase) to receive service. When a customer completes his first phase service then, with probability 1 p1 , he proceeds to the second * Corresponding author. Tel.: +30 2651098242. E-mail addresses: [email protected] (C. Langaris), [email protected] (I. Dimitriou). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.01.034
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phase, while with probability p1 , he leaves the system and joins the K 1 retrial box. This procedure is repeated in each phase and so, when a customer completes his ith phase service, then either, with probability 1 pi , he proceeds to the ði þ 1Þth phase, or with probability pi he joins the K i retrial box in which case the server starts serving the next ordinary customer (if any) in the first phase. The customers in each retrial box retry, after a random amount of time and independently to each other, to find the server available and to proceed to the next phase of their service. Note here that every customer has to receive service in all phases and so he may join more than one retrial boxes during his service procedure. Obviously the customers in the K i retrial box are originated, either by ordinary customers, or by customers from K 1 ; K 2 ; . . . ; K i1 retrial boxes, that do not succeed to proceed directly to the ði þ 1Þth phase of service. When, at the moment of a customer’s departure from the system or his entrance in a retrial box, there are no more customers waiting in the ordinary queue, then the server departs for a single vacation (update devices, maintenance, etc.) of arbitrarily distributed length. In our model, and at any time, an ordinary and n 1 retrial queues must be taken into account. Due to the n-dimensional process that arise, the analysis of the present model cannot be handled by the methodology used to analyse the two-phase model in Dimitriou and Langaris [6], as here in case of a general n, the supplementary variable technique lead us to a ðn 1Þ-dimensional partial differential equation (Eq. (24) in Section 4) that hardly can be solved. Thus, our task now is the investigation of the mean number of customers in the ordinary queue and in each retrial box, by using a special methodology used firstly in Falin [9]. We have to point out here that this is not the first time in the literature the authors of the present work are engaged in the study of multiclass queueing systems. Thus, in Langaris and Katsaros [15], a system consisting of a single queue accepting n classes of ordinary customers, with nonpreemptive priorities between them and no retrial customers at all, is considered, while in Moutzoukis and Langaris [17], the above model is extended to contain a retrial box too, accepting a number of independent classes of customers, that retry to receive only a single service and to depart from the system. Although some theoretical results in [15,17], can also be used, after suitable modifications, in the present work, it is clear from the description above that these models, and the corresponding real life applications, are completely different from the present, new in the literature, model of n phases of heterogeneous service in succession, of n 1 retrial boxes, and of n 1 classes of retrial customers that change a number of times their class, jumping from one retrial box to another, after receiving a part of their service. As we mentioned before, the present model can only be considered as a thorough generalization of the model in Dimitriou and Langaris [6]. Our system can be used to model any situation with many stages of service, where in each stage a control and a separation of the serviced units must be taking place, and if a unit satisfies some quality standards then it proceeds immediately to the next phase of service, while, if the quality of the unit is poor, then it is removed from the system and repeats its attempt to continue its service procedure later when the server is free from high quality units. For example, consider a manufacturing procedure consisting of n consecutive production levels. Any single item must be passed from and inspected in all these production levels to achieve high quality standards. Thus, if upon the end of the ith stage processing, the quality of the item is satisfactory, the item proceeds immediately to the ði þ 1Þth stage, while if it is not, the item is sent in the ith retrial box and it is characterized as an ði þ 1Þ-defective item. In the later case, the server starts processing a new item (if any) in the first production level. All ði þ 1Þ-defective items ‘‘seek” to continue their service procedure from the level of interruption (ði þ 1Þth stage) by receiving, in each of the following stages, special treatments that improve again their quality to the high standards. Thus, the processing of an ði þ 1Þ-defective item in levels ði þ 1Þ, ði þ 2Þ, etc. must be different (includes repairs for example) from the processing of an initial non-defective item in the same levels. Whenever an item, either completes the total service procedure or failed to pass an inspection and so it releases the server, and there are no new items waiting in the first stage, the server activates a maintenance of the system (vacation). Further applications of the presented model often arise in packet transmissions, in central processors, in multimedia communications etc. The article is organized as follows. A full description of the model is given in Section 2. Some, very useful for the analysis results, on the customer total service cycle and server busy and vacation periods, are given in Section 3, while a system states analysis is performed in Section 4. In Section 5 the mean number of customers in the ordinary queue and in each retrial box separately are obtained, and used to produce, in Section 6, numerical results and to compare numerically system performance under various changes of the parameters. 2. The model Consider a queueing system consisting of n phases of service and a single server who follows the customer in service when he passes from one phase to the next. Customers arrive to the system according to a Poisson process with parameter k, and are placed in a single queue (ordinary queue) waiting to start their service in first phase. When a customer finishes his service in the ith phase ði ¼ 1; 2; . . . ; n 1Þ then he either goes to the ði þ 1Þth phase with probability 1 pi , or he, with probability pi , joins the K th i retrial box from where he retries, independently to the other customers in the box, after an exponential time parameter liþ1 , to find the server idle and to proceed to the ði þ 1Þth phase of service. In case the customer chooses to join the retrial box, the server starts immediately to serve in the first phase the next customer (if any) in the queue. When a customer finishes the nth phase of service, he departs from the system. Every time the server becomes free (that is whenever the customer in service either finishes his nth phase service or joins a retrial box, and no other customers are waiting in the ordinary queue), he departs for a single vacation U 0 the length of which is arbitrarily distributed 0 and second moment about zero b ð2Þ . If with distribution function (D.F.) B0 ðxÞ, probability density function (p.d.f.) b0 ðxÞ, finite mean value b 0 the server, upon returning from a vacation, finds customers waiting for service in the first phase (ordinary queue) he starts serving them immediately, while if there are no customers waiting, he remains idle awaiting the first arrival, either from outside or from a retrial box, to start the service procedure again. From the above description, it is clear that the server’s idle period starts only when the server, upon returning from the vacation, finds no customers waiting in the ordinary queue. Moreover any retrial customer can find a position to continue his service procedure, only when the server is in the idle mode. Let us denote by P 1 , the customers who are waiting in the ordinary queue and by Pi ði ¼ 2; 3; . . . ; nÞ; those who joined the K th i1 retrial box. Any customer has to receive service in all phases before departs from the system and so he can join, during his service procedure, a number of retrial boxes. Therefore, any arriving customer may change type (class) several times during his service procedure. Thus, a Pi customer remains a Pi customer as far as he, upon emerging from the K th i1 retrial box, continues his service procedure, passing from one
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stage to the next without joining another retrial box, while if a Pi customer joins in the sequel the K th j1 retrial box ðj > iÞ then he becomes a P j customer. The service time of a P i customer in the jth phase, Bij say, must be different (contains special treatments) from the service time of a Pk ðk – iÞ customer in the same phase. Thus, it is assumed that Bij is arbitrarily distributed with D.F. Bij ðxÞ, p.d.f. bij ðxÞ , finite mean value and second moment about zero b ð2Þ (B ðxÞ; b ðxÞ; b ;b ð2Þ do not exist of course for j < i). Finally all random variables defined above b ij ij ij ij ij ij are assumed to be independent. Note finally that our model can also be described as a system of n-phases of service and a single retrial box that accepts now n 1 types Pi ; i ¼ 2; . . . ; n, say, of customers. The type of each customer in the retrial box characterizes the phase of service that he is due to receive next. Therefore, a Pi customer, upon emerging from the retrial box, will continue his service in the ith phase and if he, in the sequel, fails to proceed to the jth phase of service ðj > iÞ, then he returns into retrial box but now as a Pj customer. Thus, is the first time in the literature of the multiclass retrial queues, where every customer may join several times the retrial box, changing each time his type (class), before departs from the system. 3. General results 0 ¼ If a customer does not join any retrial box during his service procedure (with probability p P will be R0 ¼ nj¼1 B1j with Laplace-Stieltges Transform (LST) of its p.d.f.
