Abstract: We consider a single server retrial queue with batch arrivals, two phases of heterogeneous service and a general vacation time under Bernoulli ...
Quality Technology & Quantitative Management Vol. 5, No. 2, pp. 145-160, 2008
QTQM © ICAQM 2008
On the Single Server Batch Arrival Retrial Queue with General Vacation Time under Bernoulli Schedule and Two phases of Heterogeneous Service M. Senthil Kumar and R. Arumuganathan Department of Mathematics & Computer Applications, PSG College of Technology, Coimbatore, India (Received July 2006, accepted March 2007)
______________________________________________________________________ Abstract: We consider a single server retrial queue with batch arrivals, two phases of heterogeneous
service and a general vacation time under Bernoulli schedule. We carry out steady state system size distribution of number of customers in retrial group, expected number of customers in retrial group and expected waiting time of the customers in the orbit. We discuss its application of the proposed model to the analysis of a communication protocol.
Keywords: CSMA/CD protocol, essential service, generalized vacation time, Mobile IP, retrial queues, steady state distribution.
______________________________________________________________________ 1. Introduction
R
etrial queues describe operation of many telecommunication networks, e.g., the local and wide area networks with the random multiple access protocols, call centers, etc. There has been rapid growth in the literature on the queueing systems with repeated attempts which are characterized by the following feature. When an arriving customer finds that all servers are busy and no waiting position is available, the customer joins a virtual pool of blocked customer called ‘orbit’. The detailed overviews of the related references with retrial queues can be found in the recent book of Falin and Templeton [9] and the survey papers, Artalejo [1],[2]. In this paper, we focus on single server retrial queue with batch arrivals with two phase of heterogeneous service under Bernoulli schedule and a general vacation time. This first batch arrival retrial queueing model was introduced by Falin [8] who assumed the following rule: “If the server is busy at the arrival epoch, then the whole batch joins the retrial group, whereas the server is free, then one of the arriving units starts its service and the rest join the retrial group”.
The single server retrial queue with priority calls have been studied by Choi et al. [4], [5],[6] for many applications in telecommunication and mobile communication. The distribution of number of customers served in an M/G/1 retrial queue have been analysed by Lopez–Herrero [13] for finding the probability that at most k customers were served during the busy period of an M/G/1 retrial queue. Krishnakumar et al. [12] analysed the M/G/1 retrial queue with feedback and starting failures using supplementary variable technique. The overwhelming literature contributions consider queueing system with two phases of service. Madan [14] considered the classical M/G/1 queueing system in which the
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server provides the first essential service to all the arriving customers whereas some of them receive second optional service. The first essential service follows general distribution and second optional service follows exponential distribution. Medhi [16] generalized the model by considering that the second optional service, is also governed by a general distribution. Choudhury [7] investigated this queueing model by including the waiting time distribution. Madan and Choudhury [15] considered the queueing model with two phases of Heterogeneous service under Bernoulli Schedule and a General vacation time. Krishnakumar et al. [11] consider an M/G/1 retrial queue with additional phase of service. While at the first phase of service, the server may push-out the customer who is receiving such a service, to start the service of another priority arriving customer. The interrupted customers join a retrial queue and the customer at the head of this queue is allowed to conduct a repeated attempt in order to start again his first phase of service after some random time. The motivation for this model comes from some computer and communication networks where messages are processed in two stages by a single server. Recently, Artalejo et al. [3] analyze the steady state analysis of the M/G/1 queueing system with repeated attempts and two phase service using Embedded Markov chain method. In this paper, we consider a single server retrial queue with batch arrivals and two phases of essential services and general vacation time under Bernoulli Schedule. Analytical treatment of this model is obtained using supplementary variable technique. We obtain the probability generating function of number of customers in the retrial group. Our main motivation is coming from some applications to Local Area Networks (LAN), CSMA/CD protocol, client – server communication, telephone systems and electronic mail services on Internet. The interest in mobile communication on internet means that the IP protocol, originally designed for stationary devices, must be enhanced to allow the use of mobile computers, computers that move from one network to another. The detailed information regarding Mobile IP can be obtained from Forouzan and Fegan [10]. Mobile IP has two addresses for a mobile host: one home address and one care of address. The home address is permanent; the care of address changes as the mobile host moves from one network to another. To communicate with a remote host, a mobile host goes through three phases: agent discovery, registration, and data transfer. The first phase, agent discovery, involves the mobile host, the foreign agent, and the home agent. The second phase, registration, also involves the mobile host and the two agents. Finally, in the third phase (i.e. General vacation time), all four entities are involved. In this process, while receiving messages, Home agent relays messages (for the mobile host) to a foreign host. A foreign agent sends relayed messages to a mobile host. A mobile host on its home network learns the address of a home agent through a process called agent discovery. A mobile host on a foreign network learns the address of a foreign agent through agent discovery. Mobile host get connected with a remote host from foreign network (or Mobile network) through foreign agent (or Home agent) with three phases of services. While sending batch of packets from remote host to Mobile host, one of the packets be sent through three possible phases. The first phase, agent discovery, involves the mobile host, the foreign agent, and the home agent. The second phase, registration, also involves the mobile host and the two agents. Finally, in the third phase (i.e. General vacation time with probability p = 1), all four entities are involved. If one of the packets is under these three phases of service, all other packets should be kept in buffer (i.e. retrial group) on remote host and repeatedly tried to be sent to mobile host. This kind of system can be modeled as single server retrial queue with batch arrivals and two phases of essential service and general vacation time under Bernoulli Schedule.
