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1 queue with single and multiple working vacations
Computers & Industrial Engineering 65 (2013) 207–215
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Impatient customers in an M/M/1 queue with single and multiple working vacations N. Selvaraju ⇑, Cosmika Goswami Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781 039, India
a r t i c l e
i n f o
Article history: Received 7 May 2012 Received in revised form 4 December 2012 Accepted 20 February 2013 Available online 1 March 2013 Keywords: Markovian queues Impatient customers Multiple working vacations Single working vacations Performance measures Stochastic decomposition
a b s t r a c t We consider an M/M/1 queue with impatient customers and two different types of working vacations. The working vacation policy is the one in which the server serves at a lower rate during a vacation period rather than completely stop serving. The customer’s impatience is due to its arrival during a working vacation period, in which the customer service rate is lower than the normal busy period. We analyze the queue for two different working vacation termination policies, a multiple working vacation policy and a single working vacation policy. Closed-form solutions and various performance measures like, the mean queue lengths and the mean waiting times are derived. The stochastic decomposition properties are verified for both multiple and single working vacation cases. A comparison of both the models is carried out to capture their performances with the change in system parameters. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In a communication network various types of jobs, like voice packets or data files, have to wait in a queue if they do not get service immediately upon their arrival in the network. The jobs can become impatient due to high waiting times or due to uncertainty of receiving services and may leave the system unserved. This scenario often occurs in Optical Burst Switching (OBS) networks. This OBS technology is in high demand as it supports various QoS functions that facilitate low data-loss rate, low latency, survivability and is very efficient for communication networks which require high transmission rates. For example, in industrial robotics there are several protocols that integrate control services, communication services, and routing capabilities (sending documents or materials to appropriate destinations) based on Local Area Network (LAN)s and the Internet. Accessing data through OBS networks adds a further level of network management flexibility by handling increasing amount of traffic as in such industries the capability of providing access to data and information on shared storage devices is an important feature, O’Mahony, Simeonidou, Hunter, and Tzanakaki (2001). The OBS is a technology for reducing the gap between transmissions and switching speeds. In OBS, incoming traffic from clients at the edge of the network is aggregated at the ingress of the network according to a particular parameter. This is then assembled and is further transmitted ⇑ Corresponding author. Tel.: +91 361 2582611; fax: +91 361 2582649. E-mail address: [email protected] (N. Selvaraju). 0360-8352/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.02.016
through Wavelength Division Multiplexing (WDM) links. The operation of an OBS controller can be seen as a queue with reneging or impatience, studied in Bocquet (2005). In OBS networks, a control packet is sent first, on a separate signaling channel, to set up a connection. It is followed by a data burst without waiting for an acknowledgment for path establishment, Qiao and Yoo (1999). In particular, when a path is not assigned, the burst control packet is accepted to the queue and is kept waiting for a path. If its delay budget is lower than the effective processing delay, the packet becomes impatient and leaves the system unserved. To have more efficient use of such a network, the loss of packets has to be reduced for its better performance. For example, in robotics industry if packets are lost during data transferring or communicating, there can create serious issues regarding the data related performance of the computers and machines. In this paper, a study on impatience of a request in WDM links is done as it seems to be essential for the benefit of better system performance. Servi and Finn (2002) were the first to observe a reconfigurable WDM network as a new policy called Working Vacation (WV) model. This WDM optical access network with multiple wavelengths was viewed as a generalization of a multi-queue cyclic service model where a single token moves cyclically from one queue to another. Here each queue operates either at a fast service rate or a normal service rate. In a classical vacation queueing model, a server completely stops serving the customers in queue, during the vacation period and made the customer wait for service till the vacation period ends and the server gets back to its normal service period. But rather than cease the service completely during
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its vacation period, if the server serves at a lower rate, we get this new vacation policy, the WV policy, Goswami and Selvaraju (2010). There is quite a difference between WV queue and classical vacation queue. During a vacation, customers in the former may finish service and depart the system, however, customers in the latter cannot depart the system after getting service. On the other hand, during a vacation, the number of customers in the former can increase or decrease, however, the number of customers in the latter can only increase. Therefore, the WV models have more complicated modalities and the analysis of this kind of models is more difficult than the classical vacation queues. Servi and Finn considered n-queue generalization of a cyclic service queueing model where each queue can be viewed as an M/M/1 model with WV. Later, Liu, Xu, and Tian (2007) analyzed the same M/M/1/WV model using the quasi-birth death and matrix-geometric method to obtain explicit expressions of the performance measures and their stochastic decompositions. Wu and Takagi (2006) extended Servi and Finn’s work to M/G/1/WV model. Baba (2005) considered the GI/M/1/WV system with general independent arrival process and the distribution of the vacation duration times and service times as exponential. Wang (2009) studied the M/M/1 queue with WV and non-zero switching times and studied a cyclic service system in WDM-based access networks with reconfigurable delay. Tian, Zhao, and Wang (2008) have studied the M/M/1 queue with single WV. Banik (2010) analyzed single working vacation in GI/M/1/N and GI/M/1/1 queueing systems considering the service time and vacation times as exponentially distributed. Recently, Laxmi and Yesuf (2011) presented an GI/M/1 batch service queue with exponential working vacation policy. Ke, Wu, and Zhang (2010) have given a short survey on recent developments on vacation models. There are situations where the customer impatience is due to the absence of the server, more precisely, due to the server vacation and is independent of the customers in system. Altman and Yechiali (2006) studied the impatience of customers in vacation systems of M/M/1, M/G/1 and also of M/M/c queues, where each arriving customer who finds no server on duty, activates an independent random impatient timer. If a server does not return from vacation by the time the timer expires, the customer abandons the queue and never returns. If the server returns from its vacation before the timer expires, the customer stays in the system until his service is completed. Keeping in view the importance of impatience in the context of WDM networks, we consider here impatient customers in an M/M/ 1 queue under the WV policy. We analyze the queue for two different WV termination policies, the multiple working vacation (MWV) policy and the single working vacation (SWV) policy. A MWV policy requires the server to keep taking vacations until it finds at least one customer waiting in the system at a vacation completion instant. Under a SWV policy, the server takes only one vacation at the end of each busy period. As this single vacation ends, the server either serves the customers at the rate of normal busy period, if any, or stays idle. Recently, Yue, Yue, and Xu (2011) studied M/M/1 model with MWV where the customer’s impatience is due to WV. They have studied the model in which during WV all the customers in the system become impatient and if the customer’s service has not been completed before the customer’s impatient timer expires, the customer abandons the queue. We consider here a more constrained model where the customer under service, during WV, never becomes impatient but only the ones waiting in queue are reluctant to abandon the system. This imposed condition can be seen in reality as a customer receiving service seldom wants to abandon the system. This restriction turns the model given by Yue et al. into a much more complicated and analytically challenged one. We have analyzed this constrained model not only
under MWV policy but also under SWV policy. This policies corresponds to reconfiguration or wavelength assignment policies over WDM links used to control congestion in OBS networks. It is important to find the optimal reconfiguration policy for the network in order to optimize the usage of network resources since an unnecessary rearrangement or postponing a necessary reconfiguration, effects on overall performance of the network, Coudert et al. (2009). The purpose of this analysis is twofold: to obtain analytical solutions to the models and to have a comparison between the two models. We want to study the effect of packet delay and packet loss in OBS network. We seek answers to queries like – do different wavelength assignment policies in WDM links influence the system performance? If they do, which policy should be adopted for minimum packet loss and under what condition? How the parameter threshold like delay budget increases packet loss and how to have a balance between them? We first formulate the impatient M/M/1 model (for both MWV and SWV cases) as a quasi-birth–death process and derive the partial generating functions for the distribution of queue sizes when the server is in WV period and when in regular busy period. Solving the arising differential equations, we find the closed-form solutions (in terms of Beta and hypergeometric functions) and various performance measures like, the mean queue length and the mean waiting time for the both the WV termination policies. Stochastic decomposition properties are also verified for both MWV and SWV cases. The paper is organized as follows. In Section 2, we provide the description of the M/M/1 model with MWV. We formulate the model as a quasi-birth–death process and from the balance equations, the explicit expressions of the various performance measures are derived. Section 3 gives the analysis of the M/M/1 SWV model with the stochastic decomposition property. In Section 4, we have given the comparison of the distribution of number of customers when the system follows MWV policy and SWV policy.
2. Multiple working vacation (MWV) model We consider a system where the arrivals occur according to a Poisson process with rate k. The service times during the normal busy period is exponentially distributed with mean 1/lb, where we assume the stability condition that q = k/lb < 1. The server takes a WV as soon as the system becomes empty. During the vacation period, the server serves the customers at an exponential rate lv, where lv < lb. The duration of a vacation is also exponential with rate h. A MWV policy requires the server to keep taking vacations until it finds at least one customer waiting in the system at a vacation completion instant. When the server returns from its vacation and finds at least one customer in the system, the server switches its service rate from lv to lb and a non-vacation period starts; otherwise it immediately leaves for another WV. If the vacation terminates in between an ongoing service, the server switches its service rate and continues the service at the higher rate until completion. The customers in this system are assumed to be impatient as described below. Whenever arriving customer has to join a queue and finds the system in WV, the customer activates an impatient timer ‘T’, which is exponential with rate n and is independent of the number of customers in the system at that moment. If the server ends the vacation before the time T expires, the customer stays in the system till his service is completed; otherwise, the customer leaves the system and never returns. Thus, only the customers who arrive during a WV period of the server are impatient. A customer who arrives during a WV period and finds the server free, it starts getting service immediately upon it’s arrival and therefore it does not become impatient. This type of impatience under WV policy has not been studied before. This is different from the impatient
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policy studied in Altman and Yechiali (2006), in which all customers arriving during vacation period activate impatient timers, since they consider a pure vacation policy. The server serves the customers on a first-come-first-served (FCFS) basis. The interarrival times, service times, vacation duration times and the impatient times, all are taken to be mutually independent. The service system as described above can be modelled by a two-dimensional continuous-time discrete-state Markov chain for the state of the system and is denoted by D = {(Nt, Jt), t P 0}, where Nt denotes the total number of customers in the system and Jt denotes the state of the server with
where
nk k k ; n P 0; l lv n ðn þ nv Þk þ n k 1 ðn kÞ! k¼0 n lv h nk nk n n k þ þ X n n ð1Þ k k ; n P 1; HðnÞ ¼ l v lv n þ n k ðn kÞ! ðn þ n þ 1Þk k¼0 n k l h k l h and Cð1Þ ¼ K ; v 1; þ 1 : Að1Þ ¼ K ; v ; n n n n n n n X
JðnÞ¼
nk
ð1Þ
nkþ
lv n
hn
We get, E = {{(0, 0)} [ {(i, j)}, i = 1, 2, . . . , j = 0, 1} as the state space of the above Markov chain.
In the above, the quantity K is given by K(a, b, c) = B(b, c) 1 F 1 (c;b + c;a) R1 with the Beta function, Bðb; cÞ ¼ 0 t ðb1Þ ð1 tÞc1 dt; b > 0; c > 0; the P ðaÞk zk degenerate hypergeometric function, 1 F 1 ða; b; zÞ ¼ 1 k¼0 ðbÞk k! ; and the CðaþkÞ Pochhammer symbol, ðaÞk ¼ CðaÞ .
