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1 queue with disasters and working breakdowns
Applied Mathematical Modelling 38 (2014) 1788–1798
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The M/G/1 queue with disasters and working breakdowns Bo Keun Kim a, Doo Ho Lee b,⇑ a b
Department of Industrial and Systems Engineering, KAIST, Daejeon 305-701, Republic of Korea Software Research Lab, ETRI, Daejeon 305-700, Republic of Korea
a r t i c l e
i n f o
Article history: Received 3 July 2012 Received in revised form 12 July 2013 Accepted 26 September 2013 Available online 9 October 2013 Keywords: M/G/1 queue Disaster Repair period Normal service Working breakdown service
a b s t r a c t In this paper, we analyze the M/G/1 queueing system with disasters and working breakdown services. The system consists of a main server and a substitute server, and disasters only occur while the main server is in operation. The occurrence of disasters forces all customers to leave the system and causes the main server to fail. At a failure instant, the main server is sent to the repair shop and the repair period immediately begins. During the repair period, the system is equipped with the substitute server which provides the working breakdown services to arriving customers. After introducing the concept of working breakdown services, we derive the system size distribution and the sojourn time distribution. We also obtain the results of the cycle analysis. In addition, numerical works are given to examine the relation between the sojourn time and the some system parameters. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Since the first investigation of the queueing system with disasters was done by Towsley and Tripathi [1], there has been considerable attention paid to this topic due to its applications to local area networks, data/packet communication systems, and manufacturing systems. The queueing systems with disasters are characterized by the phenomenon in which the occurrence of disasters not only destroys all unfinished jobs but also breaks down the machine (processor). In that sense, disasters are referred to as ‘mass exodus’ [2], ‘queue flushing’ [1], ‘catastrophes’ [3], and ‘stochastic clearing’ [4]. The queueing systems in which disasters occur have been widely applied to model the complete server failures. Consider a storage area network (SAN), which is one of the mass storage solutions so that high reliability and high speed network services are provided. A SAN is composed of one or more servers connected to the storage devices through switching devices such as hubs, routers, and bridges. Since the connection control between the storage devices and clients is the major role of SAN servers, they are the prime targets of the distributed denial-of-service attacks (DDoS attacks). DDos attacks make all network resources and data unavailable to its intended clients, and it may be impossible to recover all the destroyed data. For another example, in email contact centers, a clearing phenomenon sometimes occurs and causes the system to be empty. In this case, all users see a server failure notification and reset their request after a random delayed time. Finally, disasters can be viewed as a machine breakdown that leads to destruction of all work in process in manufacturing systems. Towsley and Tripathi [1] studied the M/M/1 queue with disasters (DST) in order to describe the behavior of distributed database systems with site failure. This study has been extended to the M/G/1/DST queue by Jain and Sigman [5] and to the GI/M/1/DST queue by Yang and Chae [6]. Yechiali [7] discussed the M/M/1/DST queue with customer impatience where the stationary probabilities of the system state are derived. In succession, Sudhesh [8] obtained the exact transient solution for the state probabilities of the same model studied in [7]. Economou and Kapodistria [9] investigated the M/M/1/DST queue in ⇑ Corresponding author. Tel.: +82 42 860 3887; fax: +82 42 860 6699. E-mail address: [email protected] (D.H. Lee). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.09.016
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which customers become impatient and perform synchronized abandonment of their service due to the absence of the server. The state dependent M/M/c/DST queue in a random environment was analyzed by Baumann and Sandmann [10]. For this system, they considered continuous-time level dependent quasi-birth-and-death processes and obtained the stationary distribution of the number of customers in the system using an applied matrix analytic algorithm. The server failures which lead to service interruptions are quite common in many real life situations. It is well known that performance measures of unreliable queuing systems are heavily influenced by server failures. For this reason, unreliable queuing systems have been investigated extensively over the decades. As early studies on this topic, we refer Thirurengadan [11], Mitrany and Avi-Itzhak [12], and references therein. Recently, Choudhury and Ke [13], Choudhury and Tadj [14], Dimitriou [15], Dimitriou and Langaris [16], Falin [17], Ke [18–21], Ke et al. [22], Lee et al. [23], Yang et al. [24], and others considered the unreliable queueing systems with various features wherein one of the underlying assumptions is that a failed server is sent for repair at the repair shop and present customers in the system should wait for the server to be repaired without being served. However, in practical situations, the system should be equipped with a substitute (standby) server in preparation for possible main server failures. The substitute server renders services to customers while the main server is repaired. The service rate of the substitute server is different from (probably lower than) that of the main server. At the instant of the repair completion, the main server returns to the system and becomes available. This is the concept of the working breakdowns first introduced by Kalidass and Ramanath [25]. By allowing the substitute server to provide services, the system has the capability of handling emergencies which may occur during the repair period and the system utilization is optimized. Additionally, the working breakdown service can decrease complaints from the customers who should wait for the main server to be, repaired and reduces the cost of waiting customers or jobs. Therefore, the working breakdown service is a more reasonable repair policy for unreliable queueing systems. The rest of this paper is organized as follows. In Section 2, we explain the concept of the working breakdown in detail and describe the mathematical model. In Section 3, general results on the cycle analysis are presented. We derive the system size distribution and the sojourn time distribution in Sections 4 and 5, respectively. Section 6 deals with numerical works which we conducted to examine the relation between the sojourn time and some system parameters. 2. Model description In this paper, we consider a queueing system with the following features. The arrival process of customers is a Poisson process with a rate of k. The interarrival times, denoted by A, are exponentially distributed. Customers are served in the order of their arrival, i.e., first-come first-served (FCFS) discipline. We define the service rendered by the main server as normal service. The normal service times, denoted by S1, are independent and identically distributed (i.i.d.) random variables. A density and its Laplace-Stieltjes transform (LST) are respectively denoted by s1(x)dx = Pr {x < S1 < x + dx} and R1 S1 ðhÞ ¼ 0 ehx s1 ðxÞdx. Disasters only occur while the main server is in operation. The interarrival times of disasters, denoted by D, are exponentially distributed with a rate of d. Whenever a disaster occurs, the main server fails and all present customers are forced to leave the system. As soon as the main server fails, it undergoes a repair procedure. The repair times, denoted by R, have an exponential distribution with a rate of c. The repaired server is assumed to be as good as a new server. We now introduce the concept of the working breakdowns in detail. As soon as a disaster arrives at the system, the main server fails and a repair process immediately begins. During a repair period, the stream of new customer arrivals continues. The substitute server renders services to customers while the main server is repaired. The service rate of the substitute server is lower than that of the main server. We define the service rendered by the substitute server as the working breakdown service. The working breakdown service times, denoted by S0, are i.i.d. random variables. The density and its LST are respecR1 tively denoted by s0(x)dx = Pr {x < S0 < x + dx} and S0 ðhÞ ¼ 0 ehx s0 ðxÞdx. If there are customers in the system at the end of the repair, the substitute server stops service and the main server restarts and operates at its normal service rate. We assume that the service interrupted at the end of the repair is lost, and it is restarted with a different distribution at the beginning of the following normal service period. Meanwhile, if there are no customers in the system at the end of the repair, the main server returns to the system, stays idle, and waits for arriving customers. We further assume that A, S0, S1, D, and R are mutually independent. 3. Cycle analysis In this section, we perform a (regeneration) cycle analysis. To avoid terminological confusion, we define the normal busy period as the time interval during which the main server is busy, and the working breakdown busy period as the time interval during which the substitute server is busy. In this section, X⁄(h) denotes the LST of any continuous random variable X. A cycle is defined as the time interval between two successive epochs at which a disaster occurs. Let C denote the length of a cycle. There are two types of cycles, depending on customer arrivals, which occur during a repair period. The cycle is called type-1 if no customers arrive during the repair period, and type-2 if customer arrivals occur during the repair period. Let Ci denote the length of the type-i cycle and fi, i = 1, 2, denote the probability that the cycle is of type-i. Then, we have
f1 ¼ PrfA > Rg ¼
c k ; f ¼ PrfA < Rg ¼ : cþk 2 cþk
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Let BN and BR respectively denote the length of the normal busy period and the length of the working breakdown busy period. Their LSTs are given by
BN ðhÞ ¼ S1 ðh þ k kBN ðhÞÞ; BR ðhÞ ¼ S0 ðh þ k kBR ðhÞÞ:
ð1Þ
Since, in the type-1 cycle, no customers arrive during the repair period, our system becomes the standard M/G/1 queueing system immediately after the repair completion (see Fig. 1). Once a customer arrives after the repair completion, the normal busy period begins. If a disaster occurs before this normal busy period ends, the type-1 cycle ends. If not, our system behaves as the standard M/G/1 queueing system until a disaster occurs. We define the random variable F as the time duration from the end of the repair period to the next occurrence point of a disaster during the type-1 cycle. F is represented as follows:
F¼
TL;
if
D < BN ;
T S þ F; otherwise;
where TS = A + (BN | BN < D) and TL = A + (D | D < BN). T S ðhÞ and T L ðhÞ are then obtained as follows:
T S ðhÞ ¼ A ðhÞE½ehBN jBN < D ¼ T L ðhÞ ¼ A ðhÞE½ehD jD < BN ¼
k BN ðh þ dÞ ; k þ h BN ðdÞ
k 1 BN ðh þ dÞ d : k þ h 1 BN ðdÞ d þ h
From the above expression, F⁄(h) is obtained as
1 BN ðdÞ T L ðhÞ F ðhÞ ¼ : 1 BN ðdÞT S ðhÞ
Expressing C1 in terms of F, we have C1 = (R | R < A) + F. Therefore, C 1 ðhÞ and E[C1] are given by
C 1 ðhÞ ¼
kþc F ðhÞ; kþcþh
E½C 1 ¼
1 d þ k kBN ðdÞ þ : c þ k kdð1 BN ðdÞÞ
ð2Þ
Next, we consider the type-2 cycle. Assuming that at least one customer arrives during the repair period, there are two cases of the type-2 cycle. Case 1. The working breakdown busy period ends before the repair completion. Case 2. The repair is completed before the working breakdown busy period ends. Let C 2;j denote the length of the type-2 cycle in Case j. In addition, let hj, j = 1, 2, denote the probability of Case j. Then, we have
h1 ¼ PrfR > BR g ¼ BR ðcÞ; h2 ¼ PrfR < BR g ¼ 1 BR ðcÞ; C 2 ðhÞ ¼
2 X
hj C 2;j ðhÞ:
j¼1
In Case 1, the length of the remaining cycle is stochastically equivalent to the length of the unconditional cycle, due to the memoryless property (see Fig. 2). This is explained by the fact that the length of the repair period is exponentially distributed. Therefore, we have
C 2;1 ¼ ðAjA < RÞ þ ðBR jBR < RÞ þ C; C 2;1 ðhÞ ¼
k þ c BR ðc þ hÞ C ðhÞ: k þ c þ h BR ðcÞ
Fig. 1. An example of the type-1 cycle.