On the Single Server Batch Arrival Retrial Queue
147
In Mobile IP communication on internet, we analyze the clients’ batch of requests will be processed by two phases of essential service and General vacation time. The analytical treatment of this model is done by the supplementary variable technique to obtain probability generating function of the number of customers in orbit at steady state. Also, we investigated the performance characteristics of this system.
Figure 1. Home agent and foreign agent.
2. The Mathematical Model We consider the single server retrial queueing system with batch arrival. The primary calls arrive in bulk according to Poisson process with rate λ . In the batch arrival retrial queue it is assumed that at every arrival epoch a batch of k primary calls arrives with probability gk. If the server is busy at the arrival epoch, then all these calls join the orbit, whereas if the server is free, then one of the arriving calls begins its service and the others form sources of repeated calls. The server provides preliminary first essential service (FES) and second essential service (SES) to all arriving calls. As soon as the SES of a call is completed, the server may go for a vacation of random length S3 with probability ‘p’ (0< p < 1) or may continue to serve the next call, if any, with probability ‘q’ (= 1 – p). Primary calls finding the server free upon arrival automatically start their FES. However, if a primary call finds the server busy (attending FES or SES), then it joins the orbit in order to seek service again until it finds the server free. The time between two successive repeated attempts of each call in orbit is assumed to be exponentially distributed with rate ‘v’. Let S1 ( x ) ( s1 ( x )) S1 (θ ) [S10 ( x )] be the cumulative distribution function (probability density function) {Laplace – Stieltjes transform} [remaining service time] of FES. S2 ( x ) ( s2 ( x )) S2 (θ ) [S 2 0 ( x )] be cumulative distribution function (probability density function) {Laplace-Stieltjes transform} [remaining service time] of SES. S3 ( x ) ( s3 ( x )) S3 (θ ) [S30 ( x )] be cumulative distribution function (probability density function) {Laplace-Stieltjes transform} [remaining vacation time] of vacation. N (t) denotes the number of customers in the orbit at time t.
{
{
{
}
}
}
X ( z ) = ∑ ∞k =1 g k z k is denoted as the generating function of the batch size distribution. The mean batch size is denoted as E(X)= X '(1) .
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if the server is idle ⎧ 0, ⎪1 , if the server is doing FES ⎪ The server state is denoted as, C (t ) = ⎨ ⎪ 2, if the server is doing SES ⎪⎩3 , if the server is on vacation Now we define, P0,n (t )dt = Pr{N (t ) = n,C (t ) = 0} n ≥ 0 and Pi ,n ( x , t )dt = Pr{N (t ) = n,C (t ) = i , x ≤ Si 0 (t ) ≤ x + dt} ; n > 0; i = 1,2,3.