2.1. The stationary distribution
Proof. Multiplying (2) with zn and summing over n gives
Jt ¼
1; if the server is in non-vacation period at time t; 0;
if the server is in WV period at time t:
Let us define the stationary probabilities for the Markov chain D
1 1 1 X X X ½k þ lv þ h þ ðn 1Þn zn pn;0 ¼ k zn pn1;0 þ ðlv n¼1
as
pi;j ¼ PfNt ¼ i; J t ¼ jg;
i ¼ 0; 1; 2; . . . j ¼ 0; 1:
½k þ lv þ h þ ðn 1Þnpn;0
kp0;0 ¼ lv p1;0 þ lb p1;1 ;if n ¼ 0; ð1Þ ¼ kpn1;0 þ ðlv þ nnÞpnþ1;0 ;if n P 1; ð2Þ
ðk þ lb Þp1;1 ¼ hp1;0 þ lb p2;1 ;if n ¼ 1;
ð3Þ
ðk þ lb Þpn;1 ¼ kpðn1Þ;1 þ hpn;0 þ lb pnþ1;1 ;if n P 2: ð4Þ Let us define the partial probability generating functions (PGFs), for 0
n¼0
1 X P1 ðzÞ ¼ zn pn;1 ; n¼1
with P0(1) + P1(1) = 1 and P 00 ðzÞ ¼ n
n¼1
and rearranging the terms gives rise to the differential equation,
Then, the balance equations are
1 X P0 ðzÞ ¼ zn pn;0 ;
n¼1
þ nnÞzn pnþ1;0
P1
n¼1 z
nzð1 zÞP00 ðzÞ ½ð1 zÞðkz lv þ nÞ þ zhP0 ðzÞ þ ½zh ð1 zÞðlv nÞp0;0 þ zlb p1;1 ¼ 0: For n = 0, we get from the above equation
l ð1 zÞp0;0 z½lb p1;1 þ hp0;0 P0 ðzÞ ¼ v 2 ; kz zðk þ lv þ hÞ þ lv
pn;0 .
þ Remark 2.1. If n = 0, the current model reduces to M/M/1 queues with working vacations studied by Servi and Finn (2002). If we assume that the packet arrival rate into the network is lower than the fast service rate (k < lb), that the network does not explode, we can derive the relation between the total number of packets in the network, type of service provided by the network and the chance of packets being lost in the system by finding the PGFs of the sequence of joint probabilities pi, j. The following theorem derives the PGFs in terms of p0,0, the probability of the network being idle. Theorem 2.1. For q < 1 and n < lv, the (partial) PGFs can be expressed in terms of p0,0 as
ð8Þ
which is same with Servi and Finn, see Servi and Finn (2002) in Eq. (A.3). For n – 0,
P00 ðzÞ n1
ð7Þ
kz lv þ n h h ðl nÞ P0 ðzÞ þ p0;0 þ v zn nð1 zÞ nð1 zÞ zn
lb nð1 zÞ
p1;1 ¼ 0:
ð9Þ
To solve the above first order linear differential equation, an inteR k lv h lv k ½ þ dz grating factor can be found as I:F: ¼ e n zn nð1zÞ ¼ enz zð n 1Þ h
ð1 zÞn . The general solution to the differential equation is given by lv
P0 ðzÞ ¼ zð n 1Þ ð1 zÞn h
h l 1 p0;0 CðzÞ p0;0 þ b p1;1 AðzÞ ; n n n
lv
ð10Þ with
Z
z
lv
k
h
enðzxÞ x n 1 ð1 xÞn1 dx 0 Z z lv h k enðzxÞ x n 2 ð1 xÞn dx: and CðzÞ ¼
AðzÞ ¼
ð11Þ
"X 1 X n nr 1 X n nr k JðrÞ n X k z P0 ðzÞ ¼ p0;0 1 n ðn rÞ! n n n¼0 r¼0 n¼1 r¼1 1 Cð1Þ ð5Þ Jðr 1Þ þ Hðr 1Þ zn ðn rÞ! Að1Þ
To get p1,1 in terms of p0,0, let us determine (11) and (12) for limit z tending to 1. We have the identity, from Abramowitz and Stegun (1972), for Re(u) > 0, Re(v) > 0,
and
Z
lv
1 X p0;0 ðl nÞCð1Þ 1 qn n h lv P1 ðzÞ ¼ v z p0;0 1 lb Að1Þ 1 q l n 1 q b n¼1 " nsr 1 n ns 1 X n X ns XXX X k JðrÞ zn ð1 qÞs n ðn s rÞ! n¼0 s¼1 r¼0 n¼1 s¼1 r¼1 3 nsr k n Cð1Þ 7 ð6Þ Hðr 1Þ zn 5; ð1qÞs Jðr 1Þþ Að1Þ ðn s rÞ!
ð12Þ
0
w
xv 1 ðw xÞu1 ebx dx ¼ Bðu; v Þwuþv 1 1 F 1 ðv ; u þ v ; bwÞ;
ð13Þ
0
R1
tðb1Þ ð1 tÞc1 dt; b > 0; c > 0; the P ðaÞk zk degenerate hypergeometric function 1 F 1 ða; b; zÞ ¼ 1 k¼0 ðbÞ k! and with the Beta function Bðb; cÞ ¼
0
k
the Pochhammer symbol ðaÞk ¼ CCðaðþkÞ aÞ . Substituting z = 1, 1 x = t and using identity (13), in Eqs. (11) and (12), we get
Að1Þ ¼ K
k lv h k l h ; ; and Cð1Þ ¼ K ; v 1; þ 1 ; n n n n n n
ð14Þ
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h
ðlv nÞCð1Þ zp0;0 ð1 zÞ1 ð1 qzÞ1 : lb Að1Þ
where K(a, b, c) = B(b, c)1F1(c;b + c;a). C(1) is valid only if lv 1 > 0, which leads to the assumption lv > n. Now, determinn
P1 ðzÞ ¼
ing P0(z) for limit z tending to 1 gives,
Further we can expand it in series using P0(z) as
lim P 0 ðzÞ ¼ P0 ð1Þ z!1 h lv h l ¼ 1 p0;0 Cð1Þ p0;0 þ b p1;1 Að1Þ limð1zÞn : z!1 n n n
P1 ðzÞ ¼
Since 0 6 P 0 ð1Þ ¼ have the term
P1
n¼0 pn;0
h
h l p0;0 þ b p1;1 Að1Þ ¼ 0 1 p0;0 Cð1Þ n n n
lv
p1;1
ðlv nÞCð1Þ h p : ¼ lb Að1Þ lb 0;0
ð15Þ
Applying the above relation in (10) we get
P0 ðzÞ ¼ p0;0
lv n
1
lv h Cð1Þ AðzÞ zð n 1Þ ð1 zÞn : CðzÞ Að1Þ
n 1 X ð1Þn k l h k AðzÞ ¼ enz B z; n þ v ; : n n! n n n¼0
ð16Þ
Expressing the incomplete Beta function in terms of Gauss hypergeometric function and expanding the hypergeometric function in series gives h
k
lv
AðzÞ ¼ enz ð1 zÞn z n 1
1 X HðnÞznþ1 ;
ð17Þ
n¼0
"X n nr n nr X k JðrÞ k 1 ¼ p0;0 1 n ðn rÞ! n ðn rÞ! n r¼0 r¼1 Cð1Þ Jðr 1Þ þ Hðr 1Þ : Að1Þ
lv
k Cð1Þ 1 Jð0Þ þ Jð1Þ Hð0Þ 1 n Að1Þ n k h ðlv nÞCð1Þ : ¼ p0;0 lv lv Að1Þ
p1;0 ¼ p0;0
pn;1 ¼
n k þ lnv þ hn knk ð1Þnk k HðnÞ ¼ : lv ðn k þ lnv þ 1Þk n þ n k ðn kÞ! k¼0 n Similarly, we get h
lv
1 X JðnÞzn ;
ð18Þ
p0;0 ðlv nÞCð1Þ 1 qn h lv p0;0 1 lb Að1Þ 1 q lb n 1q " nrs n X ns n X ns X X k JðrÞ ð1 qÞs ð1 qÞs n ðn s rÞ! s¼1 r¼0 s¼1 r¼1 3 nrs k n Cð1Þ 7 Hðr 1Þ 5: ð21Þ Jðr 1Þ þ Að1Þ ðn s rÞ!
For n = 1, the probability p1,1 becomes
n¼0
p1;1 ¼
where
lv
This expression is the same with the one if derived from (1). Similarly, the probabilities {pn,1, n P 1}, can be obtained from P1(z) in terms of the p0,0 as
n X
k
ð20Þ
For example, the probability p1,0 can be given as
where
CðzÞ ¼ enz ð1 zÞ1þn z n 1
ð19Þ
h The probabilities {pn,0, n P 1} and {pn,1, n P 1} can be found from the PGFs P0(z) and P1(z) respectively. The PGF P0(z) gives the probabilities pn,0, for n P 1, in terms of p0,0 as
pn;0
To have P0(z) as a series, we expand the integrals A(z) and C(z) in series. The function A(z) can be written in terms of incomplete beta Rx function Bðx; a; bÞ ¼ 0 t a1 ð1 tÞb1 dt as
lb
1 X p0;0 ðlv nÞCð1Þ 1 qn n h lv z p0;0 1 lb Að1Þ 1 q l n 1 q b n¼1 " nrs 1 X n X ns X k JðrÞ zn ð1 qÞs n ðn s rÞ! n¼0 s¼1 r¼0 3 nrs k 1 X n X ns X n Cð1Þ 7 Hðr 1Þ zn 5: ð1qÞs Jðr 1Þ þ Að1Þ ðn s rÞ! n¼1 s¼1 r¼1
6 1 and limz!1 ð1 zÞn ¼ 1, so we must
and this condition gives p1,1 in terms of p0,0 as
zP 0 ðzÞ
ðlv nÞCð1Þ h p0;0 p0;0 ; lb Að1Þ lb
n k þ lnv þ hn knk ð1Þnk k JðnÞ ¼ : l lv n ðn k þ nv Þk þ n k 1 ðn kÞ! k¼0 n
which is same as in Eq. (15). Thus, we have seen that all the stationary probabilities can be derived in terms of p0,0, which can be found from the result below.