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Fig. 2. An example of the type-2 cycle: Case 1.
Fig. 3. An example of the type-2 cycle: Case 2.
Let K denote the number of existing customers at the instant of the repair completion. In Case 2, the normal busy period is the busy period which starts with K customers (see Fig. 3). If a disaster occurs before this normal busy period ends, then the cycle is terminated. If not, our system behaves as the standard M/G/1 queueing system until a disaster occurs. Let us define two random variables YS and YL as follows:
Y S ¼ ðBK jBK < DÞ; Y L ¼ ðDjD < BK Þ; where BK is the length of the normal busy period which starts with K customers. With the result of Kim et al. [26], K(z), the PGF of K, is given by
KðzÞ ¼
z BR ðcÞ S0 ðc þ k kzÞ 1 : c þ k kz 1 BR ðcÞ S0 ðc þ k kzÞ z
cz
Hence, we obtain BK ðhÞ as follows:
BK ðhÞ ¼ KðzÞjz¼B ðhÞ ¼ N
cBN ðhÞ cþk
kBN ðhÞ
BN ðhÞ BR ðcÞ S0 ðc þ k kBN ðhÞÞ 1 : 1 BR ðcÞ S0 ðc þ k kBN ðhÞÞ BN ðhÞ
ð3Þ
From (3), Y S ðhÞ and Y L ðhÞ are respectively given by
B ðh þ dÞ 1 BK ðh þ dÞ d Y S ðhÞ ¼ E½ehBK BK < D ¼ K ; Y L ðhÞ ¼ E½ehD D < BK ¼ : BK ðdÞ 1 BK ðdÞ d þ h Using F defined in the type-1 cycle, we can obtain the length of the remaining cycle after the repair completion. This is denoted as G, which is expressed as
G¼
YL;
if
D < BK ;
Y S þ F; otherwise:
From above, G⁄(h) is obtained as follows:
G ðhÞ ¼ ð1 BK ðdÞÞY L ðhÞ þ BK ðdÞY S ðhÞF ðhÞ ¼ ð1 BK ðd þ hÞÞ
d þ BK ðd þ hÞF ðhÞ: dþh
Therefore, we have
C 2;2 ¼ ðAjA < RÞ þ ðRjR < BR Þ þ G; C 2;2 ðhÞ ¼
k þ c 1 BR ðc þ hÞ c G ðhÞ: k þ c þ h 1 BR ðcÞ c þ h
Combining the two cases, we have C 2 ðhÞ and E[C2], which are determined as follows:
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k þ c BR ðc þ hÞ k þ c 1 BR ðc þ hÞ c C ðhÞ þ ð1 BR ðcÞÞ G ðhÞ; k þ c þ h BR ðcÞ k þ c þ h 1 BR ðcÞ c þ h 1 1 1 BK ðdÞ E½C 2 ¼ : þ BR ðcÞE½C þ ð1 BR ðcÞÞ þ þ cþk c d k kBN ðdÞ
C 2 ðhÞ ¼ BR ðcÞ
ð4Þ
By combining (2) and (4), we obtain
C ðhÞ ¼
2 2 X X 1 1 c þ kBK ðdÞð1 BR ðcÞÞ fi C i ðhÞ; E½C ¼ fk E½C k ¼ þ þ : d c kð1 BN ðdÞÞðc þ k kBR ðcÞÞ i¼1 k¼1
ð5Þ
Let us respectively define PR and PA as the probability that the main server is under repair and the probability that the main server is available. From the renewal reward theorem, we have
PR ¼
E½R kdð1 BN ðdÞÞðc þ k kBR ðcÞÞ ; ¼ E½C kðd þ cÞð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
PA ¼ 1 PR ¼
kcð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ : kðd þ cÞð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
ð6Þ
ð7Þ
4. System size distribution Let N(t) be the number of customers in the system at t and n(t) be an indicator random variable given by
nðtÞ ¼
0; The main server is under repair at t; 1; The main server is available at t:
Let SR,i(t) denote the remaining service time at t when n(t) = i, i e {0, 1}. Then, the vector process fNðtÞ; nðtÞ; SR;i ðtÞ; t P 0g becomes a Markov process, in which the supplementary variables are SR,i(t). To derive the PGF of the steady state system size, we define the following limiting probabilities for i e {0, 1} :
P0;i ¼ limPrfNðtÞ ¼ 0; nðtÞ ¼ ig; Pn;i ðxÞdx ¼ limPrfNðtÞ ¼ n; nðtÞ ¼ i; x < SR;i ðtÞ < x þ dxg; n P 1: t!1
t!1
Using these probabilities, the Kolmogorov equations for the system size distribution are given by
0 ¼ ðk þ cÞP0;0 þ P1;0 ð0Þ þ d
1 X Pn;1 ;
ð8Þ
n¼1
d P1;0 ðxÞ ¼ kP0;0 s0 ðxÞ ðk þ cÞP1;0 ðxÞ þ P 2;0 ð0Þs0 ðxÞ; dx
d Pn;0 ðxÞ ¼ kPn1;0 ðxÞ ðk þ cÞPn;0 ðxÞ þ Pnþ1;0 ð0Þs0 ðxÞ; n P 2; dx
ð9Þ
0 ¼ cP0;0 kP 0;1 þ P 1;1 ð0Þ;
ð10Þ ð11Þ
d P1;1 ðxÞ ¼ cP1;0 s1 ðxÞ þ kP0;1 s1 ðxÞ ðk þ dÞP1;1 ðxÞ þ P2;1 ð0Þs1 ðxÞ; dx
ð12Þ
d Pn;1 ðxÞ ¼ cPn;0 s1 ðxÞ þ kPn1;1 ðxÞ ðk þ dÞPn;1 ðxÞ þ P nþ1;1 ð0Þs1 ðxÞ; n P 2; dx
ð13Þ
where Pn;i ¼
R1 0
Pn;i ðhÞ ¼
Pn;i ðxÞdx. To solve (8)–(13), we define the following LSTs and PGFs:
Z 0
1
ehx Pn;i ðxÞdx; Pi ðz; hÞ ¼
1 1 X X P n;i ðhÞzn ; Pi ðz; 0Þ ¼ Pn;i ð0Þzn ; i 2 f0; 1g: n¼1
n¼1
Using these notations, we have the normalizing condition as follows:
P0;0 þ P0;1 þ P0 ð1; 0Þ þ P 1 ð1; 0Þ ¼ 1:
ð14Þ
Remark 1. In (14), P0,0 (P0,1) is the probability that there are no customers while the main server is under repair (available). Similarly, P 0 ð1; 0Þ (P 1 ð1; 0Þ) is the probability that the substitute server (the main server) is busy. Therefore, P P1 PR ¼ P 0;0 þ P0 ð1; 0Þ ¼ 1 n¼0 P n;0 and P A ¼ P 0;1 þ P 1 ð1; 0Þ ¼ n¼0 P n;1 .