r0 ðsÞ ¼
n Y
Qn1 i¼1
ð1 pi Þ) then his total service cycle
b1j ðsÞ;
j¼1
where bij ðÞ the LST of bij ðÞ. Let us suppose now that an arriving customer joins r retrial boxes ðr ¼ 1; 2; . . . ; n 1Þ during his service procedure, for example he joins the retrial boxes K m1 ; K m2 ; . . . ; K mr . Then it is clear that m1 ¼ 1; 2; . . . ; n r; m2 ¼ m1 þ 1; . . . ; n r þ 1; and so on, until mr ¼ mr1 þ 1; . . . ; n 1. Moreover the probability of this event is n1 Y
m1 m2 ...mr ¼ pm1 pm2 . . . pmr p
ð1 pi Þ;
i¼1 i–m1 ;...;mr
while the duration of the customer’s total service cycle in this case is (with m0 0)
Rm1 m2 ...mr ¼
!
r X
mi X
i¼1
j¼mi1 þ1
Bmi1 þ1j þ V
þ
n X
Bmr þ1j ;
j¼mr þ1
and the LST of its p.d.f.
rm1 m2 ...mr ðsÞ ¼ ðv ðsÞÞr
mi Y
rþ1 Y
bmi1 þ1j ðsÞ;
i¼1 j¼mi1 þ1
with mrþ1 n. Note here that V is the delay incurrent due to server absence (in vacations) that precedes the service of every customer emerging from a retrial box. This absence can be either of a single duration U 0 , if no customers arrive from outside during the vacation U 0 , or of a multiple duration, if at least one customer arrives during U 0 , in which case the server has to repeat the vacation as soon as he finishes the busy period of P1 customers, and before he becomes available to the customer emerging from the retrial box. Thus the p.d.f. v ðtÞ of V satisfies
v ðtÞ ¼ ekt b0 ðtÞ þ ð1 ekt Þb0 ðtÞ v ðtÞ; with LST
v ðsÞ ¼
1þ
b0 ðk þ sÞ b0 ðk þ sÞ
b0 ðsÞ
:
Thus the LST of the p.d.f. of the customer’s total service cycle R is given by
rðsÞ ¼ p 0r 0 ðsÞ þ
n1 X nr X
nrþ1 X
r¼1 m1 ¼1 m2 ¼m1 þ1
n1 X
...
m1 m2 ...mr r m1 m2 ...mr ðsÞ; p
mr ¼mr1 þ1
and if we take derivatives above, at s ¼ 0, we arrive after some algebra at
q k
n X d r ðsÞ ¼ kEðRÞ ¼ ðqj þ q0j Þ; ds s¼0 j¼1
ð1Þ
with q01 0,
0 kb p ; j ¼ 2; 3; . . . ; n; b0 ðkÞ j1 " # n k1 X Y qj ¼ kpj1 bjj þ bjk ð1 pm Þ ;
q0j ¼
k¼jþ1
ð2Þ j ¼ 1; 2; . . . ; n:
m¼j
and p0 1. Thus q must be considered as the mean number of new customers arriving during R and so, for an ergodic system, we have to assume q < 1.
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641
Let now Sj be the time interval from the epoch at which a P j customer starts his service in the jth phase until the epoch he either completes his service procedure and depart from the system or he joins another retrial box and releases the server. Let also N i ðSj Þ be the new Pi customers during Sj . Note here that during Sj we can have only new P1 customers (external arrivals) and/or one new Pjþ1 or P jþ2 or . . .or P n customer according to the retrial box that this specific Pj customer will join next (if any), while when j ¼ n we can have only new P1 customers during Sn . Define finally
aðjÞ ðt; k1 ; kjþ1 ; . . . ; kn Þdt ¼ P½Ni ðSj Þ ¼ ki i ¼ 1; j þ 1; . . . ; n; t < Sj 6 t þ dt;
aj ðz1 ; zjþ1 ; . . . ; zn Þ ¼
1 1 X X
...