Using (17) and (18) and expanding the exponential terms, eaz, in series, we get
Theorem 2.2. If q < 1 and 0 < n < lv, then the probability p0,0 is given by
n X
"X 1 X n nr 1 X n nr X k JðrÞ k 1 P0 ðzÞ ¼ p0;0 1 n ðn rÞ! n ðn rÞ! n n¼0 r¼0 n¼1 r¼1 Cð1Þ Jðr 1Þ þ Hðr 1Þ zn Að1Þ
lv
and this is the solution to the differential Eq. (9). Next we will find the generating function P1(z). Multiplying (4) with zn and summing over n gives,
ðk þ lb Þ
1 1 1 1 X X X X zn pn;1 ¼ k zn pn1;1 þ h zn pn;0 þ lb zn pnþ1;1 ; n¼2
n¼2
n¼2
p0;0 ¼
ðn þ hÞðlb kÞ ; nÞCð1Þ
hðn þ lb lv Þ þ nðlb kÞ hðlv nÞ þ ðlvhAð1Þ
where A(1) and C(1) are products of Beta function and degenerate hypergeometric series as given in (14). Proof. Applying L’Hopital’s rule in (10), we have
lv
lim P0 ðzÞ ¼ lim z!1
n¼2
which after simple algebra reduces to, for 0 < z < 1,
ð22Þ
which gives
z!1
n
1 p0;0 ð1 zÞ hn p0;0 þ lnb p1;1 z ; lv 1 ð1 zÞ hn z n
ð23Þ
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hP 0 ð1Þ ¼ hp0;0 þ lb p1;1 :
ð24Þ
Similarly, from (19), we get
EðNÞ ¼ EðN0 Þ þ EðN1 Þ 1 1 ðl nÞCð1Þ 1 v þ p0;0 : ¼ ðlb kÞ h lb hAð1Þ
Applying (24), this becomes
P1 ð1Þ ¼
hP0 ð1Þ þ
hzP 00 ð1Þ
½hp0;0 þ lb p1;1
lb k
¼
hP00 ð1Þ
¼
lb k
ð31Þ
ð32Þ
Using Little’s law, the mean waiting time in the system can be obtained as
lb k
implying that
P00 ð1Þ
h l k ðl nÞCð1Þ EðN1 Þ ¼ P01 ð1Þ ¼ EðN0 Þ ¼ b 1 v p0;0 : hAð1Þ lb lb
Hence, the steady state mean number of customers in the system is
hP 0 ðzÞ þ hzP 00 ðzÞ ½hp0;0 þ lb p1;1 : z!1 kð1 zÞ ðkz lb Þ
lim P1 ðzÞ ¼ lim z!1
EðWÞ ¼ EðNÞ=k: P1 ð1Þ:
ð25Þ
Applying L’Hopital’s rule
Another important measure of system performance is the total waiting time of a customer who actually completes their service before leaving the system. Let us denote this by Wserved. Let Wnj denote the conditional waiting time of a customer who does not abandon the system, given that the state upon arrival is (n, j). The waiting time of a customer is measured from the moment of arrival until departure, either after completion of service or due to abandonment. Then,
lim P00 ðzÞ
EðW n1 Þ ¼
h
The expression of P 00 ðzÞ can also be found from (9) as
ðkz lv þ nÞð1 zÞ þ hz P 0 ðzÞ ðhz ðlv nÞð1 zÞÞp0;0 þ lb zp1;1 : P 00 ðzÞ ¼ nzð1 zÞ ð26Þ
z!1
½kð1 2zÞ þ lv n þ hP0 ðzÞ fh þ ðlv nÞgp0;0 þ lb p1;1 ¼ lim : z!1 nð1 2zÞ ½ðkz lv þ nÞð1 zÞ þ hz
ðk lv þ nÞP0 ð1Þ þ ðlv nÞp0;0 : nþh
ð27Þ
Eqs. (25) and (27) imply that
lb k h
P1 ð1Þ ¼
ðk lv þ nÞP0 ð1Þ þ ðlv nÞp0;0 ; nþh
which simplifies to
h i l ½hðk lv þ nÞ þ ðn þ hÞðlb kÞ p0;0 þ b p1;1 h ¼ ðn þ hÞðlb kÞ þ hðlv nÞp0;0 :
;
ðn þ hÞðlb kÞ : nÞCð1Þ
hðn þ lb lv Þ þ nðlb kÞ hðlv nÞ þ ðlvhAð1Þ
h 1 þ EðW n1 Þ h þ k þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn k 1 þ þ EðW n0 Þ h þ k þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn ðn þ 1Þn n 1 þ h þ k þ lv þ ðn þ 1Þn n þ 1 h þ k þ lv þ ðn þ 1Þn lv 1 þEðW n1;0 Þ þ h þ k þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn þEðW n1;0 Þ ;
1 h þ k þ lv þ nn h þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn hðn þ 1Þ þ ðlv þ nnÞEðW n1;0 Þ : þ
EðW n0 Þ ¼
For j = 0 and n = 0,
2.2. Performance measures
EðW 00 Þ ¼
lb
From (24), the equilibrium probability that the system is in WV is
ðlv nÞCð1Þ p0;0 hAð1Þ
ð28Þ
and the probability that the system is in non-vacation period is
ðl nÞCð1Þ p0;0 : P1 ð1Þ ¼ 1 P0 ð1Þ ¼ 1 v hAð1Þ
ð29Þ
The mean number of customers when the system is on WV period is
EðN0 Þ ¼ P00 ð1Þ ¼
lb k h
ð33Þ
For j = 0 and n P 1,
h For n = 0, the customers of the system will not be impatience and the system boils down to an M/M/1/MWV model, given by Servi and Finn (2002).
P0 ð1Þ ¼
n ¼ 1; 2; 3; . . .
because, in the second term the arrival of a customer does not change the waiting time of a customer present in the system and in the third term only n customers can abandon the system as our customer is not impatient. This expression can be further rewritten as
Putting p1,1 in terms of p0,0, we get the expression for p0,0 as
p0;0 ¼
lb
EðW n0 Þ ¼
Therefore, using (24),
P00 ð1Þ ¼
nþ1
ðl nÞCð1Þ 1 v p0;0 hAð1Þ
ð30Þ
and the expected number of customers when the server is on nonvacation period is
ð34Þ
h 1 1 þ h þ k þ lv þ n h þ k þ lv þ n lb k 1 þ EðW 00 Þ þ h þ k þ lv þ n h þ k þ lv þ n lv 1 ; þ h þ k þ lv þ n h þ k þ lv þ n
which can be simplified to
EðW 00 Þ ¼
h þ k þ lv h : þ h þ lv þ n h þ k þ lv þ n lb 1
ð35Þ
Using (35) and iterating (34), we obtain for n P 0
1 h þ k þ lv þ nn ðn þ 1Þh þ h þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn lb n # n X h þ k þ lv þ ðk 1Þn kh Y lv þ in þ : ð36Þ þ h þ k þ lv þ kn lv i¼k h þ lv þ in k¼1
EðW n0 Þ ¼
212
N. Selvaraju, C. Goswami / Computers & Industrial Engineering 65 (2013) 207–215
Finally, we get the mean waiting time of customers served by the system as
EðW serv ed Þ ¼
1 1 X X pn;0 EðW n0 Þ þ pn;1 EðW n1 Þ; n¼0
n¼1
W d ðsÞ
which after using (33) becomes
EðW serv ed Þ ¼
1 X EðN1 Þ þ P1 ð1Þ pn;0 EðW n0 Þ þ : n¼0
lb
ð37Þ
It is often very much of interest to try and decompose the quantities of interest into various factors in order to have a better comparison with the existing models. A stochastic decomposition property indicates the effects of system vacation on its performance measures like queue length and waiting times and plays an important role in vacation models. We can attempt to do the same for the system under consideration as this is an M/M/1 model with MWVs. We have proved the following two results that this impatient WV model also satisfies such decomposition for stationary queue length and stationary waiting times into one that corresponds to a pure M/M/1 system plus a quantity that is due to the WV. Theorem 2.3. For q < 1, the stationary queue length N can be decomposed into a sum of two independent random variables as
N ¼ Nc þ Nd ; where Nc is the queue length of a classical M/M/1 queue without vacations and Nd is the additional queue length due to the effect of MWV with its PGF
" # p0;0 ðlb kzÞð1 zÞ hz ðlv nÞCð1Þ Nd ðzÞ¼ P0 ðzÞ þ z : p0;0 Að1Þ ðlb kÞð1 zÞ ð38Þ Proof
NðzÞ ¼ P0 ðzÞ þ P1 ðzÞ ¼ 1 þ
hz P0 ðzÞ ðkz lb Þð1 zÞ ðlv nÞCð1Þz lb k p0;0 ¼ ðkz lb Þð1 zÞAð1Þ lb kz lb kz hz P ðzÞ lb k ðlb kÞð1 zÞ 0 ðlv nÞCð1Þz 1q þ p0;0 ¼ Nd ðzÞ: ðlb kÞð1 zÞAð1Þ 1 qz
v n zn ðsayÞ;
Let s = k(1 z) which gives z ¼ 1 ks and 1 z ¼ ks . Putting these relations in (38), we get the desired expression. h
3. Single working vacation (SWV) model The M/M/1 queue with impatient customers and a SWV policy is different from the MWV model in a way that when the server returns from it’s WV period and finds no customer in the system, it does not take another vacation but remains idle until the next arrival. So, in the SWV model, the server may stay idle for some period whereas the MWV does not have any idle time for the server. In this SWV model also, the customers become impatient and activate the impatient timer ‘T’ only if they found the server in WV period servicing at a lower rate upon its arrival and if the server is not empty. For the SWV model, the Markov chain D = {(Nt, Qt), t P 0} can be defined as in MWV case but with the state space E = {(i, j), i = 0, 1, . . . , j = 0, 1}. Here j = 1 if the server is active (idle or busy) and j = 0 if the server is in WV. We have the balance equations for the equilibrium state probabilities given by
ðk þ hÞp0;0 ¼ lv p1;0 þ lb p1;1 ;if n ¼ 0; ð40Þ ðk þ lv þ h þ ðn 1ÞnÞpn;0 ¼ kpn1;0 þ ðlv þ nnÞpnþ1;0 ;if n P 1;ð41Þ kp0;1 ¼ hp0;0 ;if n ¼ 0; ð42Þ
Let us define the generating functions, for 0 < z < 1
G0 ðzÞ ¼
1 X zn pn;0 ;
G1 ðzÞ ¼
n¼0
1 X zn pn;1 :
ð44Þ
n¼0
We have G0(1) + G1(1) = 1. Multiplying Eq. (41) with zn and summing over n gives
nzð1 zÞG00 ðzÞ ½ðkz lv þ nÞð1 zÞ þ hzG0 ðzÞ þ ðn lv Þð1 zÞp0;0 þ lb zp1;1 ¼ 0:
" # 1 1 1 X 1 X X 1 h X pn;0 zn q pn;0 znþ1 þ pnþk;0 zn 1 q n¼0 lb n¼1 k¼0 n¼0 1 X
NðzÞ ¼ W ðkð1 zÞÞ:
ðk þ lb Þpn;1 ¼ kpðn1Þ;1 þ hpn;0 þ lb pnþ1;1 ;if n P 1:ð43Þ
Nd(z) can be written in series expansion as
¼
" p0;0 ðlb s kÞs hðk sÞ s ¼ P0 1 p0;0 k ðlb kÞs ðl nÞCð1Þ þðk sÞ v : Að1Þ
Proof. From the distributional form of Little’s Law, in Keilson and Servi (1988), we have the relation
2.3. Stochastic decompositions in the MWV model
Nd ðzÞ ¼
where Wc is the waiting time of a customer corresponding to classical M/M/1 queue and has exponential distribution with parameter lb(1 q) and Wd is the additional delay due to the effect of MWV with its Laplace–Stieltjes transform (LST)
ð45Þ
For limit z tending to 1, we get h G0(1) = lbp1,1. For 0 < z < 1, letting n = 0 we get,
ð39Þ
n¼0
h i P where v 0 ¼ 11 q p0;0 and v n ¼ 11 q pn;0 qpn1;0 þ lh 1 k¼0 pnþk;0 ; b n P 1. From Eqs. (20) and (22) it can be shown that vn 2 [0, 1] for P n P 0 and 1 h n¼0 v n ¼ 1. This proves that Nd(z) is a PGF.
G0 ðzÞ ¼
ðlv nÞp0;0 zðlb p1;1 þ ðlv nÞp0;0 Þ : ðkz2 zðk lv Þ þ lv
ð46Þ
For n – 0, we have G00 ðzÞ
kz lv þ n h n lv lb þ G0 ðzÞ þ p0;0 þ p ¼ 0: zn nð1 zÞ zn nð1 zÞ 1;1
ð47Þ
Solving this differential equation, as in the MWV case, we get Theorem 2.4. If q < 1, the stationary waiting time can be decomposed into the a sum of two independent random variables as
W ¼ Wc þ Wd;
lv
G0 ðzÞ ¼ zð n 1Þ ð1 zÞn h
l 1 p0;0 CðzÞ b p1;1 AðzÞ : n n
lv
b Að1Þ Here, we have G0 ð0Þ ¼ p0;0 ¼ ðllnÞCð1Þ p1;1 , which gives, v
ð48Þ
N. Selvaraju, C. Goswami / Computers & Industrial Engineering 65 (2013) 207–215
p1;1 ¼
ðlv nÞCð1Þ p0;0 : lb Að1Þ
Therefore, we get a similar expression for G0(z) as in MWV case
G0 ðzÞ ¼ p0;0
lv
1
n
lv h Cð1Þ AðzÞ zð n 1Þ ð1 zÞn : CðzÞ Að1Þ
ð49Þ
ðkz lb Þð1 zÞG1 ðzÞ ¼ hzG0 ðzÞ lb ð1 zÞp0;1 lb zp1;1 :
We state the following results concerning the stochastic decompositions in the SWV model which can be proved in a similar way as in the MWV case dealt with earlier.