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Applying LSTs to equations (9), (10), (12), and (13) yields
ðhP1;0 ðhÞ P1;0 ð0ÞÞ ¼ kP0;0 S0 ðhÞ ðk þ cÞP1;0 ðhÞ þ P 2;0 ð0ÞS0 ðhÞ;
ð15Þ
ðhPn;0 ðhÞ Pn;0 ð0ÞÞ ¼ kPn1;0 ðhÞ ðk þ cÞPn;0 ðhÞ þ Pnþ1;0 ð0ÞS0 ðhÞ:
ð16Þ
ðhP1;1 ðhÞ P1;1 ð0ÞÞ ¼ cP1;0 S1 ðhÞ þ kP0;1 S1 ðhÞ ðk þ dÞP 1;1 ðhÞ þ P2;1 ð0ÞS1 ðhÞ;
ð17Þ
ðhPn;1 ðhÞ Pn;1 ð0ÞÞ ¼ cPn;0 S1 ðhÞ þ kPn1;1 ðhÞ ðk þ dÞPn;1 ðhÞ þ Pnþ1;1 ð0ÞS1 ðhÞ; n P 2:
ð18Þ
Substituting h = 0 into (15)–(18) and then summing over n ¼ 1; 2; 3; ; we obtain the following relationships:
P1;0 ð0Þ ¼ kP0;0 cP0 ð1; 0Þ;
ð19Þ
P1;1 ð0Þ ¼ kP0;1 þ cP0 ð1; 0Þ dP1 ð1; 0Þ;
ð20Þ
and from (11) and (20), we have
dP1 ð1; 0Þ ¼ cðP0;0 þ P0 ð1; 0ÞÞ:
ð21Þ
Multiplying (15)–(18) by zn and then summing over n ¼ 1; 2; 3; ; we obtain
ðh c k þ kzÞP0 ðz; hÞ ¼ z1 P0 ðz; 0Þðz S0 ðhÞÞ S0 ðhÞðkP0;0 z P 1;0 ð0ÞÞ; ðh d k þ
kzÞP1 ðz; hÞ
¼ z1 P1 ðz; 0Þðz
S1 ðhÞÞ
S1 ðhÞð
P0 ðz; 0Þ
c
þ kP0;1 z P1;1 ð0ÞÞ:
ð22Þ ð23Þ
Inserting h ¼ c þ k kz into (22) and h ¼ d þ k kz into (23) and then simplifying them with respect to P i ðz; 0Þ, we get
P0 ðz; 0Þ ¼
zS0 ðc þ k kzÞðP1;0 ð0Þ kP0;0 zÞ ; S0 ðc þ k kzÞ z
ð24Þ
P1 ðz; 0Þ ¼
zS1 ðd þ k kzÞðP1;1 ð0Þ cP0 ðz; 0Þ kP0;1 zÞ : S1 ðd þ k kzÞ z
ð25Þ
Substituting (24) and (25) into (22) and (23), respectively, we obtain
P0 ðz; hÞ ¼
zðP1;0 ð0Þ kP0;0 zÞðS0 ðc þ k kzÞ S0 ðhÞÞ ; ðh c k þ kzÞðS0 ðc þ k kzÞ zÞ
ð26Þ
P1 ðz; hÞ ¼
zðP1;1 ð0Þ cP 0 ðz; 0Þ kP0;1 zÞðS1 ðd þ k kzÞ S1 ðhÞÞ : ðh d k þ kzÞðS1 ðd þ k kzÞ zÞ
ð27Þ
Finally, the PGF of the system size, denoted by P(z), is given by
PðzÞ ¼ P0;0 þ P0;1 þ P0 ðz; 0Þ þ P1 ðz; 0Þ ¼ P0;0 þ P0;1 þ þ
z½kP0;0 ð1 zÞ cP0 ð1; 0Þð1 S0 ðc þ k kzÞÞ ðc þ k kzÞðS0 ðc þ k kzÞ zÞ
z½cðP0 ð1; 0Þ P 0 ðz; 0ÞÞ þ kP0;1 ð1 zÞ dP1 ð1; 0Þð1 S1 ðd þ k kzÞÞ : ðd þ k kzÞðS1 ðd þ k kzÞ zÞ
ð28Þ
According to Rouche’s theorem, S0 ðc þ k kzÞ z ¼ 0 has a unique solution, denoted as z0, for |z| < 1 (see Appendix A). If the denominator of P0 ðz; 0Þ is zero when z = z0, its numerator should be zero. Similarly, S1 ðd þ k kzÞ z ¼ 0 has a unique solution, denoted as z1, for |z| < 1. If the denominator of P1 ðz; 0Þ is zero when z = z1, its numerator should be zero. From (1), we can confirm that z0 ¼ BR ðcÞ and z1 ¼ BN ðdÞ. Thus, we obtain
cP0 ð1; 0Þ ¼ kP0;0 ð1 z0 Þ;
ð29Þ
dP1 ð1; 0Þ ¼ kP0;0 ð1 z0 Þ þ kP0;1 ð1 z1 Þ cP0 ðz1 ; 0Þ:
ð30Þ
Solving (6), (14), (21), and (29) simultaneously with respect to P0,0, P0,1, P0 ð1; 0Þ, and P1 ð1; 0Þ, we have
P0;0 ¼
kdcð1 BN ðdÞÞ ; kðd þ cÞð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
P0 ð1; 0Þ ¼
kðd þ cÞð1
k2 dð1 BN ðdÞÞð1 BR ðcÞÞ ; c þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
BN ðdÞÞð
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P0;1 ¼
kðd þ cÞð1
P1 ð1; 0Þ ¼
dc½c þ kBK ðdÞð1 BR ðcÞÞ ; c þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
BN ðdÞÞð
kcð1 BN ðdÞÞðc þ k kBR ðcÞÞ : kðd þ cÞð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
Remark 2. In our study, the inequality d > 0 is a necessary and sufficient condition for the system to be stable (For details, see Appendix B).
5. Sojourn time distribution In this section, considering a test customer (TC), we derive the LST of the sojourn time distribution which is the waiting time in the queue plus the service time. Let W and W⁄(h) respectively denote the unconditional sojourn time of the TC and its LST. The TC’s arrival can belong to one of the following cases: Case Case Case Case
1. 2. 3. 4.
The The The The
TC TC TC TC
arriving arriving arriving arriving
while while while while
the the the the
main main main main
server server server server
is is is is
available finds no customer waiting. available finds that the server is busy. under repair finds no customer waiting. under repair finds that the substitute server is busy.
Let Wi, i = 1, 2, 3, 4, denote the sojourn time of the TC that arrives in Case i, and define W i ðhÞ ¼ PrfCase igE½ehW i Case i. Notice that W⁄(h) is given by
W ðhÞ ¼ W 1 ðhÞ þ W 2 ðhÞ þ W 3 ðhÞ þ W 4 ðhÞ: In Case 1, the TC arrives at the system while the main server is idle. Upon arrival, he immediately receives the normal service and leaves the system by either the occurrence of a disaster or the service completion. Therefore, we find the following relationships:
W 1 ðhÞ ¼ P0;1 ðPrfS1 < DgE½ehS1 S1 < D þ PrfS1 > DgE½ehD S1 > DÞ S ðh þ dÞ d 1 S1 ðh þ dÞ d þ hS1 ðh þ dÞ ¼ P0;1 PrfS1 < Dg 1 ¼ P0;1 : þ PrfS1 > Dg PrfS1 < Dg d þ h PrfS1 > Dg hþd
ð31Þ
In Case 2, the TC arrives at the system while the main server is busy. Let us define U and U⁄(h) as the unfinished work immediately after the arrival epoch of the TC and its LST. Using P 1 ðz; hÞ in Section 4, U⁄(h) is expressed as P U ðhÞ ¼ P1 ðS1 ðhÞ; hÞ=P1 ð1; 0Þ. Since PrfCase 2g ¼ 1 n¼1 P n;1 and W2 = min {D, U}, we have
W 2 ðhÞ ¼
1 X d þ hU ðh þ dÞ dP1 ð1; 0Þ þ hP1 ðS1 ðh þ dÞ; h þ dÞ Pn;1 E½eh minfD;Ug ¼ P1 ð1; 0Þ ¼ : hþd hþd n¼1
ð32Þ
In Case 3, the TC arriving during the repair period sees no customer in the system. Therefore, he immediately receives the working breakdown service. If the TC’s working breakdown service is completed before the repair completion, his sojourn time is simply his own working breakdown service time. Otherwise, the TC’s working breakdown service is interrupted at the end of the repair period and the new normal service is provided to him. Once he receives the normal service, his remaining sojourn time is stochastically equal to W1. Thus, the TC’s sojourn time is the remaining repair time plus W1. Note that the remaining repair time is stochastically equal to a new repair time due to the memoryless property. Considering all these, we have
W 3 ðhÞ ¼ P0;0 PrfS0 < RgE ehS0 S0 < R þ PrfS0 > RgE ehðRþW 1 Þ S0 > R
c d þ hS1 ðh þ dÞ : ¼ P0;0 S0 ðh þ cÞ þ ð1 S0 ðh þ cÞÞ hþd hþc
ð33Þ
In Case 4, the TC arrives during the repair period and finds at least one customer in the system. Suppose that the arriving TC sees n customers and the customer in working breakdown service leaves the system before the repair completion. Let H denote the sojourn time of the TC when the customer at the head in the queue is on the threshold of receiving the working breakdown service. The TC’s sojourn time is the remaining time of the ongoing working breakdown service when he arrives plus H. Let us consider the following n + 1 cases in accordance with the repair completion epochs. If kcustomers,
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k ¼ 0; 1; 2; ; n, leave the system by the time the repair is complete, there are n k customers who wait for their normal service. In this case, H is equal to the remaining repair time plus either the sum of the n k normal service times or the interarrival time of a disaster. This leads to
H ðhÞ ¼
n1 X nk n PrfSk0 < R < S0kþ1 gE½ehR Sk0 < R < S0kþ1 E½eh minfD;S1 g þ PrfSn0 < RgE½ehS0 Sn0 < R;
ð34Þ
k¼0
where Sni represents the n -fold convolution of Si with itself. The term E½ehR Sk0 < R < S0kþ1 in the right hand side of (34) is given by
E½ehR S0k1 < R < Sk0 ¼
c ð1 S0 ðh þ cÞÞðS0 ðh þ cÞÞk1 : hþc PrfSk1 < R < Sk0 g 0
Rewriting (34), we have
H ðhÞ ¼ ¼
n X cð1 S0 ðh þ cÞÞðS0 ðh þ cÞÞk1 d þ hðS1 ðh þ dÞÞnkþ1 n þ ðS0 ðh þ cÞÞ hþc hþd k¼1 n n cd½1 ðS0 ðh þ cÞÞn chð1 S0 ðh þ cÞÞ ðS1 ðh þ dÞÞ ðS0 ðh þ cÞÞ n þ ðS0 ðh þ cÞÞ : þ ðh þ cÞðh þ dÞ ðh þ cÞðh þ dÞ ðS1 ðh þ dÞ S0 ðh þ cÞÞðS1 ðh þ dÞÞ1
ð35Þ
If the remaining repair time is shorter than the remaining time of the ongoing working breakdown service, the TC’s sojourn time is the remaining repair time plus either the sum of the normal service times of the customers who are in the system or the interarrival time of a disaster. Therefore, we obtain
W 4 ðhÞ ¼
1 Z X n¼1
1
1 X P n;0 ðxÞPrfR > SR;0 SR;0 ¼ xgH ðhÞehx dx þ
x¼0
n¼1
Z
1
Pn;0 ðxÞ x¼0
Z
x
Prfy < R < y þ dyjSR;0
y¼0
nþ1
¼ xgE½eh minfD;S1 g ehy dx 1 Z 1 1 Z X X ¼ H ðhÞ P n;0 ðxÞeðhþcÞx dx þ n¼1
x¼0
n¼1
1
Pn;0 ðxÞð1 eðhþcÞx Þ
x¼0
c d þ hðS1 ðh þ dÞÞnþ1 dx hþc hþd
c d þ hðS1 ðh þ dÞÞnþ1 hþc hþd n¼1 n¼1 c dðP ð1; h þ c Þ P ðS ðh þ dÞ; h þ c ÞÞ chS1 ðh þ dÞð1 S0 ðh þ cÞÞ 0 0 1 ¼ P0 ðS0 ðh þ cÞ; h þ cÞ þ þ ðh þ cÞðh þ dÞ ðh þ cÞðh þ dÞ P ðS ðh þ dÞ; h þ cÞ P0 ðS0 ðh þ cÞ; h þ cÞ 0 1 S1 ðh þ dÞ S0 ðh þ cÞ cdðP 0 ð1; 0Þ P 0 ð1; h þ cÞÞ þ chS1 ðh þ dÞðP0 ðS1 ðh þ dÞ; 0Þ P 0 ðS1 ðh þ dÞ; h þ cÞÞ ; þ ðh þ cÞðh þ dÞ 1 1 X X ðPn;0 Pn;0 ðh þ mÞÞ ¼ H ðhÞ Pn;0 ðh þ cÞ þ
ð36Þ
where SR,0 = limt?1SR,0(t). Remark 3. Now that we deal with the Poisson arrival process queueing system, PASTA property [27] is employed to derive W i ðhÞ. Remark 4. We can confirm the result of the Little’s formula.