k1 ¼0 kjþ1 ¼0
1 X
k
jþ1 zk11 zjþ1 . . . zknn
kn ¼0
Z
1
aðjÞ ðt; k1 ; kjþ1 ; . . . ; kn Þdt:
t¼0
Then it is easy to understand that, for any j ¼ 1; 2; . . . ; n
aj ðz1 ; zjþ1 ; . . . ; zn Þ ¼
n X
pm zmþ1
m¼j
m1 Y
ð1 pl Þ
l¼j
m Y
bjk ðk kz1 Þ;
ð3Þ
k¼j
with pn 1; znþ1 1 and in general for any Q l ;
Qi
l¼j Q l
¼ 1 if i < j. Moreover from relation (2)
d a1 ðz1 ; 1; 1; . . . ; 1Þjz1 ¼1 ¼ q1 : dz1 the ð1 nÞ In general and for any xj j ¼ 1; 2; . . . ; n; let as denote, for simplicity, by x the ð1 n 1Þ vector x ¼ ðx2 ; x3 ; . . . ; xn Þ and by x ¼ ðx1 ; x2 ; . . . ; xn Þ. To proceed further we need the following Lemma the proof of which is a simple application of the well known vector x result of Takacs [18] (Lemma 1, p. 47). Lemma 1. If (i) jzk j < 1 for at least one k ¼ 2; . . . ; n, and jzm j 6 1 for all other m ¼ 2; 3; . . . ; n with m – k, or (ii) jzm j 6 1; for all m ¼ 2; 3; . . . ; n and q1 > 1, then the relation
z1 a1 ðz1 ; z2 ; . . . ; zn Þ;
ð4Þ
has one and only one zero, z1 ¼ xðzÞ say, inside the region jz1 j < 1. Specifically for z ¼ 1; xð1Þ is the smallest positive real root of (4) with xð1Þ < 1 if q1 > 1 and xð1Þ ¼ 1 for q1 6 1. Proof. See Appendix A. h Let now T ðiÞ be the duration of a busy period of P 1 customers starting with i P 1 customers and N j ðT ðiÞ Þ be the number of new P j customers ðiÞ (joining the K th j1 retrial box) during T . Define ðiÞ
g k ðtÞdt ¼ P½N j ðT ðiÞ Þ ¼ kj j ¼ 2; 3; . . . ; n; t < T ðiÞ 6 t þ dt; Z 1 1 X ðiÞ GðiÞ ðs; zÞ ¼ zk est g k ðtÞdt: k¼0
t¼0
Now one can show, by following the analysis in Takacs [18, pp. 60–63], or in Langaris and Katsaros [15] (Theorem 4), that Lemma 2. For all jzm j 6 1; m ¼ 2; 3; . . . ; n;
GðiÞ ð0; zÞ ¼ xi ðzÞ; where xðzÞ the only zero of z1 a1 ðz1 ; z2 ; . . . ; zn Þ in jz1 j < 1. Proof. Let us suppose that the busy period of P 1 customers starts with only one P1 customer. By observing in the usual way (see Takacs [18, pp. 60–63]), that the duration of the busy period and the number of customers joining the retrial boxes during it, do not affected from the order in which the P 1 customers are selected for service, we can write easily ð1Þ
g k ðtÞ ¼
n X
ekt hi ðtÞdfk¼1i g þ
i¼1
n X 1 X i¼1
X
j¼1 m1 þm2 þ...mj ¼k1i
Z 0
t
h i eku ðkuÞj ð1Þ ð1Þ hi ðuÞ g ð1Þ m1 ðt uÞ g m2 ðt uÞ g mj ðt uÞ du j!
where 1i is the ð1 n 1Þ vector with 1 in the ith position and 0 in all other entries, 1n 0, and hi ðuÞ is the total duration of the consecutive service times of a P1 customer until the epoch he joins the ith retrial box, i ¼ 1; 2; . . . ; n; where i ¼ n means that the customer ends his service procedure and departs without joining any retrial box, i.e. for i ¼ 1; 2; . . . ; n
hi ðuÞ ¼ pi
i1 Y
ð1 pl Þb11 ðuÞ b12 ðuÞ b1i ðuÞ
l¼1
with pn 1 and LST
hi ðsÞ ¼ pi
i1 Y l¼1
ð1 pl Þ
i Y k¼1
b1k ðsÞ:
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Now it is clear that
Gð1Þ ðs; zÞ ¼
1 Z X 0
j¼0
¼
Z 1 n n X ð1Þ eðkþsÞu ðkuÞj X ziþ1 hi ðuÞ½Gð1Þ ðs; zÞj du ¼ eðsþkkG ðs;zÞÞu ziþ1 hi ðuÞdu j! 0 i¼1 i¼1
1
n X
pi ziþ1
i¼1
i1 Y
ð1 pl Þ
l¼1
i Y
b1k ðs þ k kGð1Þ ðs; zÞÞ
k¼1
and using relation (3) with s ¼ 0 we arrive at
Gð1Þ ð0; zÞ ¼ a1 ðGð1Þ ð0; zÞ; z2 ; . . . ; zn Þ and from Lemma 1 Gð1Þ ð0; zÞ ¼ xðzÞ. Now it is easy to understand that if the busy period starts with i P1 customers, then it can be considered as the sum of i busy periods each of which starts with only one P 1 customer, and the lemma has been proved. Let V be the time interval from the epoch the server departs for a single vacation until the epoch he becomes idle for the first time. Denote also N j ðVÞ the number of new Pj customers during V. If we define
v k ðtÞdt ¼ P½Nj ðVÞ ¼ kj v ðs; zÞ ¼
1 X
zk
Z
1
j ¼ 2; 3; . . . ; n; t < V 6 t þ dt;
est v k ðtÞdt;
t¼0
k¼0
then
v k ðtÞ ¼ ekt b0 ðtÞdfk¼0g þ
k X 1 X ðktÞi kt e b0 ðtÞ g ðiÞ m ðtÞ v km ðtÞ; i! m¼0 i¼1
where
dfAg ¼
1 if A holds; 0 otherwise;
and so
v ð0; zÞ ¼
1þ
b0 ðkÞ
b0 ðkÞ : b0 ðk kxðzÞÞ
ð5Þ
Let DðiÞ be the time interval from the epoch a P i i ¼ 2; 3; . . . ; n retrial customer finds a position for service, until the epoch the server departs for a vacation, and denote by N j ðDðiÞ Þ, the number of the new P j customers during DðiÞ . Define finally ðiÞ
dk ðtÞdt ¼ P½Nj ðDðiÞ Þ ¼ kj j ¼ 2; 3; . . . ; n; t < DðiÞ 6 t þ dt; Z 1 1 X ðiÞ zk est dk ðtÞdt; DðiÞ ðs; zÞ ¼ k¼0
t¼0
then ðiÞ
dk ðtÞ ¼ ekt
n X
dfk¼1rþ1 g sir ðtÞ þ
r¼i
1 n X ðktÞm kt X ðmÞ sir ðtÞ g k1rþ1 ðtÞ; e m! m¼1 r¼i
ð6Þ
where 1j ¼ ð0; . . . ; 0; 1; 0; . . . ; 0Þ with the 1 in the jth position, 1nþ1 0; and
sir ðtÞ ¼ ð1 pi Þbii ðtÞ ð1 pr1 Þbir1 ðtÞ pr bir ðtÞ;
r ¼ i; i þ 1; . . . ; n;