For 0 < z < 1,
h ðl nÞCð1Þ p0;0 ðz zG0 ðzÞ ð1 zÞ þ z v k lb lb Að1Þ h
1
1
1Þ ð1 qzÞ :
ð50Þ
To find G0(z) and G1(z) explicitly, we have to find an explicit solution for p0,0. Applying L’Hopital’s rule in (50),
G00 ð1Þ ¼
lb k h
l G1 ð1Þ b p0;0 : k
ð51Þ
Applying L’Hopital’s rule in (47) gives
lim G00 ðzÞ ¼ lim z!1
z!1
½kð1 2zÞ þ lv n þ hG0 ðzÞ ðlv nÞp0;0 lb p1;1 ; nð1 2zÞ ðkz lv þ nÞð1 zÞ hz ð52Þ
which becomes
G00 ð1Þ ¼
ðk lv þ nÞG0 ð1Þ þ ðlv nÞp0;0 : nþh
ð53Þ
Equating G00 ð1Þ from (51) and (53)
ðn þ hÞðlb kÞ hðlv nÞ þ
k
hðn þ hÞ þ
ðlv hÞCð1Þ ½hðn hAð1Þ
" p0;0 ðlb kzÞð1 zÞ hz P0 ðzÞ p0;0 ðlb kÞð1 zÞ h ðl hÞCð1Þ þlb ð1 zÞ þ z v : k Að1Þ
Nd ðzÞ ¼
Theorem 3.2. If q < 1, the stationary waiting time W can be decomposed into the a sum of two independent random variables as W = Wc + Wd, where Wc is the waiting time of a customer corresponding to classical M/M/1 queue and has exponential distribution with parameter lb(1 q) and Wd is the additional delay due to the effect of SWV with its LST
W d ðsÞ ¼
We finally obtain p0,0 as lb
Theorem 3.1. For q < 1, the stationary queue length N can be decomposed into a sum of two independent random variables as N = Nc + Nd, where Nc is the queue length of a classical M/M/1 queue without vacations and Nd is the additional queue length due to the effect of SWV with its PGF given by
" p0;0 ðlb s kÞs hðk sÞ s P0 1 p0;0 k ðlb kÞs sh ðlv hÞCð1Þ þlb þ ðk sÞ : k Að1Þ
½ðk lv þ nÞ þ ðn þ hÞðlb kÞG0 ð1Þ l ¼ ðn þ hÞðlb kÞ b hðn þ hÞp0;0 : k
p0;0 ¼
The average waiting time of customers in system who are served before leaving the system, E(Wserved), can be derived similarly as in the MWV case with EðW n1 Þ ¼ nþ1 lb but for n = 0, 1, 2, . . . rather than n P 1, and E(Wn0) is as given by (34). Finally, E(Wserved) is given by (37) but with the second sum starting from n = 0. 3.2. Stochastic decompositions in the SWV model
Multiplying (43) with zn and summing over n gives
G1 ðzÞ ¼
213
þ lb lv Þ þ nðlb kÞ
:
These results can be proved similar to the MWV case in the previous section.
ð54Þ As we have given in MWV case, we give below the performance measures of this SWV model in a similar manner. 3.1. Performance measures The system performance measures of SWV model are given below. The probabilities of system being in WV and in non-vacation period are, respectively,
G0 ð1Þ ¼
ðlv hÞCð1Þ p0;0 hAð1Þ
ð55Þ
and G1 ð1Þ ¼ 1 G0 ð1Þ ¼ 1
ðlv hÞCð1Þ p0;0 : hAð1Þ
ð56Þ
The mean number of customers during WV period can be given as
EðN0 Þ ¼ G00 ð1Þ ¼
lb k h
1
ðlv hÞCð1Þ l p0;0 b p0;0 hAð1Þ k
ð57Þ
and the mean number of customers during non-vacation period as
EðN1 Þ ¼ G01 ð1Þ ¼
lb k ðl hÞCð1Þ p0;0 : 1 v hAð1Þ lb
ð58Þ
The average number of customers in the system will be
EðNÞ ¼ ðlb kÞ
1 1 ðl hÞCð1Þ l 1 v þ p0;0 b p0;0 : h lb hAð1Þ k
ð59Þ
4. Comparison of the models The model M/M/1/WV with impatient customer is studied with MWV and also with SWV policy which refers to WDM links with two different types of wavelength assignment or reconfiguration policies used in OBS networks. The performance of an OBS network depends upon its high speed data access and minimum packet loss qualities. As packet delay (the waiting in a queue) is a vital issue for packet lost, we studied the waiting time of customers in queue compared to the impatience rates of the customers for both WV policies. These give us a picture of how fast the allocation of transmission path should be to have minimum packet loss. To show the impact of packet loss or impatience on system performance, the WV policies, MWV and SWV are analytically compared. First, we have compared the probability of the system being idle in WV (p0,0) of MWV model to that of SWV model using their analytic expressions. Next, we compared the mean number of customers in the system in vacation or in non-vacation between the two models. A model can be said to be efficient than another, if its mean waiting time of the customers is less compared to the other. For a M/M/1/WV queue without impatient we get that a MWV model is always better than a SWV model in the sense that the mean queue length in MWV is always less than that in the SWV model. This result can be verified from the expressions of mean queue lengths of MWV model and SWV model found in Liu et al. (2007) and Tian et al. (2008) respectively. From the definitions of
N. Selvaraju, C. Goswami / Computers & Industrial Engineering 65 (2013) 207–215
multiple and SWV policies, it seems that, with impatient customers, SWV policy is more efficient compared to the MWV one. But because of impatience the behavior differs. To differentiate the terms of both the models we will use superscript M and S for the MWV and SWV respectively. From the expressions of p0,0 in (22) and (54) it can be seen that the probability of SWV is less that of MWV.
1 1 lb h ¼ þ ; kð l pS0;0 pM b kÞ 0;0
ð60Þ
which gives pS0;0 < pM 0;0 . The mean number of customers in the systems, when the server in non-vacation are
l k ðl hÞCð1Þ M p0;0 EM ðN 1 Þ ¼ b 1 v hAð1Þ lb lb k ðl hÞCð1Þ S S p0;0 ; 1 v and E ðN1 Þ ¼ hAð1Þ lb
1.7 EM(N ) 0
1.6
Mean Queue length in WV period
214
ES(N ) 0
1.5 1.4 1.3 1.2 1.1 1
0.9 0.8 0.7 0
0.2
0.4
0.6
0.8
1
1.2
impatience rate ξ
which implies EM ðN1 Þ < ES ðN1 Þ:
ð61Þ
Fig. 2. Mean queue lengths during WV vs. n
For the system on vacation,
lb k
M
E ðN0 Þ ¼
h
and ES ðN0 Þ ¼
"
ðlb kÞðlv hÞCð1Þ
lb k h
"
h2 Að1Þ
# pM 0;0
ðlb kÞðlv hÞCð1Þ h2 Að1Þ
þ
lb k
# pS0;0
and the total queue lengths are
1 1 ðl nÞCð1Þ M 1 v þ p0;0 ; EM ðNÞ ¼ ðlb kÞ h lb hAð1Þ 1 1 ðlv hÞCð1Þ S l S 1 þ p0;0 b pS0;0 : E ðNÞ ¼ ðlb kÞ h lb hAð1Þ k From the expressions of EM(N0) and ES(N0), it is difficult to conclude any relationship between them. Therefore, the sum, EM(N) (=EM(N0) + EM(N1)), may be greater than, equal or less than ES(N). For example, let us take the impatience model in which k = 0.7, lb = 1 and lv = 0.5. For three different values of h the mean queue lengths of the system are shown in Fig. 1. When h = 1.7 the queue length of SWV model is less than MWV and for h = 2.1 the MWV model gives lesser queue length. When h = 1.9, we see that at the point n = 0.2 both the models give the same value. So for n < 0.2 the SWV model becomes better compared to MWV one and for
mwv, θ=1.7 swv, θ=1.7 mwv, θ=1.9 swv, θ=1.9 mwv, θ=2.1 swv, θ=2.1
Mean Queue length
0.31
0.305
0.3
0.295
0.29 0.1
0.15
0.2
0.25
0.3
0.35
0.4
impatience rate ξ Fig. 1. Mean queue lengths vs. n, for different values of h
0.45
n > 0.2 the MWV model works better. That means that, when the mean of vacation duration time is less, the MWV model gives better performance than a corresponding SWV model. In Fig. 2, we have compared the mean queue lengths during a WV period of both the models. The parameter values are k = 1.8, lb = 2, lv = 1.3 and h = .09. The difference between the queue lengths increases with the increase in the impatient rate n. So, if the impatient rate is small, the MWV and SWV models give the same queue lengths during WV periods but for the higher impatient rates SWV model will give better performance. This study reveals that not only the vacation duration rate but also the impatience rate plays a major role in the performance of the model. Thus, for a given parameter values, we can choose a model to get better performance of a single server impatient queue with WVs. This underlines the fact that if the system is not in WV, average number of customers in the MWV model is always less than the SWV model. When the average vacation duration time is less, the MWV model gives better performance than the SWV model, whereas for higher impatient rates the SWV model becomes more efficient as it reduces the waiting time of the customers in the system. 5. Conclusion We have investigated the effect of impatient behavior of customers in a M/M/1/WV model. Two types of WV termination policies are taken, the multiple working vacation (MWV) policy and the single working vacation (SWV) policy. Closed-form probabilities are derived using the identities involving beta functions and degenerate hypergeometric functions. Stochastic decomposition properties are verified for both MWV and SWV cases. This work underlines the fact that if the system is not in WV, average number of customers in the MWV model is always less than the SWV model. Our study shows that packet loss is less when multiple data carrying channels are used as WDM links rather than the individual ones. Also higher is the rate of wavelength allocation the lesser will be the system congestion or loss of packets. However individual WDM links are efficient compared to multiple ones only if the data processing delay is quite high or delay budget is less than the processing delay; otherwise, multiple WDM links can ensure high speed data access with minimum data loss through OBS networks. We consider, in this paper, the arrival pattern of packets (customers) to be single arrivals. We can further extend the analysis for packets with batch arrivals, since communication networks
N. Selvaraju, C. Goswami / Computers & Industrial Engineering 65 (2013) 207–215
mostly deal with bulk arrivals of packets. This model may be extended for Markov modulated arrival process to capture correlated arrivals which is very useful in telecommunication networks. Finite buffer models can also be studied and analyzed. Not only the arrivals but the service patterns in batches are also found in networks. Considering the retransmission we can model this OBS system as a retrial (or repeated attempt) queueing model. Assigning priority to packets is another important feature to study in future. References Abramowitz, M., & Stegun, I. (1972). Handbook of mathematical functions. Dover Publication. Altman, E., & Yechiali, U. (2006). Analysis of customer’s impatience in queue with server vacations. Queueing Systems, 52, 261–279. Baba, Y. (2005). Analysis of a GI/M/1 queue with multiple working vacations. Operations Research Letters, 33, 201–209. Banik, A. (2010). Analysis of single server working vacation in GI/M/1/N and GI/M/1/ 1 queueing systems. International Journal of Operational Research, 7(3), 314–333. Bocquet, S. (2005). Queueing theory with reneging. Scientific and Technical Report of DSTO. Goswami, C., & Selvaraju, N. (2010). The discrete-time MAP/PH/1 queue with multiple working vacations. Applied Mathematical Modelling, 34(4), 931–946. Coudert, D., Huc, F., Mazauric, D., Nisse, N., Antipolis, S., & Sereni, J. S. (2009). Reconfiguration of the routing in WDM networks with two classes of services.