6. Numerical illustration In this section, we provide numerical works related to our model. We examine the relationship between the mean sojourn time and the mean interarrival time of disasters in Fig. 4. Also, we show how the mean repair time influences the mean sojourn time in Fig. 5. In each figure, we compare the mean sojourn time of the disastrous queue with working breakdown services (DST + WR) and that of the disastrous queue with non-working breakdown services (DST + NWR). We fix the system parameters in both cases. Customer arrivals are generated according to a Poisson process at a rate of 0.8. We use an exponential distribution with a rate of 0.5 for the working breakdown service time. The normal service time follows a hyperexponential distribution with the probability density function pk1 ek1 t þ ð1 pÞk2 ek2 t , where p = 0.5, k1 ¼ 4=3, and k2 ¼ 0:8. In Fig. 4, the vertical axis of the graph represents the mean sojourn time, E[W], and the horizontal axis shows the mean interarrival time of a disaster, d1. We assume that the repair time distributions follow an exponential distribution with a
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Fig. 4. Mean sojourn time of the M/G/1 over the mean interarrival time of disasters.
Fig. 5. Mean sojourn time of the M/G/1 over the mean repair time.
rate of 0.5. Fig. 4 shows that E[W] increases as d1 increases. We can expect that a larger value of d1 causes fewer customers to be forced to leave the system by a disaster, thus increasing the probability of a larger system size and a longer sojourn time. In Fig. 5, the vertical axis is E[W] and the horizontal axis is the mean repair time, c1. We assume that the interarrival time of a disaster follows an exponential distribution with a rate of 0.5. Fig. 5 confirms that E[W] increases with an increase of c1. This indicates that a longer mean repair time leads to a greater number of customers who are provided with the working breakdown service with a rate that is lower than that of the normal service. Finally, we verify that the working breakdown service model is superior to the non-working breakdown service model from both figures. Especially in Fig. 5, as c1 increases, the gap of E[W] between the two models becomes larger. This is very intuitive as, in the working breakdown service model, customers can leave the system during the repair period via the completion of their service. In addition, we observe the mean sojourn time of a TC under the Case i, which is defined in Section 5. We examine the relationship between E½W i jCase i and d1 in Fig. 6 and that between E½W i jCase i and c1 in Fig. 7, respectively. In Fig. 6, the vertical axis of the graph represents the E½W i jCase i and the horizontal axis shows d1. We assume that the repair time distributions follow an exponential distribution with a rate of 0.5. Fig. 6 shows that E½W 1 jCase 1 and E½W 2 jCase 2 increase as d1 increases. We can expect that a larger value of d1 causes fewer customers to be forced to leave the system by a disaster. On the other hand, E½W 3 jCase 3 and E½W 4 jCase 4 are almost unchanged as d1 increases. This indicates that d1 hardly affects the sojourn time of the customers who arrive at the system during repair period. In Fig. 7, the vertical axis is E½W i jCase i and the horizontal axis is c1. We assume that the interarrival time of a disaster follows an exponential distribution with a rate of 0.5. Fig. 7 reveals that E½W 3 jCase 3 and E½W 4 jCase 4 increase with an increase of c1 because a longer mean repair time leads to a greater number of customers who are provided with the working breakdown service at a rate that is lower than that of the normal service. On the other hand, E½W 1 jCase 1 and E½W 2 jCase 2 are almost unchanged as c1 increases. We can expect that a longer mean repair time has little effect on the sojourn time of the customers who arrive at the system during a normal service period.
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Fig. 6. Mean sojourn times of four cases over the mean interarrival time of disasters.
Fig. 7. Mean sojourn times of four cases over the mean repair time.
7. Conclusion In this paper, we analyzed an M/G/1 queue with disasters and the working breakdown service. We derived the results of the important system characteristics of the PGF of the system size and the LST of the FCFS sojourn time. We also carried out a cycle analysis. Finally, numerical works were performed to show the relationship between the mean sojourn time and other system parameters, such as the mean interarrival time of a disaster or the mean repair time. Our study presents an extension of the repairable queueing system and its results may provide a better decision-making tool for the repair arising policies in many practical systems. For future studies, we can extend this model to more complex situations such as the repairable queueing systems with a Markovian arrival process of customers or disasters and a Markovian service process of normal services and working breakdown services. Acknowledgement This research was funded by the MSIP (Ministry of Science, ICT & Future Planning), Korea in the ICT R&D Program 2013 (No. 2013-005-021-001). Appendix A. Application of Rouche’s theorem Let us define S0 ðc þ k kzÞ z as /(z), which is an analytic function in the unit circle |z| < 1. Suppose f(z) = z and gðzÞ ¼ S0 ðc þ k kzÞ, which are all analytic. It can be shown that |g(z)| < |f(z)| on the contour of the circle because
jf ðzÞj ¼ jzj ¼ 1; jgðzÞj 6 gðjzjÞ ¼ S0 ðc þ k kjzjÞ ¼ S0 ðcÞ: Hence, from Rouche’s theorem, it follows that f(z) and f(z) + g(z) will have the same number of zeros inside |z| < 1. Since f(z) has only one zero inside this circle, f(z) + g(z) /(z) will also have only one zero inside |z| < 1. In the same manner, we can show that S1 ðd þ k kzÞ z ¼ 0 has a unique solution for |z| < 1.
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Appendix B. Proof of the stability condition As long as d > 0, then the system under our study is stable. Let us define mk as the joint probability that k customers arrive at the system during a normal service time and a disaster does not occur while this normal service time is ongoing. If we P k assume that MðzÞ ¼ 1 k¼0 z mk , mk and M(z) can be respectively expressed as
mk ¼
Z 0
1
1 X ðkxÞk eðkþdÞx s1 ðxÞdx; k P 0; MðzÞ ¼ zk mk ¼ S1 ðd þ k kzÞ; k! k¼0
Note that Mð0Þ ¼ S1 ðd þ kÞ and Mð1Þ ¼ S1 ðdÞ. For all z within (0, 1), we have 2
1 1 dMðzÞ X d MðzÞ X k1 ¼ kz mk > 0; ¼ kðk 1Þzk2 mk > 0: 2 dz dz k¼1 k¼2
ðB:1Þ
(B.1) implies that M(z) is a convex increasing function. We can also confirm that M(0) < M(1), as dS1 ðhÞ=dh < 0. Thus, from (1), BN ðdÞ is the unique root of MðBN ðdÞÞ ¼ BN ðdÞ within (0, 1) if the following inequality holds:
0 < Mð0Þ < Mð1Þ < 1:
ðB:2Þ
Therefore, d should be a non-zero positive number for (B.2) to hold. This is a sufficient condition. On the other hand, if 0 < BN ðdÞ < 1, then 0 < MðBN ðdÞÞ < 1. This implies that d þ kð1 BN ðdÞÞ > 0. Therefore, under the condition of d > 0, it is proven that E[C] in (5) has a finite value because the related values of c1, BR ðcÞ, and BK ðdÞ in (5) are all positive. This is a necessary condition for stability. In fact, all customers in the system are flushed out whenever disasters occur, which means that the number of customers at arbitrary epochs does not go to infinity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
D. Towsley, S.K. Tripathi, A single server priority queue with server failures and queue flushing, Oper. Res. Lett. 10 (1991) 353–362. A. Chen, E. Renshaw, The M/M/1 queue with mass exodus and mass arrivals when empty, J. Appl. Probab. 34 (1997) 192–207. P.J. Brockwell, J. Gani, S.I. Resnick, Birth, immigration and catastrophe processes, Adv. Appl. Prob. 14 (1982) 709–731. J.R. Artalejo, A. Gómez-Corral, Analysis of a stochastic clearing system with repeated attempts, in: Commun. Statist.-Stoch. Models. 14 (1998) 623–645. G. Jain, K. Sigman, A pollaczek-khintchine formula for M/G/1 queues with disasters, J. Appl. Probab. 33 (1996) 1191–1200. W.S. Yang, K.C. Chae, A note on the GI/M/1 queue with poisson negative arrivals, J. Appl. Probab. 38 (2001) 1081–1085. U. Yechiali, Queues with system disasters and impatient customers when system is down, Queueing Syst. 56 (2007) 195–202. R. Sudhesh, Transient analysis of a queue with system disasters and customer impatience, Queueing Syst. 66 (2010) 95–105. A. Economou, S. Kapodistria, Synchronized abandonments in a single server unreliable queue, Eur. J. Oper. Res. 203 (2010) 143–155. H. Baumann, W. Sandmann, Steady state analysis of lever dependent quasi-birth-and-death processes with catastrophes, Comput. Oper. Res. 39 (2012) 413–423. K. Thirurengadan, Queueing with breakdowns, Oper. Res. 11 (1963) 303–320. I.L. Mitrany, B. Avi-Itzhak, A many server queue with service interruptions, Oper. Res. 16 (1968) 628–638. G. Choudhury, J.C. Ke, A batch arrival retrial queue with general retrial times under bernoulli vacation schedule for unreliable server and delaying repair, Appl. Math. Model. 36 (2012) 255–269. G. Choudhury, L. Tadj, The optimal control of an Mx/G/1 unreliable server queue with two phases of service and Bernoulli vacation schedule, Math. Comput. Model. 54 (2011) 673–688. I. Dimitriou, A preemptive resume priority retrial queue with state dependent arrivals, unreliable server and negative customers, TOP. doi: 10.1007/ s11750-011-0198-4. I. Dimitriou, C. Langaris, A repairable queueing model with two-phase service, start-up times and retrial customers, Comput. Oper. Res. 37 (2010) 1181–1190. G. Falin, An M/G/1 retrial queue with an unreliable server and general repair times, Perform. Evaluation 67 (2010) 569–582. J.C. Ke, Bi-level control for batch arrival queues with an early startup and un-reliable server, Appl. Math. Model. 28 (2004) 469–485. J.C. Ke, Modified T vacation policy for an M/G/1 queueing system with an un-reliable server and startup, Math. Comput. Model. 41 (2005) 1267–1277. J.C. Ke, On M/G/1 system under NT policies with breakdowns, startup and closedown, Appl. Math. Model. 30 (2006) 49–66. J.C. Ke, An M/G/1 queue under hysteretic vacation policy with an early startup and un-reliable server, Math. Method. Oper. Res. 63 (2006) 357–369. J.C. Ke, C.H. Lin, J.Y. Yang, Z.G. Zhang, Optimal (d, c) vacation policy for a finite buffer M/M/c queue with unreliable servers and repairs, Appl. Math. Model. 33 (2009) 3949–3962. D.H. Lee, W.S. Yang, H.M. Park, Geo/G/1 queues with disasters and general repair times, Appl. Math. Model. 35 (2011) 1561–1570. W.S. Yang, J.D. Kim, K.C. Chae, Analysis of M/G/1 stochastic clearing systems, Stoch. Anal. Appl. 20 (2002) 1083–1100. K. Kalidass, K. Ramanath, A queue with working breakdowns, Comput. Ind. Eng. doi: 10.1016/j.cie.2012.04.018. J.D. Kim, D.W. Choi, K.C. Chae, Analysis of queue-length distribution of the M/G/1 queue with working vacations, in: Hawaii International Conference on Statistics and Related Fields, June 5–8, 2003. R.W. Wolff, Poisson arrivals see time averages, Oper. Res. 30 (1982) 223–231.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
The M/G/1 queue with disasters and working breakdowns Bo Keun Kim a, Doo Ho Lee b,⇑ a b
Department of Industrial and Systems Engineering, KAIST, Daejeon 305-701, Republic of Korea Software Research Lab, ETRI, Daejeon 305-700, Republic of Korea
a r t i c l e
i n f o
Article history: Received 3 July 2012 Received in revised form 12 July 2013 Accepted 26 September 2013 Available online 9 October 2013 Keywords: M/G/1 queue Disaster Repair period Normal service Working breakdown service