is in fact the total time the P i retrial customer holds the server from the epoch he finds a position for service until the epoch he joins the K th r retrial box and becomes a Prþ1 customer or departs from the system (case r ¼ n). By taking LST in (6) above we arrive at
DðiÞ ð0; zÞ ¼ ai ðxðzÞ; ziþ1 ; . . . ; zn Þ;
ð7Þ
where the function ai ðxðzÞ; ziþ1 ; . . . ; zn Þ has been defined in (3). Define finally C ðiÞ as the time interval from the epoch at which a P i customer finds a position for service until the epoch the server becomes for the first time idle and ready to accept the next customer from outside or from a retrial box. If N j ðC ðiÞ Þ is the number of new P j customers during C ðiÞ and define ðiÞ
ck ðtÞdt ¼ P½Nj ðC ðiÞ Þ ¼ kj j ¼ 2; 3; . . . ; n; t < C ðiÞ 6 t þ dt; Z 1 1 X ðiÞ C ðiÞ ðs; zÞ ¼ zk est ck ðtÞdt; k¼0
t¼0
then it is easy to realize that C ðiÞ ¼ DðiÞ þ V and from (7)
C ðiÞ ð0; zÞ ¼ ai ðxðzÞ; ziþ1 ; . . . ; zn Þv ð0; zÞ:
ð8Þ
C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
643
We have to state here the following theorem. The proof is similar to the proof of Theorem 3.2 in Moutzoukis and Langaris [17]. h Theorem 3. For any permutation ði2 ; i3 ; . . . ; in Þ of the set ð2; 3; . . . ; nÞ and for j ¼ 2; 3; . . . ; n; if (a) jzim j < 1 for at least one m ¼ j þ 1; . . . ; n, and ij1 > 1ðq i1 q1 Þ, or (c) jzir j 6 1, for all jzir j 6 1, for all other r ¼ j þ 1; . . . ; n with r – m, or (b) jzir j 6 1, for all r ¼ j þ 1; . . . ; n, and q ij1 , the equation ij > 1 P q r ¼ j þ 1; . . . ; n, and q
zij C ðij Þ ð0; wij1 ðzij ; zijþ1 ; . . . ; zin ÞÞ ¼ 0;
ð9Þ
has one and only one root, zij ¼ xij ðzijþ1 ; . . . ; zin Þ; j – n; zin ¼ xin say, inside the region jzij j < 1, where the vector wij ðzijþ1 ; zijþ2 ; . . . ; zin Þ is defined by
wi1 ðzi2 ; zi3 ; . . . ; zin Þ ¼ ðz2 ; z3 ; . . . ; zn Þ; wi2 ðzi3 ; zi4 ; . . . ; zin Þ ¼ wi1 ðxi2 ðzi3 ; . . . ; zin Þ; zi3 ; . . . ; zin Þ; wik ðzikþ1 ; zikþ2 ; . . . ; zin Þ ¼ wik1 ðxik ðzikþ1 ; . . . ; zin Þ; zikþ1 ; . . . ; zin Þ;
k ¼ 2; . . . ; n 1;
while for j ¼ 2; 3; . . . ; n
q ij ¼
@ ðij Þ C ð0; wij1 ðzij ; zijþ1 ; . . . ; zin ÞÞjzi ¼zi ¼¼zi ¼1 : n j jþ1 @zij
ð10Þ
ij1 6 1 the root xij ð1; . . . ; 1Þ is the smallest positive real root of (9) with xij ð1; . . . ; 1Þ < 1 if Moreover for real zir ¼ 1; r ¼ j þ 1; . . . ; n, and q
q ij > 1 and xij ð1; . . . ; 1Þ ¼ 1 for q ij 6 1. Proof. See Appendix A. h in in (10) is given by, One can show here that, for any permutation ði2 ; i3 ; . . . ; in Þ of the set ð2; 3; . . . ; nÞ, the final term q
P
q in ¼
qin þ q0in þ dfin
ð11Þ
in (and all other q ij ) is always less than one. and so it is clear, by comparing relations (1) and (11) that, for q < 1; q 4. Steady states analysis Let us assume that a state of statistical equilibrium exists and let N 1 ; N i ; i ¼ 2; . . . ; n denote the number of P1 ; P i , customers in the ordinary queue (not in service) and in the K th i1 retrial box, respectively. Let also
8 if server on vacation; > <0 n ¼ ði; jÞ if server busy on jth phase with P i customer; > : id if server idle;
and
qðkÞ ¼ Pðn ¼ id; N 1 ¼ 0; Nm ¼ km ; m ¼ 2; 3; . . . ; nÞ; xÞdx ¼ Pðn ¼ 0; N m ¼ km ; m ¼ 1; 2; . . . ; n; x < U 0 6 x þ dxÞ; p0 ðk; xÞdx ¼ Pðn ¼ ði; jÞ; N m ¼ km ; m ¼ 1; 2; . . . ; n; x < U ij 6 x þ dxÞ; p ðk; ij
¼ ðk1 ; k2 ; . . . ; kn Þ ðk1 ; kÞ, and U ; U 0 the elapsed service or vacation time, respectively. If where, as it is stated before, k ¼ ðk2 ; . . . ; kn Þ; k ij finally
Q ðzÞ ¼
X
qðkÞzk
kP0
P0 ðz; xÞ ¼
X
X
...
k2 P0
X
qðk2 ; . . . ; kn Þzk22 zk33 . . . zknn ;
kn P0
xÞzk ; p0 ðk;
0 kP
Pij ðz; xÞ ¼
X
xÞzk ; pij ðk;
i; j ¼ 1; 2;
0 kP
then we arrive easily, for x > 0, at
P0 ðz; xÞ ¼ P0 ðz; 0Þð1 B0 ðxÞÞ exp½ðk kz1 Þx; Pij ðz; xÞ ¼ Pij ðz; 0Þð1 Bij ðxÞÞ exp½ðk kz1 Þx;
ð12Þ
and n X j¼2
lj zj
@ Q ðzÞ þ kQ ðzÞ ¼ P0 ðð0; zÞ; 0Þb0 ðkÞ; @zj
with b0 ð:Þ the LST of b0 ð:Þ. For the boundary conditions ðx ¼ 0Þ we obtain in a similar way
ð13Þ
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C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
P0 ðð0; zÞ; 0Þ ¼
n1 X
pm zmþ1
m¼1
m X
Pim ðð0; zÞ; 0Þbim ðkÞ þ
i¼1
n X
Pin ðð0; zÞ; 0Þbin ðkÞ;
ð14Þ
i¼1
@ Q ðzÞ; j ¼ 2; . . . ; n; @zj j1 Y ð1 pm Þb1m ðk kz1 Þ; P1j ðz; 0Þ ¼ P11 ðz; 0Þ
Pjj ðð0; zÞ; 0Þ ¼ lj
j ¼ 2; . . . ; n;
m¼1
Pij ðz; 0Þ ¼ li
j1 Y @ Q ðzÞ ð1 pm Þbim ðk kz1 Þ; @zi m¼i
i ¼ 2; . . . ; n j ¼ i þ 1; . . . ; n;
ð15Þ
while for the P11 ðz; 0Þ we obtain using relations (14) and (15)
( P11 ðz; 0Þ ¼
kz1 QðzÞ þ
n X
), aj ðz1 ; zjþ1 ; . . . ; zn ÞP jj ðð0; zÞ; 0Þ P0 ðð0; zÞ; 0Þ½1 þ b0 ðkÞ b0 ðk kz1 Þ
½z1 a1 ðzÞ:
ð16Þ
j¼2
Replacing now in the numerator of (16) the zero xðzÞ of the denominator (see Lemma 1), we arrive at
P0 ðð0; zÞ; 0Þ ¼
P kxðzÞQðzÞ þ nj¼2 aj ðxðzÞ; zjþ1 ; . . . ; zn ÞPjj ðð0; zÞ; 0Þ ; 1 þ b0 ðkÞ b0 ðk kxðzÞÞ
ð17Þ