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Impatient customers in an M/M/1 queue with single and multiple working vacations N. Selvaraju ⇑, Cosmika Goswami Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781 039, India
a r t i c l e
i n f o
Article history: Received 7 May 2012 Received in revised form 4 December 2012 Accepted 20 February 2013 Available online 1 March 2013 Keywords: Markovian queues Impatient customers Multiple working vacations Single working vacations Performance measures Stochastic decomposition
a b s t r a c t We consider an M/M/1 queue with impatient customers and two different types of working vacations. The working vacation policy is the one in which the server serves at a lower rate during a vacation period rather than completely stop serving. The customer’s impatience is due to its arrival during a working vacation period, in which the customer service rate is lower than the normal busy period. We analyze the queue for two different working vacation termination policies, a multiple working vacation policy and a single working vacation policy. Closed-form solutions and various performance measures like, the mean queue lengths and the mean waiting times are derived. The stochastic decomposition properties are verified for both multiple and single working vacation cases. A comparison of both the models is carried out to capture their performances with the change in system parameters. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In a communication network various types of jobs, like voice packets or data files, have to wait in a queue if they do not get service immediately upon their arrival in the network. The jobs can become impatient due to high waiting times or due to uncertainty of receiving services and may leave the system unserved. This scenario often occurs in Optical Burst Switching (OBS) networks. This OBS technology is in high demand as it supports various QoS functions that facilitate low data-loss rate, low latency, survivability and is very efficient for communication networks which require high transmission rates. For example, in industrial robotics there are several protocols that integrate control services, communication services, and routing capabilities (sending documents or materials to appropriate destinations) based on Local Area Network (LAN)s and the Internet. Accessing data through OBS networks adds a further level of network management flexibility by handling increasing amount of traffic as in such industries the capability of providing access to data and information on shared storage devices is an important feature, O’Mahony, Simeonidou, Hunter, and Tzanakaki (2001). The OBS is a technology for reducing the gap between transmissions and switching speeds. In OBS, incoming traffic from clients at the edge of the network is aggregated at the ingress of the network according to a particular parameter. This is then assembled and is further transmitted ⇑ Corresponding author. Tel.: +91 361 2582611; fax: +91 361 2582649. E-mail address: [email protected] (N. Selvaraju). 0360-8352/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.02.016
through Wavelength Division Multiplexing (WDM) links. The operation of an OBS controller can be seen as a queue with reneging or impatience, studied in Bocquet (2005). In OBS networks, a control packet is sent first, on a separate signaling channel, to set up a connection. It is followed by a data burst without waiting for an acknowledgment for path establishment, Qiao and Yoo (1999). In particular, when a path is not assigned, the burst control packet is accepted to the queue and is kept waiting for a path. If its delay budget is lower than the effective processing delay, the packet becomes impatient and leaves the system unserved. To have more efficient use of such a network, the loss of packets has to be reduced for its better performance. For example, in robotics industry if packets are lost during data transferring or communicating, there can create serious issues regarding the data related performance of the computers and machines. In this paper, a study on impatience of a request in WDM links is done as it seems to be essential for the benefit of better system performance. Servi and Finn (2002) were the first to observe a reconfigurable WDM network as a new policy called Working Vacation (WV) model. This WDM optical access network with multiple wavelengths was viewed as a generalization of a multi-queue cyclic service model where a single token moves cyclically from one queue to another. Here each queue operates either at a fast service rate or a normal service rate. In a classical vacation queueing model, a server completely stops serving the customers in queue, during the vacation period and made the customer wait for service till the vacation period ends and the server gets back to its normal service period. But rather than cease the service completely during
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its vacation period, if the server serves at a lower rate, we get this new vacation policy, the WV policy, Goswami and Selvaraju (2010). There is quite a difference between WV queue and classical vacation queue. During a vacation, customers in the former may finish service and depart the system, however, customers in the latter cannot depart the system after getting service. On the other hand, during a vacation, the number of customers in the former can increase or decrease, however, the number of customers in the latter can only increase. Therefore, the WV models have more complicated modalities and the analysis of this kind of models is more difficult than the classical vacation queues. Servi and Finn considered n-queue generalization of a cyclic service queueing model where each queue can be viewed as an M/M/1 model with WV. Later, Liu, Xu, and Tian (2007) analyzed the same M/M/1/WV model using the quasi-birth death and matrix-geometric method to obtain explicit expressions of the performance measures and their stochastic decompositions. Wu and Takagi (2006) extended Servi and Finn’s work to M/G/1/WV model. Baba (2005) considered the GI/M/1/WV system with general independent arrival process and the distribution of the vacation duration times and service times as exponential. Wang (2009) studied the M/M/1 queue with WV and non-zero switching times and studied a cyclic service system in WDM-based access networks with reconfigurable delay. Tian, Zhao, and Wang (2008) have studied the M/M/1 queue with single WV. Banik (2010) analyzed single working vacation in GI/M/1/N and GI/M/1/1 queueing systems considering the service time and vacation times as exponentially distributed. Recently, Laxmi and Yesuf (2011) presented an GI/M/1 batch service queue with exponential working vacation policy. Ke, Wu, and Zhang (2010) have given a short survey on recent developments on vacation models. There are situations where the customer impatience is due to the absence of the server, more precisely, due to the server vacation and is independent of the customers in system. Altman and Yechiali (2006) studied the impatience of customers in vacation systems of M/M/1, M/G/1 and also of M/M/c queues, where each arriving customer who finds no server on duty, activates an independent random impatient timer. If a server does not return from vacation by the time the timer expires, the customer abandons the queue and never returns. If the server returns from its vacation before the timer expires, the customer stays in the system until his service is completed. Keeping in view the importance of impatience in the context of WDM networks, we consider here impatient customers in an M/M/ 1 queue under the WV policy. We analyze the queue for two different WV termination policies, the multiple working vacation (MWV) policy and the single working vacation (SWV) policy. A MWV policy requires the server to keep taking vacations until it finds at least one customer waiting in the system at a vacation completion instant. Under a SWV policy, the server takes only one vacation at the end of each busy period. As this single vacation ends, the server either serves the customers at the rate of normal busy period, if any, or stays idle. Recently, Yue, Yue, and Xu (2011) studied M/M/1 model with MWV where the customer’s impatience is due to WV. They have studied the model in which during WV all the customers in the system become impatient and if the customer’s service has not been completed before the customer’s impatient timer expires, the customer abandons the queue. We consider here a more constrained model where the customer under service, during WV, never becomes impatient but only the ones waiting in queue are reluctant to abandon the system. This imposed condition can be seen in reality as a customer receiving service seldom wants to abandon the system. This restriction turns the model given by Yue et al. into a much more complicated and analytically challenged one. We have analyzed this constrained model not only
under MWV policy but also under SWV policy. This policies corresponds to reconfiguration or wavelength assignment policies over WDM links used to control congestion in OBS networks. It is important to find the optimal reconfiguration policy for the network in order to optimize the usage of network resources since an unnecessary rearrangement or postponing a necessary reconfiguration, effects on overall performance of the network, Coudert et al. (2009). The purpose of this analysis is twofold: to obtain analytical solutions to the models and to have a comparison between the two models. We want to study the effect of packet delay and packet loss in OBS network. We seek answers to queries like – do different wavelength assignment policies in WDM links influence the system performance? If they do, which policy should be adopted for minimum packet loss and under what condition? How the parameter threshold like delay budget increases packet loss and how to have a balance between them? We first formulate the impatient M/M/1 model (for both MWV and SWV cases) as a quasi-birth–death process and derive the partial generating functions for the distribution of queue sizes when the server is in WV period and when in regular busy period. Solving the arising differential equations, we find the closed-form solutions (in terms of Beta and hypergeometric functions) and various performance measures like, the mean queue length and the mean waiting time for the both the WV termination policies. Stochastic decomposition properties are also verified for both MWV and SWV cases. The paper is organized as follows. In Section 2, we provide the description of the M/M/1 model with MWV. We formulate the model as a quasi-birth–death process and from the balance equations, the explicit expressions of the various performance measures are derived. Section 3 gives the analysis of the M/M/1 SWV model with the stochastic decomposition property. In Section 4, we have given the comparison of the distribution of number of customers when the system follows MWV policy and SWV policy.
2. Multiple working vacation (MWV) model We consider a system where the arrivals occur according to a Poisson process with rate k. The service times during the normal busy period is exponentially distributed with mean 1/lb, where we assume the stability condition that q = k/lb < 1. The server takes a WV as soon as the system becomes empty. During the vacation period, the server serves the customers at an exponential rate lv, where lv < lb. The duration of a vacation is also exponential with rate h. A MWV policy requires the server to keep taking vacations until it finds at least one customer waiting in the system at a vacation completion instant. When the server returns from its vacation and finds at least one customer in the system, the server switches its service rate from lv to lb and a non-vacation period starts; otherwise it immediately leaves for another WV. If the vacation terminates in between an ongoing service, the server switches its service rate and continues the service at the higher rate until completion. The customers in this system are assumed to be impatient as described below. Whenever arriving customer has to join a queue and finds the system in WV, the customer activates an impatient timer ‘T’, which is exponential with rate n and is independent of the number of customers in the system at that moment. If the server ends the vacation before the time T expires, the customer stays in the system till his service is completed; otherwise, the customer leaves the system and never returns. Thus, only the customers who arrive during a WV period of the server are impatient. A customer who arrives during a WV period and finds the server free, it starts getting service immediately upon it’s arrival and therefore it does not become impatient. This type of impatience under WV policy has not been studied before. This is different from the impatient
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policy studied in Altman and Yechiali (2006), in which all customers arriving during vacation period activate impatient timers, since they consider a pure vacation policy. The server serves the customers on a first-come-first-served (FCFS) basis. The interarrival times, service times, vacation duration times and the impatient times, all are taken to be mutually independent. The service system as described above can be modelled by a two-dimensional continuous-time discrete-state Markov chain for the state of the system and is denoted by D = {(Nt, Jt), t P 0}, where Nt denotes the total number of customers in the system and Jt denotes the state of the server with
where
nk k k ; n P 0; l lv n ðn þ nv Þk þ n k 1 ðn kÞ! k¼0 n lv h nk nk n n k þ þ X n n ð1Þ k k ; n P 1; HðnÞ ¼ l v lv n þ n k ðn kÞ! ðn þ n þ 1Þk k¼0 n k l h k l h and Cð1Þ ¼ K ; v 1; þ 1 : Að1Þ ¼ K ; v ; n n n n n n n X
JðnÞ¼
nk
ð1Þ
nkþ
lv n
hn
We get, E = {{(0, 0)} [ {(i, j)}, i = 1, 2, . . . , j = 0, 1} as the state space of the above Markov chain.
In the above, the quantity K is given by K(a, b, c) = B(b, c) 1 F 1 (c;b + c;a) R1 with the Beta function, Bðb; cÞ ¼ 0 t ðb1Þ ð1 tÞc1 dt; b > 0; c > 0; the P ðaÞk zk degenerate hypergeometric function, 1 F 1 ða; b; zÞ ¼ 1 k¼0 ðbÞk k! ; and the CðaþkÞ Pochhammer symbol, ðaÞk ¼ CðaÞ .