a b s t r a c t In this paper, we analyze the M/G/1 queueing system with disasters and working breakdown services. The system consists of a main server and a substitute server, and disasters only occur while the main server is in operation. The occurrence of disasters forces all customers to leave the system and causes the main server to fail. At a failure instant, the main server is sent to the repair shop and the repair period immediately begins. During the repair period, the system is equipped with the substitute server which provides the working breakdown services to arriving customers. After introducing the concept of working breakdown services, we derive the system size distribution and the sojourn time distribution. We also obtain the results of the cycle analysis. In addition, numerical works are given to examine the relation between the sojourn time and the some system parameters. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Since the first investigation of the queueing system with disasters was done by Towsley and Tripathi [1], there has been considerable attention paid to this topic due to its applications to local area networks, data/packet communication systems, and manufacturing systems. The queueing systems with disasters are characterized by the phenomenon in which the occurrence of disasters not only destroys all unfinished jobs but also breaks down the machine (processor). In that sense, disasters are referred to as ‘mass exodus’ [2], ‘queue flushing’ [1], ‘catastrophes’ [3], and ‘stochastic clearing’ [4]. The queueing systems in which disasters occur have been widely applied to model the complete server failures. Consider a storage area network (SAN), which is one of the mass storage solutions so that high reliability and high speed network services are provided. A SAN is composed of one or more servers connected to the storage devices through switching devices such as hubs, routers, and bridges. Since the connection control between the storage devices and clients is the major role of SAN servers, they are the prime targets of the distributed denial-of-service attacks (DDoS attacks). DDos attacks make all network resources and data unavailable to its intended clients, and it may be impossible to recover all the destroyed data. For another example, in email contact centers, a clearing phenomenon sometimes occurs and causes the system to be empty. In this case, all users see a server failure notification and reset their request after a random delayed time. Finally, disasters can be viewed as a machine breakdown that leads to destruction of all work in process in manufacturing systems. Towsley and Tripathi [1] studied the M/M/1 queue with disasters (DST) in order to describe the behavior of distributed database systems with site failure. This study has been extended to the M/G/1/DST queue by Jain and Sigman [5] and to the GI/M/1/DST queue by Yang and Chae [6]. Yechiali [7] discussed the M/M/1/DST queue with customer impatience where the stationary probabilities of the system state are derived. In succession, Sudhesh [8] obtained the exact transient solution for the state probabilities of the same model studied in [7]. Economou and Kapodistria [9] investigated the M/M/1/DST queue in ⇑ Corresponding author. Tel.: +82 42 860 3887; fax: +82 42 860 6699. E-mail address: [email protected] (D.H. Lee). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.09.016
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which customers become impatient and perform synchronized abandonment of their service due to the absence of the server. The state dependent M/M/c/DST queue in a random environment was analyzed by Baumann and Sandmann [10]. For this system, they considered continuous-time level dependent quasi-birth-and-death processes and obtained the stationary distribution of the number of customers in the system using an applied matrix analytic algorithm. The server failures which lead to service interruptions are quite common in many real life situations. It is well known that performance measures of unreliable queuing systems are heavily influenced by server failures. For this reason, unreliable queuing systems have been investigated extensively over the decades. As early studies on this topic, we refer Thirurengadan [11], Mitrany and Avi-Itzhak [12], and references therein. Recently, Choudhury and Ke [13], Choudhury and Tadj [14], Dimitriou [15], Dimitriou and Langaris [16], Falin [17], Ke [18–21], Ke et al. [22], Lee et al. [23], Yang et al. [24], and others considered the unreliable queueing systems with various features wherein one of the underlying assumptions is that a failed server is sent for repair at the repair shop and present customers in the system should wait for the server to be repaired without being served. However, in practical situations, the system should be equipped with a substitute (standby) server in preparation for possible main server failures. The substitute server renders services to customers while the main server is repaired. The service rate of the substitute server is different from (probably lower than) that of the main server. At the instant of the repair completion, the main server returns to the system and becomes available. This is the concept of the working breakdowns first introduced by Kalidass and Ramanath [25]. By allowing the substitute server to provide services, the system has the capability of handling emergencies which may occur during the repair period and the system utilization is optimized. Additionally, the working breakdown service can decrease complaints from the customers who should wait for the main server to be, repaired and reduces the cost of waiting customers or jobs. Therefore, the working breakdown service is a more reasonable repair policy for unreliable queueing systems. The rest of this paper is organized as follows. In Section 2, we explain the concept of the working breakdown in detail and describe the mathematical model. In Section 3, general results on the cycle analysis are presented. We derive the system size distribution and the sojourn time distribution in Sections 4 and 5, respectively. Section 6 deals with numerical works which we conducted to examine the relation between the sojourn time and some system parameters. 2. Model description In this paper, we consider a queueing system with the following features. The arrival process of customers is a Poisson process with a rate of k. The interarrival times, denoted by A, are exponentially distributed. Customers are served in the order of their arrival, i.e., first-come first-served (FCFS) discipline. We define the service rendered by the main server as normal service. The normal service times, denoted by S1, are independent and identically distributed (i.i.d.) random variables. A density and its Laplace-Stieltjes transform (LST) are respectively denoted by s1(x)dx = Pr {x < S1 < x + dx} and R1 S1 ðhÞ ¼ 0 ehx s1 ðxÞdx. Disasters only occur while the main server is in operation. The interarrival times of disasters, denoted by D, are exponentially distributed with a rate of d. Whenever a disaster occurs, the main server fails and all present customers are forced to leave the system. As soon as the main server fails, it undergoes a repair procedure. The repair times, denoted by R, have an exponential distribution with a rate of c. The repaired server is assumed to be as good as a new server. We now introduce the concept of the working breakdowns in detail. As soon as a disaster arrives at the system, the main server fails and a repair process immediately begins. During a repair period, the stream of new customer arrivals continues. The substitute server renders services to customers while the main server is repaired. The service rate of the substitute server is lower than that of the main server. We define the service rendered by the substitute server as the working breakdown service. The working breakdown service times, denoted by S0, are i.i.d. random variables. The density and its LST are respecR1 tively denoted by s0(x)dx = Pr {x < S0 < x + dx} and S0 ðhÞ ¼ 0 ehx s0 ðxÞdx. If there are customers in the system at the end of the repair, the substitute server stops service and the main server restarts and operates at its normal service rate. We assume that the service interrupted at the end of the repair is lost, and it is restarted with a different distribution at the beginning of the following normal service period. Meanwhile, if there are no customers in the system at the end of the repair, the main server returns to the system, stays idle, and waits for arriving customers. We further assume that A, S0, S1, D, and R are mutually independent. 3. Cycle analysis In this section, we perform a (regeneration) cycle analysis. To avoid terminological confusion, we define the normal busy period as the time interval during which the main server is busy, and the working breakdown busy period as the time interval during which the substitute server is busy. In this section, X⁄(h) denotes the LST of any continuous random variable X. A cycle is defined as the time interval between two successive epochs at which a disaster occurs. Let C denote the length of a cycle. There are two types of cycles, depending on customer arrivals, which occur during a repair period. The cycle is called type-1 if no customers arrive during the repair period, and type-2 if customer arrivals occur during the repair period. Let Ci denote the length of the type-i cycle and fi, i = 1, 2, denote the probability that the cycle is of type-i. Then, we have
f1 ¼ PrfA > Rg ¼
c k ; f ¼ PrfA < Rg ¼ : cþk 2 cþk
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Let BN and BR respectively denote the length of the normal busy period and the length of the working breakdown busy period. Their LSTs are given by
BN ðhÞ ¼ S1 ðh þ k kBN ðhÞÞ; BR ðhÞ ¼ S0 ðh þ k kBR ðhÞÞ:
ð1Þ
Since, in the type-1 cycle, no customers arrive during the repair period, our system becomes the standard M/G/1 queueing system immediately after the repair completion (see Fig. 1). Once a customer arrives after the repair completion, the normal busy period begins. If a disaster occurs before this normal busy period ends, the type-1 cycle ends. If not, our system behaves as the standard M/G/1 queueing system until a disaster occurs. We define the random variable F as the time duration from the end of the repair period to the next occurrence point of a disaster during the type-1 cycle. F is represented as follows:
F¼
TL;
if
D < BN ;
T S þ F; otherwise;
where TS = A + (BN | BN < D) and TL = A + (D | D < BN). T S ðhÞ and T L ðhÞ are then obtained as follows:
T S ðhÞ ¼ A ðhÞE½ehBN jBN < D ¼ T L ðhÞ ¼ A ðhÞE½ehD jD < BN ¼
k BN ðh þ dÞ ; k þ h BN ðdÞ
k 1 BN ðh þ dÞ d : k þ h 1 BN ðdÞ d þ h
From the above expression, F⁄(h) is obtained as
1 BN ðdÞ T L ðhÞ F ðhÞ ¼ : 1 BN ðdÞT S ðhÞ
Expressing C1 in terms of F, we have C1 = (R | R < A) + F. Therefore, C 1 ðhÞ and E[C1] are given by
C 1 ðhÞ ¼
kþc F ðhÞ; kþcþh
E½C 1 ¼
1 d þ k kBN ðdÞ þ : c þ k kdð1 BN ðdÞÞ
ð2Þ
Next, we consider the type-2 cycle. Assuming that at least one customer arrives during the repair period, there are two cases of the type-2 cycle. Case 1. The working breakdown busy period ends before the repair completion. Case 2. The repair is completed before the working breakdown busy period ends. Let C 2;j denote the length of the type-2 cycle in Case j. In addition, let hj, j = 1, 2, denote the probability of Case j. Then, we have
h1 ¼ PrfR > BR g ¼ BR ðcÞ; h2 ¼ PrfR < BR g ¼ 1 BR ðcÞ; C 2 ðhÞ ¼
2 X
hj C 2;j ðhÞ:
j¼1
In Case 1, the length of the remaining cycle is stochastically equivalent to the length of the unconditional cycle, due to the memoryless property (see Fig. 2). This is explained by the fact that the length of the repair period is exponentially distributed. Therefore, we have
C 2;1 ¼ ðAjA < RÞ þ ðBR jBR < RÞ þ C; C 2;1 ðhÞ ¼
k þ c BR ðc þ hÞ C ðhÞ: k þ c þ h BR ðcÞ
Fig. 1. An example of the type-1 cycle.