and substituting back from (14)–(17) in (12) and (13) and integrating with respect to x we obtain for j ¼ 2; . . . ; n
e1j ðz1 ÞP1j ðzÞ ¼ e11 ðz1 ÞP11 ðzÞ
j1 Y
ð1 pm Þb1m ðk kz1 Þ;
ð18Þ
m¼1
ejj ðz1 ÞPjj ðzÞ ¼ lj
@ Q ðzÞ; @zj
ð19Þ
j1 Y i ¼ 2; . . . ; n @ Q ðzÞ ð1 pm Þbim ðk kz1 Þ; @zi j ¼ i þ 1; . . . ; n; m¼i Pn kxðz2 ; . . . ; zn ÞQ ðzÞ þ j¼2 ejj ðz1 Þaj ðxðz2 ; . . . ; zn Þ; zjþ1 ; . . . ; zn ÞPjj ðzÞ ; e0 ðz1 ÞP0 ðzÞ ¼ 1 þ b0 ðkÞ b0 ðk kxðzÞÞ Xn e ðz1 Þ½aj ðz1 ; zjþ1 ; . . . ; zn Þ aj ðxðzÞ; zjþ1 ; . . . ; zn ÞPjj ðzÞ e11 ðz1 ÞP11 ðzÞ ¼ fk½z1 xðzÞQ ðzÞ þ j¼2 jj
eij ðz1 ÞPij ðzÞ ¼ li
þ e0 ðz1 Þ½b0 ðk kz1 Þ b0 ðk kxðzÞÞP0 ðzÞg=½z1 a1 ðzÞ; n X
lj zj
j¼2
@ Q ðzÞ þ kQ ðzÞ ¼ e0 ðz1 ÞP0 ðzÞb0 ðkÞ; @zj
where in general P ðzÞ ¼
eij ðz1 Þ ¼
R
ð20Þ ð21Þ
ð22Þ ð23Þ
P ðz; xÞdx and
k kz1 ; 1 bij ðk kz1 Þ
1 eij ð1Þ ¼ : bij
By substituting finally from (21) to (23) and using relations (5), (8), (19), we arrive easily at the ðn 1Þ-dimensional partial differential equation n X
lj ½zj C ðjÞ ð0; zÞ
j¼2
@ Q ðzÞ þ k½1 xðzÞv ð0; zÞQ ðzÞ ¼ 0 @zj
ð24Þ
Now to obtain, from (18)–(22) the generating functions, one has to solve first Eq. (24) to obtain Q ðzÞ, and it is easy to understand that Eq. (24) hardly can be solved. In the sequel we will use the relations obtained so far to investigate, using the methodology in Falin [9], the mean number of customers in the ordinary queue and in each retrial box separately. 5. Mean number of ordinary and retrial customers To start the analysis we need to calculate the generating functions at the point z ¼ 1: are given by z¼1 Theorem 4. For q < 1 the generating functions Pij ð:Þ; P 0 ð:Þ; Q ð:Þ at the point
11 ; ¼ kb P11 ð1Þ ¼ kp b Pij ð1Þ i1 ij
¼ kp b Pjj ð1Þ j1 jj ; j1 Y m¼i
ð1 pm Þ;
i ¼ 1; . . . ; n j ¼ i þ 1; . . . ; n
1q 0 =b ðkÞ ; 1 þ kb " 0 # n 0 X k b ¼ p þ Qð1Þ : P0 ð1Þ b0 ðkÞ j¼2 j1
Q ð1Þ ¼
j ¼ 2; . . . ; n
ð25Þ
C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
645
Proof. Let us define
N ðzÞ ¼ Q ðzÞ þ P 0 ðzÞ þ
n X n X i¼1
Pij ðzÞ;
ð26Þ
j¼i
then from relations (18)–(20)
Pij ðzÞ ¼
j1 Y i ¼ 1; 2; . . . ; n eii ðz1 Þ Pii ðzÞ ð1 pm Þbim ðk kz1 Þ; eij ðz1 Þ j ¼ i þ 1; . . . ; n m¼i
i.e. j1 Y ¼ P ii ð1Þ bij Pij ð1Þ ð1 pm Þ; b ii m¼i
i ¼ 1; 2; . . . ; n
ð27Þ
j ¼ i þ 1; . . . ; n
¼ 1 we obtain after manipulations and substituting back in (26), using (21) and observing that N ð1Þ
! " # n n k1 0 0 X Y X b kb Pjj ð1Þ Q ð1Þ þ ¼ 1: b b P ð 1Þ þ 1 þ þ ð1 p Þ þ jj jk m jj 11 11 b0 ðkÞ b0 ðkÞ b kb j¼2 m¼j k¼jþ1
q1
ð28Þ
we arrive at Using now (21) in (22) and putting z¼1
P11 ð1Þ k ¼ 1 q1 b11
( 1þ
! " #) n n k1 0 0 X X Y kb Pjj ð1Þ b þ Q ð1Þ þ ; ð1 p Þ þ b b jj jk m b0 ðkÞ b0 ðkÞ b jj j¼2 m¼j k¼jþ1
ð29Þ
where q1 has been defined in (2), and so from (27)–(29) above we conclude
; ¼ kb P11 ð1Þ 11
¼ kb P1j ð1Þ 1j
j1 Y
ð1 pm Þ;
j ¼ 2; 3; . . . ; n:
ð30Þ
m¼1
Multiplying relation (19) by zj , adding for all j, subtracting from (23) and using (21) we arrive at n X ejj ðz1 ÞPjj ðzÞ ½Dj ðzÞ zj ¼ k½1 xðzÞf ðxðzÞÞ; Q ðzÞ j¼2
ð31Þ
with
f ðxðzÞÞ ¼
b0 ðkÞ ; 1 þ b0 ðkÞ b0 ðk kxðzÞÞ
ð32Þ
Dj ðzÞ ¼ f ðxðzÞÞaj ðxðzÞ; zjþ1 ; . . . ; zn Þ:
ð33Þ
Here the great importance of the analysis and the results obtained in Section 2 is apparent. By comparing relations (32) and (33) above with relations (5) and (8) in Section 2 it is clear that
Di ðzÞ C ðiÞ ð0; zÞ; and so for q < 1, the main Theorem 3 can be used directly in (31). Thus by putting in (31) z1 ¼ 1, and, for any permutation ði2 ; i3 ; . . . ; in Þ of ð2; 3; . . . ; nÞ; replacing in succession zi2 ; zi3 ; . . . ; zin1 by the corresponding zeros xi2 ðÞ; xi3 ðÞ; . . . ; xin1 ðÞ we succeed to eliminate all except the final term of the sum in the left hand side of (31). Thus the only unknown remaining in (31) is P in in ðÞ and this for all in ¼ 2; 3; . . . ; n; and so finally
¼ Pjj ð1Þ
! 0 kpj1 b kb jj Q ð1Þ; 1 þ 1 q b0 ðkÞ
j ¼ 2; 3; . . . ; n;
ð34Þ
and replacing in (28) we obtain after manipulations
Q ð1Þ ¼
1 q
0 1 þ bkbðkÞ
ð35Þ
;
0
which is the third of (25). in (27) and (21) we arrive at the second and forth of (25), From (35) and (34) we obtain the first of (25) and putting back Pjj ð1Þ respectively and the theorem has been proved. h For p0 1 and j ¼ 1; . . . ; n, let us define now
(
pj ¼ pj1
ð2Þ þ b jj
n X k¼jþ1
ð2Þ b jk
k1 Y m¼j
ð1 pm Þ þ 2
n1 Y r X r¼j m¼j
then, for the mean length of the ordinary queue,
" #) n k1 X Y ð1 pm Þbjr bjrþ1 þ ð1 pm Þ ; bjk k¼rþ2
m¼rþ1
ð36Þ
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C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
Theorem 5. The mean number of P1 customers, in the ordinary queue is given by,
( ) " # n n 0 X X k2 b EðN1 Þ ¼ p þ Qð1Þ þ pk ; 2ð1 q1 Þ b0 ðkÞ k¼2 k1 k¼1
where Q ð1Þ and
ð37Þ
pk are given by (35) and (36), respectively.