2.1. The stationary distribution
Proof. Multiplying (2) with zn and summing over n gives
Jt ¼
1; if the server is in non-vacation period at time t; 0;
if the server is in WV period at time t:
Let us define the stationary probabilities for the Markov chain D
1 1 1 X X X ½k þ lv þ h þ ðn 1Þn zn pn;0 ¼ k zn pn1;0 þ ðlv n¼1
as
pi;j ¼ PfNt ¼ i; J t ¼ jg;
i ¼ 0; 1; 2; . . . j ¼ 0; 1:
½k þ lv þ h þ ðn 1Þnpn;0
kp0;0 ¼ lv p1;0 þ lb p1;1 ;if n ¼ 0; ð1Þ ¼ kpn1;0 þ ðlv þ nnÞpnþ1;0 ;if n P 1; ð2Þ
ðk þ lb Þp1;1 ¼ hp1;0 þ lb p2;1 ;if n ¼ 1;
ð3Þ
ðk þ lb Þpn;1 ¼ kpðn1Þ;1 þ hpn;0 þ lb pnþ1;1 ;if n P 2: ð4Þ Let us define the partial probability generating functions (PGFs), for 0
n¼0
1 X P1 ðzÞ ¼ zn pn;1 ; n¼1
with P0(1) + P1(1) = 1 and P 00 ðzÞ ¼ n
n¼1
and rearranging the terms gives rise to the differential equation,
Then, the balance equations are
1 X P0 ðzÞ ¼ zn pn;0 ;
n¼1
þ nnÞzn pnþ1;0
P1
n¼1 z
nzð1 zÞP00 ðzÞ ½ð1 zÞðkz lv þ nÞ þ zhP0 ðzÞ þ ½zh ð1 zÞðlv nÞp0;0 þ zlb p1;1 ¼ 0: For n = 0, we get from the above equation
l ð1 zÞp0;0 z½lb p1;1 þ hp0;0 P0 ðzÞ ¼ v 2 ; kz zðk þ lv þ hÞ þ lv
pn;0 .
þ Remark 2.1. If n = 0, the current model reduces to M/M/1 queues with working vacations studied by Servi and Finn (2002). If we assume that the packet arrival rate into the network is lower than the fast service rate (k < lb), that the network does not explode, we can derive the relation between the total number of packets in the network, type of service provided by the network and the chance of packets being lost in the system by finding the PGFs of the sequence of joint probabilities pi, j. The following theorem derives the PGFs in terms of p0,0, the probability of the network being idle. Theorem 2.1. For q < 1 and n < lv, the (partial) PGFs can be expressed in terms of p0,0 as
ð8Þ
which is same with Servi and Finn, see Servi and Finn (2002) in Eq. (A.3). For n – 0,
P00 ðzÞ n1
ð7Þ
kz lv þ n h h ðl nÞ P0 ðzÞ þ p0;0 þ v zn nð1 zÞ nð1 zÞ zn
lb nð1 zÞ
p1;1 ¼ 0:
ð9Þ
To solve the above first order linear differential equation, an inteR k lv h lv k ½ þ dz grating factor can be found as I:F: ¼ e n zn nð1zÞ ¼ enz zð n 1Þ h
ð1 zÞn . The general solution to the differential equation is given by lv
P0 ðzÞ ¼ zð n 1Þ ð1 zÞn h
h l 1 p0;0 CðzÞ p0;0 þ b p1;1 AðzÞ ; n n n
lv
ð10Þ with
Z
z
lv
k
h
enðzxÞ x n 1 ð1 xÞn1 dx 0 Z z lv h k enðzxÞ x n 2 ð1 xÞn dx: and CðzÞ ¼
AðzÞ ¼
ð11Þ
"X 1 X n nr 1 X n nr k JðrÞ n X k z P0 ðzÞ ¼ p0;0 1 n ðn rÞ! n n n¼0 r¼0 n¼1 r¼1 1 Cð1Þ ð5Þ Jðr 1Þ þ Hðr 1Þ zn ðn rÞ! Að1Þ
To get p1,1 in terms of p0,0, let us determine (11) and (12) for limit z tending to 1. We have the identity, from Abramowitz and Stegun (1972), for Re(u) > 0, Re(v) > 0,
and
Z
lv
1 X p0;0 ðl nÞCð1Þ 1 qn n h lv P1 ðzÞ ¼ v z p0;0 1 lb Að1Þ 1 q l n 1 q b n¼1 " nsr 1 n ns 1 X n X ns XXX X k JðrÞ zn ð1 qÞs n ðn s rÞ! n¼0 s¼1 r¼0 n¼1 s¼1 r¼1 3 nsr k n Cð1Þ 7 ð6Þ Hðr 1Þ zn 5; ð1qÞs Jðr 1Þþ Að1Þ ðn s rÞ!
ð12Þ
0
w
xv 1 ðw xÞu1 ebx dx ¼ Bðu; v Þwuþv 1 1 F 1 ðv ; u þ v ; bwÞ;
ð13Þ
0
R1
tðb1Þ ð1 tÞc1 dt; b > 0; c > 0; the P ðaÞk zk degenerate hypergeometric function 1 F 1 ða; b; zÞ ¼ 1 k¼0 ðbÞ k! and with the Beta function Bðb; cÞ ¼
0
k
the Pochhammer symbol ðaÞk ¼ CCðaðþkÞ aÞ . Substituting z = 1, 1 x = t and using identity (13), in Eqs. (11) and (12), we get
Að1Þ ¼ K
k lv h k l h ; ; and Cð1Þ ¼ K ; v 1; þ 1 ; n n n n n n
ð14Þ
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h
ðlv nÞCð1Þ zp0;0 ð1 zÞ1 ð1 qzÞ1 : lb Að1Þ
where K(a, b, c) = B(b, c)1F1(c;b + c;a). C(1) is valid only if lv 1 > 0, which leads to the assumption lv > n. Now, determinn
P1 ðzÞ ¼
ing P0(z) for limit z tending to 1 gives,
Further we can expand it in series using P0(z) as
lim P 0 ðzÞ ¼ P0 ð1Þ z!1 h lv h l ¼ 1 p0;0 Cð1Þ p0;0 þ b p1;1 Að1Þ limð1zÞn : z!1 n n n
P1 ðzÞ ¼
Since 0 6 P 0 ð1Þ ¼ have the term
P1
n¼0 pn;0
h
h l p0;0 þ b p1;1 Að1Þ ¼ 0 1 p0;0 Cð1Þ n n n
lv
p1;1
ðlv nÞCð1Þ h p : ¼ lb Að1Þ lb 0;0
ð15Þ
Applying the above relation in (10) we get
P0 ðzÞ ¼ p0;0
lv n
1
lv h Cð1Þ AðzÞ zð n 1Þ ð1 zÞn : CðzÞ Að1Þ
n 1 X ð1Þn k l h k AðzÞ ¼ enz B z; n þ v ; : n n! n n n¼0
ð16Þ
Expressing the incomplete Beta function in terms of Gauss hypergeometric function and expanding the hypergeometric function in series gives h
k
lv
AðzÞ ¼ enz ð1 zÞn z n 1
1 X HðnÞznþ1 ;
ð17Þ
n¼0
"X n nr n nr X k JðrÞ k 1 ¼ p0;0 1 n ðn rÞ! n ðn rÞ! n r¼0 r¼1 Cð1Þ Jðr 1Þ þ Hðr 1Þ : Að1Þ
lv
k Cð1Þ 1 Jð0Þ þ Jð1Þ Hð0Þ 1 n Að1Þ n k h ðlv nÞCð1Þ : ¼ p0;0 lv lv Að1Þ
p1;0 ¼ p0;0
pn;1 ¼
n k þ lnv þ hn knk ð1Þnk k HðnÞ ¼ : lv ðn k þ lnv þ 1Þk n þ n k ðn kÞ! k¼0 n Similarly, we get h
lv
1 X JðnÞzn ;
ð18Þ
p0;0 ðlv nÞCð1Þ 1 qn h lv p0;0 1 lb Að1Þ 1 q lb n 1q " nrs n X ns n X ns X X k JðrÞ ð1 qÞs ð1 qÞs n ðn s rÞ! s¼1 r¼0 s¼1 r¼1 3 nrs k n Cð1Þ 7 Hðr 1Þ 5: ð21Þ Jðr 1Þ þ Að1Þ ðn s rÞ!
For n = 1, the probability p1,1 becomes
n¼0
p1;1 ¼
where
lv
This expression is the same with the one if derived from (1). Similarly, the probabilities {pn,1, n P 1}, can be obtained from P1(z) in terms of the p0,0 as
n X
k
ð20Þ
For example, the probability p1,0 can be given as
where
CðzÞ ¼ enz ð1 zÞ1þn z n 1
ð19Þ
h The probabilities {pn,0, n P 1} and {pn,1, n P 1} can be found from the PGFs P0(z) and P1(z) respectively. The PGF P0(z) gives the probabilities pn,0, for n P 1, in terms of p0,0 as
pn;0
To have P0(z) as a series, we expand the integrals A(z) and C(z) in series. The function A(z) can be written in terms of incomplete beta Rx function Bðx; a; bÞ ¼ 0 t a1 ð1 tÞb1 dt as
lb
1 X p0;0 ðlv nÞCð1Þ 1 qn n h lv z p0;0 1 lb Að1Þ 1 q l n 1 q b n¼1 " nrs 1 X n X ns X k JðrÞ zn ð1 qÞs n ðn s rÞ! n¼0 s¼1 r¼0 3 nrs k 1 X n X ns X n Cð1Þ 7 Hðr 1Þ zn 5: ð1qÞs Jðr 1Þ þ Að1Þ ðn s rÞ! n¼1 s¼1 r¼1
6 1 and limz!1 ð1 zÞn ¼ 1, so we must
and this condition gives p1,1 in terms of p0,0 as
zP 0 ðzÞ
ðlv nÞCð1Þ h p0;0 p0;0 ; lb Að1Þ lb
n k þ lnv þ hn knk ð1Þnk k JðnÞ ¼ : l lv n ðn k þ nv Þk þ n k 1 ðn kÞ! k¼0 n
which is same as in Eq. (15). Thus, we have seen that all the stationary probabilities can be derived in terms of p0,0, which can be found from the result below.