B.K. Kim, D.H. Lee / Applied Mathematical Modelling 38 (2014) 1788–1798
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Fig. 2. An example of the type-2 cycle: Case 1.
Fig. 3. An example of the type-2 cycle: Case 2.
Let K denote the number of existing customers at the instant of the repair completion. In Case 2, the normal busy period is the busy period which starts with K customers (see Fig. 3). If a disaster occurs before this normal busy period ends, then the cycle is terminated. If not, our system behaves as the standard M/G/1 queueing system until a disaster occurs. Let us define two random variables YS and YL as follows:
Y S ¼ ðBK jBK < DÞ; Y L ¼ ðDjD < BK Þ; where BK is the length of the normal busy period which starts with K customers. With the result of Kim et al. [26], K(z), the PGF of K, is given by
KðzÞ ¼
z BR ðcÞ S0 ðc þ k kzÞ 1 : c þ k kz 1 BR ðcÞ S0 ðc þ k kzÞ z
cz
Hence, we obtain BK ðhÞ as follows:
BK ðhÞ ¼ KðzÞjz¼B ðhÞ ¼ N
cBN ðhÞ cþk
kBN ðhÞ
BN ðhÞ BR ðcÞ S0 ðc þ k kBN ðhÞÞ 1 : 1 BR ðcÞ S0 ðc þ k kBN ðhÞÞ BN ðhÞ
ð3Þ
From (3), Y S ðhÞ and Y L ðhÞ are respectively given by
B ðh þ dÞ 1 BK ðh þ dÞ d Y S ðhÞ ¼ E½ehBK BK < D ¼ K ; Y L ðhÞ ¼ E½ehD D < BK ¼ : BK ðdÞ 1 BK ðdÞ d þ h Using F defined in the type-1 cycle, we can obtain the length of the remaining cycle after the repair completion. This is denoted as G, which is expressed as
G¼
YL;
if
D < BK ;
Y S þ F; otherwise:
From above, G⁄(h) is obtained as follows:
G ðhÞ ¼ ð1 BK ðdÞÞY L ðhÞ þ BK ðdÞY S ðhÞF ðhÞ ¼ ð1 BK ðd þ hÞÞ
d þ BK ðd þ hÞF ðhÞ: dþh
Therefore, we have
C 2;2 ¼ ðAjA < RÞ þ ðRjR < BR Þ þ G; C 2;2 ðhÞ ¼
k þ c 1 BR ðc þ hÞ c G ðhÞ: k þ c þ h 1 BR ðcÞ c þ h
Combining the two cases, we have C 2 ðhÞ and E[C2], which are determined as follows:
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k þ c BR ðc þ hÞ k þ c 1 BR ðc þ hÞ c C ðhÞ þ ð1 BR ðcÞÞ G ðhÞ; k þ c þ h BR ðcÞ k þ c þ h 1 BR ðcÞ c þ h 1 1 1 BK ðdÞ E½C 2 ¼ : þ BR ðcÞE½C þ ð1 BR ðcÞÞ þ þ cþk c d k kBN ðdÞ
C 2 ðhÞ ¼ BR ðcÞ
ð4Þ
By combining (2) and (4), we obtain
C ðhÞ ¼
2 2 X X 1 1 c þ kBK ðdÞð1 BR ðcÞÞ fi C i ðhÞ; E½C ¼ fk E½C k ¼ þ þ : d c kð1 BN ðdÞÞðc þ k kBR ðcÞÞ i¼1 k¼1
ð5Þ
Let us respectively define PR and PA as the probability that the main server is under repair and the probability that the main server is available. From the renewal reward theorem, we have
PR ¼
E½R kdð1 BN ðdÞÞðc þ k kBR ðcÞÞ ; ¼ E½C kðd þ cÞð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
PA ¼ 1 PR ¼
kcð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ : kðd þ cÞð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
ð6Þ
ð7Þ
4. System size distribution Let N(t) be the number of customers in the system at t and n(t) be an indicator random variable given by
nðtÞ ¼
0; The main server is under repair at t; 1; The main server is available at t:
Let SR,i(t) denote the remaining service time at t when n(t) = i, i e {0, 1}. Then, the vector process fNðtÞ; nðtÞ; SR;i ðtÞ; t P 0g becomes a Markov process, in which the supplementary variables are SR,i(t). To derive the PGF of the steady state system size, we define the following limiting probabilities for i e {0, 1} :
P0;i ¼ limPrfNðtÞ ¼ 0; nðtÞ ¼ ig; Pn;i ðxÞdx ¼ limPrfNðtÞ ¼ n; nðtÞ ¼ i; x < SR;i ðtÞ < x þ dxg; n P 1: t!1
t!1
Using these probabilities, the Kolmogorov equations for the system size distribution are given by
0 ¼ ðk þ cÞP0;0 þ P1;0 ð0Þ þ d
1 X Pn;1 ;
ð8Þ
n¼1
d P1;0 ðxÞ ¼ kP0;0 s0 ðxÞ ðk þ cÞP1;0 ðxÞ þ P 2;0 ð0Þs0 ðxÞ; dx
d Pn;0 ðxÞ ¼ kPn1;0 ðxÞ ðk þ cÞPn;0 ðxÞ þ Pnþ1;0 ð0Þs0 ðxÞ; n P 2; dx
ð9Þ
0 ¼ cP0;0 kP 0;1 þ P 1;1 ð0Þ;
ð10Þ ð11Þ
d P1;1 ðxÞ ¼ cP1;0 s1 ðxÞ þ kP0;1 s1 ðxÞ ðk þ dÞP1;1 ðxÞ þ P2;1 ð0Þs1 ðxÞ; dx
ð12Þ
d Pn;1 ðxÞ ¼ cPn;0 s1 ðxÞ þ kPn1;1 ðxÞ ðk þ dÞPn;1 ðxÞ þ P nþ1;1 ð0Þs1 ðxÞ; n P 2; dx
ð13Þ
where Pn;i ¼
R1 0
Pn;i ðhÞ ¼
Pn;i ðxÞdx. To solve (8)–(13), we define the following LSTs and PGFs:
Z 0
1
ehx Pn;i ðxÞdx; Pi ðz; hÞ ¼
1 1 X X P n;i ðhÞzn ; Pi ðz; 0Þ ¼ Pn;i ð0Þzn ; i 2 f0; 1g: n¼1
n¼1
Using these notations, we have the normalizing condition as follows:
P0;0 þ P0;1 þ P0 ð1; 0Þ þ P 1 ð1; 0Þ ¼ 1:
ð14Þ
Remark 1. In (14), P0,0 (P0,1) is the probability that there are no customers while the main server is under repair (available). Similarly, P 0 ð1; 0Þ (P 1 ð1; 0Þ) is the probability that the substitute server (the main server) is busy. Therefore, P P1 PR ¼ P 0;0 þ P0 ð1; 0Þ ¼ 1 n¼0 P n;0 and P A ¼ P 0;1 þ P 1 ð1; 0Þ ¼ n¼0 P n;1 .