Proof. Differentiating relations (19)–(21), with respect to z1 , and setting zl ¼ 1; l ¼ 1; 2; . . . ; n, we arrive easily at,
ð2Þ k2 pi1 b @Pii ðzÞ ii ¼ ; @z1 z¼1 2 ð2Þ Y j1 j1 j1 X Y k2 pi1 b @Pij ðzÞ ij ; ¼ ð1 pm Þ þ k2 pi1 b ð1 pm Þ b ij im @z1 z¼1 2 m¼i m¼i m¼i " # n ð2Þ X @P0 ðzÞ k2 b ¼ 0 p þ Q ð1Þ : @z1 z¼1 2b0 ðkÞ k¼2 k1
i ¼ 2; 3; . . . ; n; j ¼ i þ 1; . . . ; n;
ð38Þ
In a similar way, from (22) and (18),
" # " # n n ð2Þ X ð2Þ X @P11 ðzÞ k3 b k2 b b 11 0 ¼ pk1 þ Q ð1Þ þ pk þ 11 ; @z1 z¼1 2ð1 q1 Þ b0 ðkÞ k¼2 2 k¼1 " # " # j1 ð2Þ n n 3 X Y X b @P1j ðzÞ k b 1j 0 ¼ ð1 pm Þ p þ Q ð1Þ þ pk @z1 z¼1 2ð1 q1 Þ m¼1 b0 ðkÞ k¼2 k1 k¼1 " # j1 j1 ð2Þ Y X þ b1j ; j ¼ 2; 3; . . . ; n: ð1 pm Þ b b þ k2 1j 1m 2 m¼1 m¼1
ð39Þ
Observing now that
EðN1 Þ ¼
n X n X @N ðzÞ @Pij ðzÞ @P0 ðzÞ ¼ þ ; @z1 z¼1 @z1 z¼1 @z1 z¼1 i¼1 j¼i
and replacing from (38) and (39) we obtain relation (37) and the theorem has been proved. h Before giving the mean queue lengths for the retrial customers we have to state some preliminary results. Let, for k; j ¼ 2; . . . ; n; m ¼ 1; 2, ðmÞ
hk
¼
m d xðzÞ ; m dzk z¼1
q^ ðmÞ jk ¼
m d aj ðxðzÞ; . . . ; zk ; . . . 1Þ m ; dzk z¼1
^ ðuÞ ^ ðuÞ where xðzÞ is defined in Lemma 1 (note that q jk – qkj ; k – j). Then after some algebra we obtain, ð1Þ
hk ¼ ð2Þ
k2 pk1 Y ð1 pm Þ; ð1 q1 Þ m¼1
ð1Þ k1 X k2 ðhk Þ2 p1 þ 2kðhkð1Þ Þ2 b1m ; ð1 q1 Þ m¼1 ! n r1 X Y ð1Þ ¼ kh ð1 p Þ þ d b þ b
hk ¼
q^ jkð1Þ
k
jj
jr
þ q^ jkð2Þ ¼ khð2Þ b jj k
m¼j
n X
r1 Y
r¼jþ1
b jr
fk>jg
m
r¼jþ1
"
! ð1 pm Þ þ
m¼j
# k2 Y pk1 ð1 pm Þ ; m¼j
ð1Þ ðkhk Þ2
pj1
pj þ dfk>jg 2khð1Þ k pk1
k2 Y m¼j
ð1 pm Þ
k1 X
b jr
k; j ¼ 2; . . . ; n:
r¼j
Moreover define
sk ¼
ð2Þ n h i X kQ ð1; . . . ; 1Þhk k2 pk1 P0 ð1; . . . ; 1Þ k ð2Þ þ 0 hð2Þ þ ðkhð1Þ Þ2 b ^ ð2Þ þ pj1 q þ kb Qk2 Qk2 Qk2 0 k k jk l ð1 q Þ 2pk1 m¼1 ð1 pm Þ 2pk1 b0 m¼1 ð1 pm Þ 2pk1 m¼1 ð1 pm Þ j¼2 1 k ! ð1Þ n 0 X khk b k2 b kp k 0 ð1Þ ; k ¼ 2; . . . ; n: þ q0j q^ ð1Þ ð1 q Þhk þ k1 þ jk þ ð1 q1 Þ j¼2 ð1 q1 Þb0 ðkÞ lk b0 ðkÞ
and denote
@2 Dkl ¼ Q ðzÞ @zk @zl
;
k; l ¼ 2; 3; . . . ; n:
z¼1
Note here that Dkl ¼ Dlk 8 k; l. Then we state the following theorem. Theorem 6. The quantities Dkl ; k; l ¼ 2; . . . ; n can be found as the solution of the system of linear equations,
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C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
ðlk þ ll ÞDkl ¼
n1 X
!
lj Dkj pl1
j¼2
þ pl1
l2 n1 k2 X Y Y ð1 pm Þ dfl>jg þ lj Dlj dfk>jg pk1 ð1 pm Þ m¼j
l2 Y
ð1 pm Þ
m¼1
" n X j¼2
j¼2
m¼j
#
!