Using (17) and (18) and expanding the exponential terms, eaz, in series, we get
Theorem 2.2. If q < 1 and 0 < n < lv, then the probability p0,0 is given by
n X
"X 1 X n nr 1 X n nr X k JðrÞ k 1 P0 ðzÞ ¼ p0;0 1 n ðn rÞ! n ðn rÞ! n n¼0 r¼0 n¼1 r¼1 Cð1Þ Jðr 1Þ þ Hðr 1Þ zn Að1Þ
lv
and this is the solution to the differential Eq. (9). Next we will find the generating function P1(z). Multiplying (4) with zn and summing over n gives,
ðk þ lb Þ
1 1 1 1 X X X X zn pn;1 ¼ k zn pn1;1 þ h zn pn;0 þ lb zn pnþ1;1 ; n¼2
n¼2
n¼2
p0;0 ¼
ðn þ hÞðlb kÞ ; nÞCð1Þ
hðn þ lb lv Þ þ nðlb kÞ hðlv nÞ þ ðlvhAð1Þ
where A(1) and C(1) are products of Beta function and degenerate hypergeometric series as given in (14). Proof. Applying L’Hopital’s rule in (10), we have
lv
lim P0 ðzÞ ¼ lim z!1
n¼2
which after simple algebra reduces to, for 0 < z < 1,
ð22Þ
which gives
z!1
n
1 p0;0 ð1 zÞ hn p0;0 þ lnb p1;1 z ; lv 1 ð1 zÞ hn z n
ð23Þ
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N. Selvaraju, C. Goswami / Computers & Industrial Engineering 65 (2013) 207–215
hP 0 ð1Þ ¼ hp0;0 þ lb p1;1 :
ð24Þ
Similarly, from (19), we get
EðNÞ ¼ EðN0 Þ þ EðN1 Þ 1 1 ðl nÞCð1Þ 1 v þ p0;0 : ¼ ðlb kÞ h lb hAð1Þ
Applying (24), this becomes
P1 ð1Þ ¼
hP0 ð1Þ þ
hzP 00 ð1Þ
½hp0;0 þ lb p1;1
lb k
¼
hP00 ð1Þ
¼
lb k
ð31Þ
ð32Þ
Using Little’s law, the mean waiting time in the system can be obtained as
lb k
implying that
P00 ð1Þ
h l k ðl nÞCð1Þ EðN1 Þ ¼ P01 ð1Þ ¼ EðN0 Þ ¼ b 1 v p0;0 : hAð1Þ lb lb
Hence, the steady state mean number of customers in the system is
hP 0 ðzÞ þ hzP 00 ðzÞ ½hp0;0 þ lb p1;1 : z!1 kð1 zÞ ðkz lb Þ
lim P1 ðzÞ ¼ lim z!1
EðWÞ ¼ EðNÞ=k: P1 ð1Þ:
ð25Þ
Applying L’Hopital’s rule
Another important measure of system performance is the total waiting time of a customer who actually completes their service before leaving the system. Let us denote this by Wserved. Let Wnj denote the conditional waiting time of a customer who does not abandon the system, given that the state upon arrival is (n, j). The waiting time of a customer is measured from the moment of arrival until departure, either after completion of service or due to abandonment. Then,
lim P00 ðzÞ
EðW n1 Þ ¼
h
The expression of P 00 ðzÞ can also be found from (9) as
ðkz lv þ nÞð1 zÞ þ hz P 0 ðzÞ ðhz ðlv nÞð1 zÞÞp0;0 þ lb zp1;1 : P 00 ðzÞ ¼ nzð1 zÞ ð26Þ
z!1
½kð1 2zÞ þ lv n þ hP0 ðzÞ fh þ ðlv nÞgp0;0 þ lb p1;1 ¼ lim : z!1 nð1 2zÞ ½ðkz lv þ nÞð1 zÞ þ hz
ðk lv þ nÞP0 ð1Þ þ ðlv nÞp0;0 : nþh
ð27Þ
Eqs. (25) and (27) imply that
lb k h
P1 ð1Þ ¼
ðk lv þ nÞP0 ð1Þ þ ðlv nÞp0;0 ; nþh
which simplifies to
h i l ½hðk lv þ nÞ þ ðn þ hÞðlb kÞ p0;0 þ b p1;1 h ¼ ðn þ hÞðlb kÞ þ hðlv nÞp0;0 :
;
ðn þ hÞðlb kÞ : nÞCð1Þ
hðn þ lb lv Þ þ nðlb kÞ hðlv nÞ þ ðlvhAð1Þ
h 1 þ EðW n1 Þ h þ k þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn k 1 þ þ EðW n0 Þ h þ k þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn ðn þ 1Þn n 1 þ h þ k þ lv þ ðn þ 1Þn n þ 1 h þ k þ lv þ ðn þ 1Þn lv 1 þEðW n1;0 Þ þ h þ k þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn þEðW n1;0 Þ ;
1 h þ k þ lv þ nn h þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn hðn þ 1Þ þ ðlv þ nnÞEðW n1;0 Þ : þ
EðW n0 Þ ¼
For j = 0 and n = 0,
2.2. Performance measures
EðW 00 Þ ¼
lb
From (24), the equilibrium probability that the system is in WV is
ðlv nÞCð1Þ p0;0 hAð1Þ
ð28Þ
and the probability that the system is in non-vacation period is
ðl nÞCð1Þ p0;0 : P1 ð1Þ ¼ 1 P0 ð1Þ ¼ 1 v hAð1Þ
ð29Þ
The mean number of customers when the system is on WV period is
EðN0 Þ ¼ P00 ð1Þ ¼
lb k h
ð33Þ
For j = 0 and n P 1,
h For n = 0, the customers of the system will not be impatience and the system boils down to an M/M/1/MWV model, given by Servi and Finn (2002).
P0 ð1Þ ¼
n ¼ 1; 2; 3; . . .
because, in the second term the arrival of a customer does not change the waiting time of a customer present in the system and in the third term only n customers can abandon the system as our customer is not impatient. This expression can be further rewritten as
Putting p1,1 in terms of p0,0, we get the expression for p0,0 as
p0;0 ¼
lb
EðW n0 Þ ¼
Therefore, using (24),
P00 ð1Þ ¼
nþ1
ðl nÞCð1Þ 1 v p0;0 hAð1Þ
ð30Þ
and the expected number of customers when the server is on nonvacation period is
ð34Þ
h 1 1 þ h þ k þ lv þ n h þ k þ lv þ n lb k 1 þ EðW 00 Þ þ h þ k þ lv þ n h þ k þ lv þ n lv 1 ; þ h þ k þ lv þ n h þ k þ lv þ n
which can be simplified to
EðW 00 Þ ¼
h þ k þ lv h : þ h þ lv þ n h þ k þ lv þ n lb 1
ð35Þ
Using (35) and iterating (34), we obtain for n P 0
1 h þ k þ lv þ nn ðn þ 1Þh þ h þ lv þ ðn þ 1Þn h þ k þ lv þ ðn þ 1Þn lb n # n X h þ k þ lv þ ðk 1Þn kh Y lv þ in þ : ð36Þ þ h þ k þ lv þ kn lv i¼k h þ lv þ in k¼1
EðW n0 Þ ¼
212
N. Selvaraju, C. Goswami / Computers & Industrial Engineering 65 (2013) 207–215
Finally, we get the mean waiting time of customers served by the system as
EðW serv ed Þ ¼
1 1 X X pn;0 EðW n0 Þ þ pn;1 EðW n1 Þ; n¼0
n¼1
W d ðsÞ
which after using (33) becomes
EðW serv ed Þ ¼
1 X EðN1 Þ þ P1 ð1Þ pn;0 EðW n0 Þ þ : n¼0
lb
ð37Þ
It is often very much of interest to try and decompose the quantities of interest into various factors in order to have a better comparison with the existing models. A stochastic decomposition property indicates the effects of system vacation on its performance measures like queue length and waiting times and plays an important role in vacation models. We can attempt to do the same for the system under consideration as this is an M/M/1 model with MWVs. We have proved the following two results that this impatient WV model also satisfies such decomposition for stationary queue length and stationary waiting times into one that corresponds to a pure M/M/1 system plus a quantity that is due to the WV. Theorem 2.3. For q < 1, the stationary queue length N can be decomposed into a sum of two independent random variables as
N ¼ Nc þ Nd ; where Nc is the queue length of a classical M/M/1 queue without vacations and Nd is the additional queue length due to the effect of MWV with its PGF
" # p0;0 ðlb kzÞð1 zÞ hz ðlv nÞCð1Þ Nd ðzÞ¼ P0 ðzÞ þ z : p0;0 Að1Þ ðlb kÞð1 zÞ ð38Þ Proof
NðzÞ ¼ P0 ðzÞ þ P1 ðzÞ ¼ 1 þ
hz P0 ðzÞ ðkz lb Þð1 zÞ ðlv nÞCð1Þz lb k p0;0 ¼ ðkz lb Þð1 zÞAð1Þ lb kz lb kz hz P ðzÞ lb k ðlb kÞð1 zÞ 0 ðlv nÞCð1Þz 1q þ p0;0 ¼ Nd ðzÞ: ðlb kÞð1 zÞAð1Þ 1 qz
v n zn ðsayÞ;
Let s = k(1 z) which gives z ¼ 1 ks and 1 z ¼ ks . Putting these relations in (38), we get the desired expression. h
3. Single working vacation (SWV) model The M/M/1 queue with impatient customers and a SWV policy is different from the MWV model in a way that when the server returns from it’s WV period and finds no customer in the system, it does not take another vacation but remains idle until the next arrival. So, in the SWV model, the server may stay idle for some period whereas the MWV does not have any idle time for the server. In this SWV model also, the customers become impatient and activate the impatient timer ‘T’ only if they found the server in WV period servicing at a lower rate upon its arrival and if the server is not empty. For the SWV model, the Markov chain D = {(Nt, Qt), t P 0} can be defined as in MWV case but with the state space E = {(i, j), i = 0, 1, . . . , j = 0, 1}. Here j = 1 if the server is active (idle or busy) and j = 0 if the server is in WV. We have the balance equations for the equilibrium state probabilities given by
ðk þ hÞp0;0 ¼ lv p1;0 þ lb p1;1 ;if n ¼ 0; ð40Þ ðk þ lv þ h þ ðn 1ÞnÞpn;0 ¼ kpn1;0 þ ðlv þ nnÞpnþ1;0 ;if n P 1;ð41Þ kp0;1 ¼ hp0;0 ;if n ¼ 0; ð42Þ
Let us define the generating functions, for 0 < z < 1
G0 ðzÞ ¼
1 X zn pn;0 ;
G1 ðzÞ ¼
n¼0
1 X zn pn;1 :
ð44Þ
n¼0
We have G0(1) + G1(1) = 1. Multiplying Eq. (41) with zn and summing over n gives
nzð1 zÞG00 ðzÞ ½ðkz lv þ nÞð1 zÞ þ hzG0 ðzÞ þ ðn lv Þð1 zÞp0;0 þ lb zp1;1 ¼ 0:
" # 1 1 1 X 1 X X 1 h X pn;0 zn q pn;0 znþ1 þ pnþk;0 zn 1 q n¼0 lb n¼1 k¼0 n¼0 1 X
NðzÞ ¼ W ðkð1 zÞÞ:
ðk þ lb Þpn;1 ¼ kpðn1Þ;1 þ hpn;0 þ lb pnþ1;1 ;if n P 1:ð43Þ
Nd(z) can be written in series expansion as
¼
" p0;0 ðlb s kÞs hðk sÞ s ¼ P0 1 p0;0 k ðlb kÞs ðl nÞCð1Þ þðk sÞ v : Að1Þ
Proof. From the distributional form of Little’s Law, in Keilson and Servi (1988), we have the relation
2.3. Stochastic decompositions in the MWV model
Nd ðzÞ ¼
where Wc is the waiting time of a customer corresponding to classical M/M/1 queue and has exponential distribution with parameter lb(1 q) and Wd is the additional delay due to the effect of MWV with its Laplace–Stieltjes transform (LST)
ð45Þ
For limit z tending to 1, we get h G0(1) = lbp1,1. For 0 < z < 1, letting n = 0 we get,
ð39Þ
n¼0
h i P where v 0 ¼ 11 q p0;0 and v n ¼ 11 q pn;0 qpn1;0 þ lh 1 k¼0 pnþk;0 ; b n P 1. From Eqs. (20) and (22) it can be shown that vn 2 [0, 1] for P n P 0 and 1 h n¼0 v n ¼ 1. This proves that Nd(z) is a PGF.
G0 ðzÞ ¼
ðlv nÞp0;0 zðlb p1;1 þ ðlv nÞp0;0 Þ : ðkz2 zðk lv Þ þ lv
ð46Þ
For n – 0, we have G00 ðzÞ
kz lv þ n h n lv lb þ G0 ðzÞ þ p0;0 þ p ¼ 0: zn nð1 zÞ zn nð1 zÞ 1;1
ð47Þ
Solving this differential equation, as in the MWV case, we get Theorem 2.4. If q < 1, the stationary waiting time can be decomposed into the a sum of two independent random variables as
W ¼ Wc þ Wd;
lv
G0 ðzÞ ¼ zð n 1Þ ð1 zÞn h
l 1 p0;0 CðzÞ b p1;1 AðzÞ : n n
lv
b Að1Þ Here, we have G0 ð0Þ ¼ p0;0 ¼ ðllnÞCð1Þ p1;1 , which gives, v
ð48Þ
N. Selvaraju, C. Goswami / Computers & Industrial Engineering 65 (2013) 207–215
p1;1 ¼
ðlv nÞCð1Þ p0;0 : lb Að1Þ
Therefore, we get a similar expression for G0(z) as in MWV case
G0 ðzÞ ¼ p0;0
lv
1
n
lv h Cð1Þ AðzÞ zð n 1Þ ð1 zÞn : CðzÞ Að1Þ
ð49Þ
ðkz lb Þð1 zÞG1 ðzÞ ¼ hzG0 ðzÞ lb ð1 zÞp0;1 lb zp1;1 :
We state the following results concerning the stochastic decompositions in the SWV model which can be proved in a similar way as in the MWV case dealt with earlier.