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Applying LSTs to equations (9), (10), (12), and (13) yields
ðhP1;0 ðhÞ P1;0 ð0ÞÞ ¼ kP0;0 S0 ðhÞ ðk þ cÞP1;0 ðhÞ þ P 2;0 ð0ÞS0 ðhÞ;
ð15Þ
ðhPn;0 ðhÞ Pn;0 ð0ÞÞ ¼ kPn1;0 ðhÞ ðk þ cÞPn;0 ðhÞ þ Pnþ1;0 ð0ÞS0 ðhÞ:
ð16Þ
ðhP1;1 ðhÞ P1;1 ð0ÞÞ ¼ cP1;0 S1 ðhÞ þ kP0;1 S1 ðhÞ ðk þ dÞP 1;1 ðhÞ þ P2;1 ð0ÞS1 ðhÞ;
ð17Þ
ðhPn;1 ðhÞ Pn;1 ð0ÞÞ ¼ cPn;0 S1 ðhÞ þ kPn1;1 ðhÞ ðk þ dÞPn;1 ðhÞ þ Pnþ1;1 ð0ÞS1 ðhÞ; n P 2:
ð18Þ
Substituting h = 0 into (15)–(18) and then summing over n ¼ 1; 2; 3; ; we obtain the following relationships:
P1;0 ð0Þ ¼ kP0;0 cP0 ð1; 0Þ;
ð19Þ
P1;1 ð0Þ ¼ kP0;1 þ cP0 ð1; 0Þ dP1 ð1; 0Þ;
ð20Þ
and from (11) and (20), we have
dP1 ð1; 0Þ ¼ cðP0;0 þ P0 ð1; 0ÞÞ:
ð21Þ
Multiplying (15)–(18) by zn and then summing over n ¼ 1; 2; 3; ; we obtain
ðh c k þ kzÞP0 ðz; hÞ ¼ z1 P0 ðz; 0Þðz S0 ðhÞÞ S0 ðhÞðkP0;0 z P 1;0 ð0ÞÞ; ðh d k þ
kzÞP1 ðz; hÞ
¼ z1 P1 ðz; 0Þðz
S1 ðhÞÞ
S1 ðhÞð
P0 ðz; 0Þ
c
þ kP0;1 z P1;1 ð0ÞÞ:
ð22Þ ð23Þ
Inserting h ¼ c þ k kz into (22) and h ¼ d þ k kz into (23) and then simplifying them with respect to P i ðz; 0Þ, we get
P0 ðz; 0Þ ¼
zS0 ðc þ k kzÞðP1;0 ð0Þ kP0;0 zÞ ; S0 ðc þ k kzÞ z
ð24Þ
P1 ðz; 0Þ ¼
zS1 ðd þ k kzÞðP1;1 ð0Þ cP0 ðz; 0Þ kP0;1 zÞ : S1 ðd þ k kzÞ z
ð25Þ
Substituting (24) and (25) into (22) and (23), respectively, we obtain
P0 ðz; hÞ ¼
zðP1;0 ð0Þ kP0;0 zÞðS0 ðc þ k kzÞ S0 ðhÞÞ ; ðh c k þ kzÞðS0 ðc þ k kzÞ zÞ
ð26Þ
P1 ðz; hÞ ¼
zðP1;1 ð0Þ cP 0 ðz; 0Þ kP0;1 zÞðS1 ðd þ k kzÞ S1 ðhÞÞ : ðh d k þ kzÞðS1 ðd þ k kzÞ zÞ
ð27Þ
Finally, the PGF of the system size, denoted by P(z), is given by
PðzÞ ¼ P0;0 þ P0;1 þ P0 ðz; 0Þ þ P1 ðz; 0Þ ¼ P0;0 þ P0;1 þ þ
z½kP0;0 ð1 zÞ cP0 ð1; 0Þð1 S0 ðc þ k kzÞÞ ðc þ k kzÞðS0 ðc þ k kzÞ zÞ
z½cðP0 ð1; 0Þ P 0 ðz; 0ÞÞ þ kP0;1 ð1 zÞ dP1 ð1; 0Þð1 S1 ðd þ k kzÞÞ : ðd þ k kzÞðS1 ðd þ k kzÞ zÞ
ð28Þ
According to Rouche’s theorem, S0 ðc þ k kzÞ z ¼ 0 has a unique solution, denoted as z0, for |z| < 1 (see Appendix A). If the denominator of P0 ðz; 0Þ is zero when z = z0, its numerator should be zero. Similarly, S1 ðd þ k kzÞ z ¼ 0 has a unique solution, denoted as z1, for |z| < 1. If the denominator of P1 ðz; 0Þ is zero when z = z1, its numerator should be zero. From (1), we can confirm that z0 ¼ BR ðcÞ and z1 ¼ BN ðdÞ. Thus, we obtain
cP0 ð1; 0Þ ¼ kP0;0 ð1 z0 Þ;
ð29Þ
dP1 ð1; 0Þ ¼ kP0;0 ð1 z0 Þ þ kP0;1 ð1 z1 Þ cP0 ðz1 ; 0Þ:
ð30Þ
Solving (6), (14), (21), and (29) simultaneously with respect to P0,0, P0,1, P0 ð1; 0Þ, and P1 ð1; 0Þ, we have
P0;0 ¼
kdcð1 BN ðdÞÞ ; kðd þ cÞð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
P0 ð1; 0Þ ¼
kðd þ cÞð1
k2 dð1 BN ðdÞÞð1 BR ðcÞÞ ; c þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
BN ðdÞÞð
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P0;1 ¼
kðd þ cÞð1
P1 ð1; 0Þ ¼
dc½c þ kBK ðdÞð1 BR ðcÞÞ ; c þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
BN ðdÞÞð
kcð1 BN ðdÞÞðc þ k kBR ðcÞÞ : kðd þ cÞð1 BN ðdÞÞðc þ k kBR ðcÞÞ þ dc½c þ kBK ðdÞð1 BR ðcÞÞ
Remark 2. In our study, the inequality d > 0 is a necessary and sufficient condition for the system to be stable (For details, see Appendix B).
5. Sojourn time distribution In this section, considering a test customer (TC), we derive the LST of the sojourn time distribution which is the waiting time in the queue plus the service time. Let W and W⁄(h) respectively denote the unconditional sojourn time of the TC and its LST. The TC’s arrival can belong to one of the following cases: Case Case Case Case
1. 2. 3. 4.
The The The The
TC TC TC TC
arriving arriving arriving arriving
while while while while
the the the the
main main main main
server server server server
is is is is
available finds no customer waiting. available finds that the server is busy. under repair finds no customer waiting. under repair finds that the substitute server is busy.
Let Wi, i = 1, 2, 3, 4, denote the sojourn time of the TC that arrives in Case i, and define W i ðhÞ ¼ PrfCase igE½ehW i Case i. Notice that W⁄(h) is given by
W ðhÞ ¼ W 1 ðhÞ þ W 2 ðhÞ þ W 3 ðhÞ þ W 4 ðhÞ: In Case 1, the TC arrives at the system while the main server is idle. Upon arrival, he immediately receives the normal service and leaves the system by either the occurrence of a disaster or the service completion. Therefore, we find the following relationships:
W 1 ðhÞ ¼ P0;1 ðPrfS1 < DgE½ehS1 S1 < D þ PrfS1 > DgE½ehD S1 > DÞ S ðh þ dÞ d 1 S1 ðh þ dÞ d þ hS1 ðh þ dÞ ¼ P0;1 PrfS1 < Dg 1 ¼ P0;1 : þ PrfS1 > Dg PrfS1 < Dg d þ h PrfS1 > Dg hþd
ð31Þ
In Case 2, the TC arrives at the system while the main server is busy. Let us define U and U⁄(h) as the unfinished work immediately after the arrival epoch of the TC and its LST. Using P 1 ðz; hÞ in Section 4, U⁄(h) is expressed as P U ðhÞ ¼ P1 ðS1 ðhÞ; hÞ=P1 ð1; 0Þ. Since PrfCase 2g ¼ 1 n¼1 P n;1 and W2 = min {D, U}, we have
W 2 ðhÞ ¼
1 X d þ hU ðh þ dÞ dP1 ð1; 0Þ þ hP1 ðS1 ðh þ dÞ; h þ dÞ Pn;1 E½eh minfD;Ug ¼ P1 ð1; 0Þ ¼ : hþd hþd n¼1
ð32Þ
In Case 3, the TC arriving during the repair period sees no customer in the system. Therefore, he immediately receives the working breakdown service. If the TC’s working breakdown service is completed before the repair completion, his sojourn time is simply his own working breakdown service time. Otherwise, the TC’s working breakdown service is interrupted at the end of the repair period and the new normal service is provided to him. Once he receives the normal service, his remaining sojourn time is stochastically equal to W1. Thus, the TC’s sojourn time is the remaining repair time plus W1. Note that the remaining repair time is stochastically equal to a new repair time due to the memoryless property. Considering all these, we have
W 3 ðhÞ ¼ P0;0 PrfS0 < RgE ehS0 S0 < R þ PrfS0 > RgE ehðRþW 1 Þ S0 > R
c d þ hS1 ðh þ dÞ : ¼ P0;0 S0 ðh þ cÞ þ ð1 S0 ðh þ cÞÞ hþd hþc
ð33Þ
In Case 4, the TC arrives during the repair period and finds at least one customer in the system. Suppose that the arriving TC sees n customers and the customer in working breakdown service leaves the system before the repair completion. Let H denote the sojourn time of the TC when the customer at the head in the queue is on the threshold of receiving the working breakdown service. The TC’s sojourn time is the remaining time of the ongoing working breakdown service when he arrives plus H. Let us consider the following n + 1 cases in accordance with the repair completion epochs. If kcustomers,
B.K. Kim, D.H. Lee / Applied Mathematical Modelling 38 (2014) 1788–1798
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k ¼ 0; 1; 2; ; n, leave the system by the time the repair is complete, there are n k customers who wait for their normal service. In this case, H is equal to the remaining repair time plus either the sum of the n k normal service times or the interarrival time of a disaster. This leads to
H ðhÞ ¼
n1 X nk n PrfSk0 < R < S0kþ1 gE½ehR Sk0 < R < S0kþ1 E½eh minfD;S1 g þ PrfSn0 < RgE½ehS0 Sn0 < R;
ð34Þ
k¼0
where Sni represents the n -fold convolution of Si with itself. The term E½ehR Sk0 < R < S0kþ1 in the right hand side of (34) is given by
E½ehR S0k1 < R < Sk0 ¼
c ð1 S0 ðh þ cÞÞðS0 ðh þ cÞÞk1 : hþc PrfSk1 < R < Sk0 g 0
Rewriting (34), we have
H ðhÞ ¼ ¼
n X cð1 S0 ðh þ cÞÞðS0 ðh þ cÞÞk1 d þ hðS1 ðh þ dÞÞnkþ1 n þ ðS0 ðh þ cÞÞ hþc hþd k¼1 n n cd½1 ðS0 ðh þ cÞÞn chð1 S0 ðh þ cÞÞ ðS1 ðh þ dÞÞ ðS0 ðh þ cÞÞ n þ ðS0 ðh þ cÞÞ : þ ðh þ cÞðh þ dÞ ðh þ cÞðh þ dÞ ðS1 ðh þ dÞ S0 ðh þ cÞÞðS1 ðh þ dÞÞ1
ð35Þ
If the remaining repair time is shorter than the remaining time of the ongoing working breakdown service, the TC’s sojourn time is the remaining repair time plus either the sum of the normal service times of the customers who are in the system or the interarrival time of a disaster. Therefore, we obtain
W 4 ðhÞ ¼
1 Z X n¼1
1
1 X P n;0 ðxÞPrfR > SR;0 SR;0 ¼ xgH ðhÞehx dx þ
x¼0
n¼1
Z
1
Pn;0 ðxÞ x¼0
Z
x
Prfy < R < y þ dyjSR;0
y¼0
nþ1
¼ xgE½eh minfD;S1 g ehy dx 1 Z 1 1 Z X X ¼ H ðhÞ P n;0 ðxÞeðhþcÞx dx þ n¼1
x¼0
n¼1
1
Pn;0 ðxÞð1 eðhþcÞx Þ
x¼0
c d þ hðS1 ðh þ dÞÞnþ1 dx hþc hþd