"
#
k2 n X Y lj ðqj þ q0j Þ lj ðqj þ q0j Þ ð1 pm Þ Dkj þ sk þ pk1 Dlj þ sl : pj1 ð1 q1 Þ pj1 ð1 q1 Þ m¼1 j¼2
ð40Þ
Proof. Replacing (21) to (22) we arrive at
e0 ðz1 ÞP0 ðzÞ ¼
kz1 Q ðzÞ þ
Pn
z1 Þ
j¼2 ejj ðz1 Þaj ðz1 ; zjþ1 ; . . . ; zn ÞP jj ðz Þ þ e11 ðz1 ÞP 11 ðz Þða1 ðz Þ 1 þ b0 ðkÞ b0 ðk kz1 Þ
;
and replacing to (23) and setting z1 ¼ 1 we obtain n X
lj zj
j¼2
n X @ Pjj ð1; zÞ P11 ð1; zÞ Q ðzÞ ¼ jj aj ð1; zjþ1 ; . . . ; zn Þ þ b 11 ða1 ð1; zÞ 1Þ: @zj b j¼2
ð41Þ
Now adding relation (19) for all j ¼ 2; . . . ; n, putting z1 ¼ 1 and subtracting from (41) we arrive at our basic equation, n X
lj ðzj 1Þ
j¼2
n1 X @ Pjj ð1; zÞ QðzÞ ¼ ½aj ð1; zjþ1 ; . . . ; zn Þ 1: @zj b jj j¼1
ð42Þ
Taking finally derivatives above with respect to zk ; zl , and using the fact that from (19) after some algebra at relation (40). h
@P jj ð zÞ jz¼1 @zk
; 8 k; j ¼ 2; 3; . . . ; n; we arrive ¼ lj Djk b jj
Now we are ready to give the mean number of customers in the retrial boxes. Lemma 7. The mean number of Pk ; k ¼ 2; 3; . . . ; n customers in K th k1 retrial box is given by
EðNk Þ ¼
ð1Þ n 0 khk b lm ðqm þ q0m Þ sk q1 X kb kp kp 0 ð1Þ q0m ðq^ ð1Þ Dmk þ þ Þþ ð1 q Þhk þ k1 þ k1 : mk þ kp ð1 q Þ k b ðkÞ b ðkÞ l lk m1 1 k 0 0 m¼2 m¼2 n X
ð43Þ
where Dkl are found in Theorem 6. Proof. Differentiating (22) and (18), with respect to zk , and setting zl ¼ 1; l ¼ 1; 2; . . . ; n, we obtain after manipulations,
" # n X @P 11 ðzÞ lm ðqm þ q0m Þ 11 ¼ b þ s D mk k ; @zk z¼1 pm1 ð1 q1 Þ m¼2 j1 Q ð1 p Þ b 1j r @P 1j ðzÞ @P11 ðzÞ r¼1 ¼ ; j ¼ 2; . . . ; n: 11 @zk z¼1 @zk b
ð44Þ
z¼1
In a similar way from (19)–(21),
@Pii ðzÞ @zk z ¼1
@Pij ðzÞ @zk
z¼1
@P0 ðzÞ @zk
z¼1
; ¼ li Dik b ii Q j1 b ð1p ij r Þ @P ii ð z Þ r¼i ¼ @z b k
ii
¼
n P lm q0m D m¼2
kpm1
mk
z¼1
i ¼ 2; . . . ; n;
; h
j ¼ i þ 1; . . . ; n; ð1Þ
ð45Þ i
0 þ bkbðkÞ ð1 q Þhk þ kplk1 þ 0
k
n P m¼2
q0m
ð1Þ khk b 0 q^ ð1Þ : mk þ b ðkÞ 0
Observing now that, for any k ¼ 2; 3; . . . ; n;
EðNk Þ ¼
n X n X @N ðzÞ @Q ðzÞ @Pij ðzÞ @P0 ðzÞ ¼ þ þ ; @zk z¼1 @zk z¼1 i¼1 j¼i @zk z¼1 @zk z¼1
ð46Þ
and replacing from (44), (45) to (46) we arrive easily at (43) and the theorem has been proved. h
6. Numerical results In this section we consider a system of n ¼ 4 phases of service and so three retrial boxes and use the formulae derived previously to obtain numerical results and to investigate the way the mean number of customers in the retrial boxes EðN i Þ i ¼ 2; 3; 4 are affected when 11 , and the mean retrial interval in the first box 0 , the mean service time of the ordinary customers b we vary the mean vacation time b Eðretrial K 1 Þ ¼ 1=l2 , always for increasing values of the mean arrival rate k. To construct the tables we assume that the vacation time U 0 and the service times follow exponential distributions with p.d.f.’s respectively,
1 b0 ðxÞ ¼ eð1=b0 Þx ; b0
1 bij ðxÞ ¼ eð1=bij Þx ; bij
i ¼ 1; . . . ; 4 j ¼ i; . . . ; 4:
648
C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
14 ¼ 0:2, b 12 ¼ b 22 ¼ b 23 ¼ b 33 ¼ 0:33, b 13 ¼ b 24 ¼ b 34 ¼ b 44 ¼ 0:25; p ¼ 0:7, Moreover we assume that in all tables below b 1 p2 ¼ 0:5; p3 ¼ 0:1. Finally l3 ¼ 0:5; l4 ¼ 2. Table 1 shows the way EðN i Þ i ¼ 2; 3; 4 changes when we vary the mean vacation time, for increasing values of the mean arrival rate k. Here one can observe the crucial role that the vacation plays on the number of retrial customers. Thus, even for a small value of k; k ¼ 0:15 0 ¼ 0Þ to the system with for example, EðN2 Þ increases from 0.1488 to 46.026 when we pass from a system without vacation period (b
Table 1 Values of EðN i Þ; i ¼ 2; 3; 4; for
l2 ¼ 1; b11 ¼ 0:5.
knb 0
0
0.2
0.6
1.3
2.7
0.1488 0.2029 0.0112
0.1926 0.2451 0.0161
0.3084 0.3602 0.0282
0.6816 0.7605 0.0623
46.026 64.83 3.316
0.27
0.3798 0.4958 0.0287
0.5273 0.6693 0.0435
1.1548 1.3906 0.0954
120.04 170.57 8.599
0.42
1.045 1.3022 0.0768
1.8373 2.2913 0.1395
267.02 380.45 19.077
0.59
4.1926 5.2556 0.2956
221.4 314.4 15.814
0.71
126.8 179.3 9.0327
0.15
EðN 2 Þ EðN3 Þ EðN4 Þ
Table 2 Values of EðN i Þ; i ¼ 2; 3; 4; for
l2 ¼ 1; b0 ¼ 0:2.