For 0 < z < 1,
h ðl nÞCð1Þ p0;0 ðz zG0 ðzÞ ð1 zÞ þ z v k lb lb Að1Þ h
1
1
1Þ ð1 qzÞ :
ð50Þ
To find G0(z) and G1(z) explicitly, we have to find an explicit solution for p0,0. Applying L’Hopital’s rule in (50),
G00 ð1Þ ¼
lb k h
l G1 ð1Þ b p0;0 : k
ð51Þ
Applying L’Hopital’s rule in (47) gives
lim G00 ðzÞ ¼ lim z!1
z!1
½kð1 2zÞ þ lv n þ hG0 ðzÞ ðlv nÞp0;0 lb p1;1 ; nð1 2zÞ ðkz lv þ nÞð1 zÞ hz ð52Þ
which becomes
G00 ð1Þ ¼
ðk lv þ nÞG0 ð1Þ þ ðlv nÞp0;0 : nþh
ð53Þ
Equating G00 ð1Þ from (51) and (53)
ðn þ hÞðlb kÞ hðlv nÞ þ
k
hðn þ hÞ þ
ðlv hÞCð1Þ ½hðn hAð1Þ
" p0;0 ðlb kzÞð1 zÞ hz P0 ðzÞ p0;0 ðlb kÞð1 zÞ h ðl hÞCð1Þ þlb ð1 zÞ þ z v : k Að1Þ
Nd ðzÞ ¼
Theorem 3.2. If q < 1, the stationary waiting time W can be decomposed into the a sum of two independent random variables as W = Wc + Wd, where Wc is the waiting time of a customer corresponding to classical M/M/1 queue and has exponential distribution with parameter lb(1 q) and Wd is the additional delay due to the effect of SWV with its LST
W d ðsÞ ¼
We finally obtain p0,0 as lb
Theorem 3.1. For q < 1, the stationary queue length N can be decomposed into a sum of two independent random variables as N = Nc + Nd, where Nc is the queue length of a classical M/M/1 queue without vacations and Nd is the additional queue length due to the effect of SWV with its PGF given by
" p0;0 ðlb s kÞs hðk sÞ s P0 1 p0;0 k ðlb kÞs sh ðlv hÞCð1Þ þlb þ ðk sÞ : k Að1Þ
½ðk lv þ nÞ þ ðn þ hÞðlb kÞG0 ð1Þ l ¼ ðn þ hÞðlb kÞ b hðn þ hÞp0;0 : k
p0;0 ¼
The average waiting time of customers in system who are served before leaving the system, E(Wserved), can be derived similarly as in the MWV case with EðW n1 Þ ¼ nþ1 lb but for n = 0, 1, 2, . . . rather than n P 1, and E(Wn0) is as given by (34). Finally, E(Wserved) is given by (37) but with the second sum starting from n = 0. 3.2. Stochastic decompositions in the SWV model
Multiplying (43) with zn and summing over n gives
G1 ðzÞ ¼
213
þ lb lv Þ þ nðlb kÞ
:
These results can be proved similar to the MWV case in the previous section.
ð54Þ As we have given in MWV case, we give below the performance measures of this SWV model in a similar manner. 3.1. Performance measures The system performance measures of SWV model are given below. The probabilities of system being in WV and in non-vacation period are, respectively,
G0 ð1Þ ¼
ðlv hÞCð1Þ p0;0 hAð1Þ
ð55Þ
and G1 ð1Þ ¼ 1 G0 ð1Þ ¼ 1
ðlv hÞCð1Þ p0;0 : hAð1Þ
ð56Þ
The mean number of customers during WV period can be given as
EðN0 Þ ¼ G00 ð1Þ ¼
lb k h
1
ðlv hÞCð1Þ l p0;0 b p0;0 hAð1Þ k
ð57Þ
and the mean number of customers during non-vacation period as
EðN1 Þ ¼ G01 ð1Þ ¼
lb k ðl hÞCð1Þ p0;0 : 1 v hAð1Þ lb
ð58Þ
The average number of customers in the system will be
EðNÞ ¼ ðlb kÞ
1 1 ðl hÞCð1Þ l 1 v þ p0;0 b p0;0 : h lb hAð1Þ k
ð59Þ
4. Comparison of the models The model M/M/1/WV with impatient customer is studied with MWV and also with SWV policy which refers to WDM links with two different types of wavelength assignment or reconfiguration policies used in OBS networks. The performance of an OBS network depends upon its high speed data access and minimum packet loss qualities. As packet delay (the waiting in a queue) is a vital issue for packet lost, we studied the waiting time of customers in queue compared to the impatience rates of the customers for both WV policies. These give us a picture of how fast the allocation of transmission path should be to have minimum packet loss. To show the impact of packet loss or impatience on system performance, the WV policies, MWV and SWV are analytically compared. First, we have compared the probability of the system being idle in WV (p0,0) of MWV model to that of SWV model using their analytic expressions. Next, we compared the mean number of customers in the system in vacation or in non-vacation between the two models. A model can be said to be efficient than another, if its mean waiting time of the customers is less compared to the other. For a M/M/1/WV queue without impatient we get that a MWV model is always better than a SWV model in the sense that the mean queue length in MWV is always less than that in the SWV model. This result can be verified from the expressions of mean queue lengths of MWV model and SWV model found in Liu et al. (2007) and Tian et al. (2008) respectively. From the definitions of
N. Selvaraju, C. Goswami / Computers & Industrial Engineering 65 (2013) 207–215
multiple and SWV policies, it seems that, with impatient customers, SWV policy is more efficient compared to the MWV one. But because of impatience the behavior differs. To differentiate the terms of both the models we will use superscript M and S for the MWV and SWV respectively. From the expressions of p0,0 in (22) and (54) it can be seen that the probability of SWV is less that of MWV.
1 1 lb h ¼ þ ; kð l pS0;0 pM b kÞ 0;0
ð60Þ
which gives pS0;0 < pM 0;0 . The mean number of customers in the systems, when the server in non-vacation are
l k ðl hÞCð1Þ M p0;0 EM ðN 1 Þ ¼ b 1 v hAð1Þ lb lb k ðl hÞCð1Þ S S p0;0 ; 1 v and E ðN1 Þ ¼ hAð1Þ lb
1.7 EM(N ) 0
1.6
Mean Queue length in WV period
214
ES(N ) 0
1.5 1.4 1.3 1.2 1.1 1
0.9 0.8 0.7 0
0.2
0.4
0.6
0.8
1
1.2
impatience rate ξ
which implies EM ðN1 Þ < ES ðN1 Þ:
ð61Þ
Fig. 2. Mean queue lengths during WV vs. n
For the system on vacation,
lb k
M
E ðN0 Þ ¼
h
and ES ðN0 Þ ¼
"
ðlb kÞðlv hÞCð1Þ
lb k h
"
h2 Að1Þ
# pM 0;0
ðlb kÞðlv hÞCð1Þ h2 Að1Þ
þ
lb k
# pS0;0
and the total queue lengths are
1 1 ðl nÞCð1Þ M 1 v þ p0;0 ; EM ðNÞ ¼ ðlb kÞ h lb hAð1Þ 1 1 ðlv hÞCð1Þ S l S 1 þ p0;0 b pS0;0 : E ðNÞ ¼ ðlb kÞ h lb hAð1Þ k From the expressions of EM(N0) and ES(N0), it is difficult to conclude any relationship between them. Therefore, the sum, EM(N) (=EM(N0) + EM(N1)), may be greater than, equal or less than ES(N). For example, let us take the impatience model in which k = 0.7, lb = 1 and lv = 0.5. For three different values of h the mean queue lengths of the system are shown in Fig. 1. When h = 1.7 the queue length of SWV model is less than MWV and for h = 2.1 the MWV model gives lesser queue length. When h = 1.9, we see that at the point n = 0.2 both the models give the same value. So for n < 0.2 the SWV model becomes better compared to MWV one and for
mwv, θ=1.7 swv, θ=1.7 mwv, θ=1.9 swv, θ=1.9 mwv, θ=2.1 swv, θ=2.1
Mean Queue length
0.31
0.305
0.3
0.295
0.29 0.1
0.15
0.2
0.25
0.3
0.35
0.4
impatience rate ξ Fig. 1. Mean queue lengths vs. n, for different values of h
0.45
n > 0.2 the MWV model works better. That means that, when the mean of vacation duration time is less, the MWV model gives better performance than a corresponding SWV model. In Fig. 2, we have compared the mean queue lengths during a WV period of both the models. The parameter values are k = 1.8, lb = 2, lv = 1.3 and h = .09. The difference between the queue lengths increases with the increase in the impatient rate n. So, if the impatient rate is small, the MWV and SWV models give the same queue lengths during WV periods but for the higher impatient rates SWV model will give better performance. This study reveals that not only the vacation duration rate but also the impatience rate plays a major role in the performance of the model. Thus, for a given parameter values, we can choose a model to get better performance of a single server impatient queue with WVs. This underlines the fact that if the system is not in WV, average number of customers in the MWV model is always less than the SWV model. When the average vacation duration time is less, the MWV model gives better performance than the SWV model, whereas for higher impatient rates the SWV model becomes more efficient as it reduces the waiting time of the customers in the system. 5. Conclusion We have investigated the effect of impatient behavior of customers in a M/M/1/WV model. Two types of WV termination policies are taken, the multiple working vacation (MWV) policy and the single working vacation (SWV) policy. Closed-form probabilities are derived using the identities involving beta functions and degenerate hypergeometric functions. Stochastic decomposition properties are verified for both MWV and SWV cases. This work underlines the fact that if the system is not in WV, average number of customers in the MWV model is always less than the SWV model. Our study shows that packet loss is less when multiple data carrying channels are used as WDM links rather than the individual ones. Also higher is the rate of wavelength allocation the lesser will be the system congestion or loss of packets. However individual WDM links are efficient compared to multiple ones only if the data processing delay is quite high or delay budget is less than the processing delay; otherwise, multiple WDM links can ensure high speed data access with minimum data loss through OBS networks. We consider, in this paper, the arrival pattern of packets (customers) to be single arrivals. We can further extend the analysis for packets with batch arrivals, since communication networks
N. Selvaraju, C. Goswami / Computers & Industrial Engineering 65 (2013) 207–215
mostly deal with bulk arrivals of packets. This model may be extended for Markov modulated arrival process to capture correlated arrivals which is very useful in telecommunication networks. Finite buffer models can also be studied and analyzed. Not only the arrivals but the service patterns in batches are also found in networks. Considering the retransmission we can model this OBS system as a retrial (or repeated attempt) queueing model. Assigning priority to packets is another important feature to study in future. References Abramowitz, M., & Stegun, I. (1972). Handbook of mathematical functions. Dover Publication. Altman, E., & Yechiali, U. (2006). Analysis of customer’s impatience in queue with server vacations. Queueing Systems, 52, 261–279. Baba, Y. (2005). Analysis of a GI/M/1 queue with multiple working vacations. Operations Research Letters, 33, 201–209. Banik, A. (2010). Analysis of single server working vacation in GI/M/1/N and GI/M/1/ 1 queueing systems. International Journal of Operational Research, 7(3), 314–333. Bocquet, S. (2005). Queueing theory with reneging. Scientific and Technical Report of DSTO. Goswami, C., & Selvaraju, N. (2010). The discrete-time MAP/PH/1 queue with multiple working vacations. Applied Mathematical Modelling, 34(4), 931–946. Coudert, D., Huc, F., Mazauric, D., Nisse, N., Antipolis, S., & Sereni, J. S. (2009). Reconfiguration of the routing in WDM networks with two classes of services.
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