c d þ hðS1 ðh þ dÞÞnþ1 hþc hþd n¼1 n¼1 c dðP ð1; h þ c Þ P ðS ðh þ dÞ; h þ c ÞÞ chS1 ðh þ dÞð1 S0 ðh þ cÞÞ 0 0 1 ¼ P0 ðS0 ðh þ cÞ; h þ cÞ þ þ ðh þ cÞðh þ dÞ ðh þ cÞðh þ dÞ P ðS ðh þ dÞ; h þ cÞ P0 ðS0 ðh þ cÞ; h þ cÞ 0 1 S1 ðh þ dÞ S0 ðh þ cÞ cdðP 0 ð1; 0Þ P 0 ð1; h þ cÞÞ þ chS1 ðh þ dÞðP0 ðS1 ðh þ dÞ; 0Þ P 0 ðS1 ðh þ dÞ; h þ cÞÞ ; þ ðh þ cÞðh þ dÞ 1 1 X X ðPn;0 Pn;0 ðh þ mÞÞ ¼ H ðhÞ Pn;0 ðh þ cÞ þ
ð36Þ
where SR,0 = limt?1SR,0(t). Remark 3. Now that we deal with the Poisson arrival process queueing system, PASTA property [27] is employed to derive W i ðhÞ. Remark 4. We can confirm the result of the Little’s formula.
6. Numerical illustration In this section, we provide numerical works related to our model. We examine the relationship between the mean sojourn time and the mean interarrival time of disasters in Fig. 4. Also, we show how the mean repair time influences the mean sojourn time in Fig. 5. In each figure, we compare the mean sojourn time of the disastrous queue with working breakdown services (DST + WR) and that of the disastrous queue with non-working breakdown services (DST + NWR). We fix the system parameters in both cases. Customer arrivals are generated according to a Poisson process at a rate of 0.8. We use an exponential distribution with a rate of 0.5 for the working breakdown service time. The normal service time follows a hyperexponential distribution with the probability density function pk1 ek1 t þ ð1 pÞk2 ek2 t , where p = 0.5, k1 ¼ 4=3, and k2 ¼ 0:8. In Fig. 4, the vertical axis of the graph represents the mean sojourn time, E[W], and the horizontal axis shows the mean interarrival time of a disaster, d1. We assume that the repair time distributions follow an exponential distribution with a
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Fig. 4. Mean sojourn time of the M/G/1 over the mean interarrival time of disasters.
Fig. 5. Mean sojourn time of the M/G/1 over the mean repair time.
rate of 0.5. Fig. 4 shows that E[W] increases as d1 increases. We can expect that a larger value of d1 causes fewer customers to be forced to leave the system by a disaster, thus increasing the probability of a larger system size and a longer sojourn time. In Fig. 5, the vertical axis is E[W] and the horizontal axis is the mean repair time, c1. We assume that the interarrival time of a disaster follows an exponential distribution with a rate of 0.5. Fig. 5 confirms that E[W] increases with an increase of c1. This indicates that a longer mean repair time leads to a greater number of customers who are provided with the working breakdown service with a rate that is lower than that of the normal service. Finally, we verify that the working breakdown service model is superior to the non-working breakdown service model from both figures. Especially in Fig. 5, as c1 increases, the gap of E[W] between the two models becomes larger. This is very intuitive as, in the working breakdown service model, customers can leave the system during the repair period via the completion of their service. In addition, we observe the mean sojourn time of a TC under the Case i, which is defined in Section 5. We examine the relationship between E½W i jCase i and d1 in Fig. 6 and that between E½W i jCase i and c1 in Fig. 7, respectively. In Fig. 6, the vertical axis of the graph represents the E½W i jCase i and the horizontal axis shows d1. We assume that the repair time distributions follow an exponential distribution with a rate of 0.5. Fig. 6 shows that E½W 1 jCase 1 and E½W 2 jCase 2 increase as d1 increases. We can expect that a larger value of d1 causes fewer customers to be forced to leave the system by a disaster. On the other hand, E½W 3 jCase 3 and E½W 4 jCase 4 are almost unchanged as d1 increases. This indicates that d1 hardly affects the sojourn time of the customers who arrive at the system during repair period. In Fig. 7, the vertical axis is E½W i jCase i and the horizontal axis is c1. We assume that the interarrival time of a disaster follows an exponential distribution with a rate of 0.5. Fig. 7 reveals that E½W 3 jCase 3 and E½W 4 jCase 4 increase with an increase of c1 because a longer mean repair time leads to a greater number of customers who are provided with the working breakdown service at a rate that is lower than that of the normal service. On the other hand, E½W 1 jCase 1 and E½W 2 jCase 2 are almost unchanged as c1 increases. We can expect that a longer mean repair time has little effect on the sojourn time of the customers who arrive at the system during a normal service period.
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Fig. 6. Mean sojourn times of four cases over the mean interarrival time of disasters.
Fig. 7. Mean sojourn times of four cases over the mean repair time.
7. Conclusion In this paper, we analyzed an M/G/1 queue with disasters and the working breakdown service. We derived the results of the important system characteristics of the PGF of the system size and the LST of the FCFS sojourn time. We also carried out a cycle analysis. Finally, numerical works were performed to show the relationship between the mean sojourn time and other system parameters, such as the mean interarrival time of a disaster or the mean repair time. Our study presents an extension of the repairable queueing system and its results may provide a better decision-making tool for the repair arising policies in many practical systems. For future studies, we can extend this model to more complex situations such as the repairable queueing systems with a Markovian arrival process of customers or disasters and a Markovian service process of normal services and working breakdown services. Acknowledgement This research was funded by the MSIP (Ministry of Science, ICT & Future Planning), Korea in the ICT R&D Program 2013 (No. 2013-005-021-001). Appendix A. Application of Rouche’s theorem Let us define S0 ðc þ k kzÞ z as /(z), which is an analytic function in the unit circle |z| < 1. Suppose f(z) = z and gðzÞ ¼ S0 ðc þ k kzÞ, which are all analytic. It can be shown that |g(z)| < |f(z)| on the contour of the circle because
jf ðzÞj ¼ jzj ¼ 1; jgðzÞj 6 gðjzjÞ ¼ S0 ðc þ k kjzjÞ ¼ S0 ðcÞ: Hence, from Rouche’s theorem, it follows that f(z) and f(z) + g(z) will have the same number of zeros inside |z| < 1. Since f(z) has only one zero inside this circle, f(z) + g(z) /(z) will also have only one zero inside |z| < 1. In the same manner, we can show that S1 ðd þ k kzÞ z ¼ 0 has a unique solution for |z| < 1.
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Appendix B. Proof of the stability condition As long as d > 0, then the system under our study is stable. Let us define mk as the joint probability that k customers arrive at the system during a normal service time and a disaster does not occur while this normal service time is ongoing. If we P k assume that MðzÞ ¼ 1 k¼0 z mk , mk and M(z) can be respectively expressed as
mk ¼
Z 0
1
1 X ðkxÞk eðkþdÞx s1 ðxÞdx; k P 0; MðzÞ ¼ zk mk ¼ S1 ðd þ k kzÞ; k! k¼0
Note that Mð0Þ ¼ S1 ðd þ kÞ and Mð1Þ ¼ S1 ðdÞ. For all z within (0, 1), we have 2
1 1 dMðzÞ X d MðzÞ X k1 ¼ kz mk > 0; ¼ kðk 1Þzk2 mk > 0: 2 dz dz k¼1 k¼2
ðB:1Þ
(B.1) implies that M(z) is a convex increasing function. We can also confirm that M(0) < M(1), as dS1 ðhÞ=dh < 0. Thus, from (1), BN ðdÞ is the unique root of MðBN ðdÞÞ ¼ BN ðdÞ within (0, 1) if the following inequality holds:
0 < Mð0Þ < Mð1Þ < 1:
ðB:2Þ
Therefore, d should be a non-zero positive number for (B.2) to hold. This is a sufficient condition. On the other hand, if 0 < BN ðdÞ < 1, then 0 < MðBN ðdÞÞ < 1. This implies that d þ kð1 BN ðdÞÞ > 0. Therefore, under the condition of d > 0, it is proven that E[C] in (5) has a finite value because the related values of c1, BR ðcÞ, and BK ðdÞ in (5) are all positive. This is a necessary condition for stability. In fact, all customers in the system are flushed out whenever disasters occur, which means that the number of customers at arbitrary epochs does not go to infinity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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