knb 11
0.2
0.8
1.3
2.1
2.8
5.5
0.1711 0.2257 0.0145
0.2233 0.271 0.0181
0.3014 0.3291 0.0226
0.5481 0.4882 0.0346
1.0297 0.7737 0.0595
18522.3 26410.6 1321.15
0.25
0.3646 0.4783 0.0306
0.6088 0.7108 0.047
1.1043 1.1131 0.0755
4.3585 3.97 0.25
193.69 263.19 13.378
0.3
0.5051 0.662 0.042
0.9877 1.1428 0.0742
2.267 2.354 0.1496
64.9 85.09 4.423
0.4
0.9447 1.2382 0.0765
3.034 3.6085 0.2161
112.03 156.15 7.9094
0.5
1.8448 2.4349 0.1444
108.39 152.81 7.717
0.71
92.598 131.62 6.632
0.15
EðN 2 Þ EðN 3 Þ EðN 4 Þ
Table 3 ¼ 0:2; b ¼ 0:5. Values of EðN i Þ; i ¼ 2; 3; 4 for b 0 11 0.02
0.2
1
2
10
0.0452 0.2539 0.0177
0.0727 0.2515 0.0171
0.1926 0.2454 0.0161
0.3397 0.2409 0.0157
1.4994 0.2329 0.0151
0.27
0.1423 0.7229 0.0521
0.2185 0.7087 0.0488
0.5373 0.6693 0.0435
0.9183 0.6443 0.413
3.862 0.5987 0.0382
0.42
0.4837 2.7032 0.1904
0.761 2.5895 0.1699
1.8373 2.2913 0.1395
3.045 2.1153 0.1271
11.953 1.8179 0.1093
0.59
8.33 464.3 23.5
60.25 419.03 21.12
221.4 314.4 15.814
371.7 265.23 13.31
1347.5 192.75 9.6763
k n Eðretrial K 1 Þ 0.15
EðN 2 Þ EðN 3 Þ EðN 4 Þ
C. Langaris, I. Dimitriou / European Journal of Operational Research 205 (2010) 638–649
649
0 ¼ 2:7; while the corresponding value for EðN 3 Þ increases from 0.2029 to 64.83. When now the arrival rate k becomes k ¼ 0:42 then even a b 0 ¼ 0:6 increases dramatically the mean number of retrial customers to 267.02 in retrial box K 1 and to 380.45 0 ¼ 0 to b small change from b in retrial box K 2 , respectively. Thus we must be very careful on the vacation period that we must allow, to avoid overcrowded retrial boxes. The behavior of the third retrial box K 3 ðEðN 4 ÞÞ is smoother, and it shows us a way to reduce this dramatic effect of the vacation period by allowing faster retrials (Eðretrial K 3 Þ ¼ 1=l4 ¼ 0:5 here) and/or less preference of the box ðp3 ¼ 0:1Þ. 11 . Similar observations can be deduced from Table 2 that contains values of EðNi Þ i ¼ 2; 3; 4 when we vary the mean first stage service b One can observe again the way the mean number of retrial customers in each box increases when b11 increases. An increase that depends on how fast or slow the mean retrial Eðretrial K i Þ is and/or on the preference that customers show to the corresponding box pi . Table 3 finally depicts the way the EðN i Þ i ¼ 2; 3; 4 are affected when we vary the mean retrial rate in the first box Eðretrial K 1 Þ ¼ 1=l2 . One can observe here not only the increase of EðN 2 Þ, but mainly the reduction of EðN 3 Þ and EðN 4 Þ in the other two boxes when we increase 1=l2 , a reduction which is more apparent when k increases. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.ejor.2010.01.034. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
J.R. Artalejo, A classified bibliography of research on retrial queues: progress in 1990–1999, Top 7 (2) (1999) 187–211. J.R. Artalejo, A. Gomez-Corral, Retrial queueing systems, A Computational Approach, Springer-Verlag, Berlin, Heidelberg, 2008. D.I. Choi, T. Kim, Analysis of a two-phase queueing system with vacations and Bernoulli feedback, Stochastic Analysis and Applications 21 (5) (2003) 1009–1019. G. Choudhury, Steady state analysis of a M/G/1 queue with linear retrial policy and two-phase service under Bernoulli vacation schedule, Applied Mathematical Modelling 32 (12) (2008) 2480–2489. G. Choudhury, K.C. Madan, A two stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy, Mathematical and Computer Modelling 42 (2005) 71–85. I. Dimitriou, C. Langaris, Analysis of a retrial queue with two-phase service and server vacations, Queueing Systems 60 (1–2) (2008) 111–129. B.T. Doshi, Analysis of a two-phase queueing system with general service times, Operation Research Letters 10 (1991) 265–272. G.I. Falin, J.G.C. Templeton, Retrial Queues, Chapman and Hall, London, 1997. G.I. Falin, On a multiclass batch arrival retrial queue, Advances in Applied Probability 20 (1988) 483–487. T. Katayama, K. Kobayashi, Sojourn time analysis of a queueing system with two-phase service and server vacations, Naval Research Logistics 54 (1) (2006) 59–65. J.-C. Ke, An M[x]/G/1 system with startup server and J additional options for service, Applied Mathematical Modelling 32 (4) (2008) 443–458. C.M. Krishna, Y.H. Lee, A study of a two-phase service, Operation Research Letters 9 (1990) 91–97. V.G. Kulkarni, H.M. Liang, Retrial queues revisited, in: J.H. Dshalalow (Ed.), Frontiers in Queueing, CRP Press, Boca Raton, 1997, pp. 19–34. A. Krishna Kumar, A. Vijayakumar, D. Arivudainambi, An M/G/1 retrial queueing system with two-phases of service and preemptive resume, Annals of Operation Research 113 (2002) 61–79. C. Langaris, A. Katsaros, Time depended analysis of a queue with batch arrivals and N levels of non preemptive priority, Queueing Systems 19 (1995) 269–288. K.C. Madan, On a single server queue with two stage heterogeneous service and deterministic server vacations, International Journal of Systems Science 32 (7) (2001) 837–844. E. Moutzoukis, C. Langaris, Non-preemptive priorities and vacations in a multiclass retrial queueing system, Stochastic Models 12 (3) (1996) 455–472. L. Takacs, Introduction to the Theory of Queues, Oxford University Press, New York, 1962.