3. Steady State System Size Distribution The following equations are obtained for the queueing system, using supplementary variable technique. P0, j (t + Δt ) = P0, j (t )(1 − λΔt − jv Δt ) + P3, j (0, t ) Δt + q P2, j (0, t ) Δt j +1
P1, j ( x − Δt , t + Δt ) = P1, j ( x , t )(1 − λΔt ) + λΔt ∑ g k P0, j − k +1 (t ) s1 ( x ) k =1
j
+ ( j + 1)vP0, j +1 (t ) s1 ( x ) Δt + λΔt ∑ g k P1, j − k ( x , t ) k =1
j
P2, j ( x − Δt , t + Δt ) = P2, j ( x , t )(1 − λΔt ) + λΔt ∑ g k P2, j − k ( x , t ) + P1, j (0, t ) s2 ( x ) Δt k =1 j
P3, j ( x − Δt , t + Δt ) = P3, j ( x , t )(1 − λΔt ) + λΔt ∑ g k P3, j − k ( x , t ) + p P2, j (0, t ) s3 ( x ) k =1
The Steady State equations of the above equations are, (λ + jv ) P0, j = P3, j (0) + qP2, j (0) , −
(1)
j +1 j d P1, j ( x ) = −λ P1, j ( x ) + λ ∑ g k P0, j − k +1 s1 ( x ) + ( j + 1) v P0, j +1 s1 ( x ) + λ ∑ g k P1, j − k ( x ) , dx k =1 k =1
(2)
−
j d P2, j ( x ) = −λ P2, j ( x ) + λ ∑ g k P2, j − k ( x ) + P1, j (0) s2 ( x ) , dx k =1
(3)
−
j d P3, j ( x ) = −λ P3, j ( x ) + λ ∑ g k P3, j − k ( x ) + pP2, j (0) s3 ( x ) . dx k =1
(4)
Let LST {P1, j ( x )} = P1, j (θ ) ; LST {P2, j ( x )} = P2, j (θ ) ; LST {P3, j ( x )} = P3, j (θ ) . Taking LST of steady state Equations (2), (3) and (4), we have, j +1
j
k =1
k =1
θ P1, j (θ ) − P1, j (0) = λ P1, j (θ ) − λ ∑ g k P0, j − k +1 S1 (θ ) − ( j + 1)vP0, j +1S1 (θ ) − λ ∑ g k P1, j − k (θ ) ,
(5)
On the Single Server Batch Arrival Retrial Queue
149 j
θ P2, j (θ ) − P2, j (0) = λ P2, j (θ ) − λ ∑ g k P2, j − k (θ ) − P1, j (0)S2 (θ ) , k =1
j
θ P3, j (θ ) − P3, j (0) = λ P3, j (θ ) − λ ∑ g k P3, j −k (θ ) − p P2, j (0)S3 (θ ) . k =1
(6)
(7)
Now, we define the following probability generating functions (PGF) ∞
P0 ( z ) = ∑ P0 j z j , j =0
∞
Pi ( z ,θ ) = ∑ Pi , j (θ ) z j , j =0
∞
Pi ( z ,0) = ∑ Pi , j (0) z j ; i =1, 2, 3.
(8)
j =0
Using PGF Equations (1), (5), (6) and (7) can be written as follows,
λ P0 ( z ) + vzP0 '( z ) = P3 ( z ,0) + qP2 ( z ,0) , (θ − λ + λ X ( z )) P1 ( z ,θ ) = P1 ( z ,0) − λ
(9)
X (z ) P0 ( z )S1 (θ ) − vP0 '( z )S1 (θ ) , z
(10)
(θ − λ + λ X ( z )) P2 ( z ,θ ) = P2 ( z ,0) − P1 ( z ,0)S2 (θ ) ,
(11)
(θ − λ + λ X ( z )) P3 ( z ,θ ) = P3 ( z ,0) − pP2 ( z ,0)S3 (θ ) ,
(12)
Substituting θ = λ − λ X ( z ) in the Equations (10), (11) and (12), we have P1 ( z ,0) = λ
X (z ) P0 ( z )S1 (λ − λ X ( z )) + vP0 '( z )S1 (λ − λ X ( z )) , z
(13)
P2 ( z ,0) = P1 ( z ,0)S2 (λ − λ X ( z )) ,
(14)
P3 ( z ,0) = pP2 ( z ,0)S3 (λ − λ X ( z )) .
(15)
Substituting P1 ( z ,0), P2 ( z ,0) and P3 ( z ,0) in the Equation (9), we have,
(
)
λ P0 ( z ) + vzP0 '( z ) = pS3 (λ − λ X ( z )) + q S2 (λ − λ X ( z )) ⎛ X (z ) ⎞ P0 ( z ) + vP0 '( z ) ⎟ S1 (λ − λ X ( z )) ⎜λ z ⎝ ⎠ From the Equation (16), we get, ⎛
X (z ) ⎞ ⎟ ⎟ P0 ( z ) . ⎟ ⎠
λ ⎜ 1 − [S1 (λ − λ X ( z ))S2 (λ − λ X ( z ))( pS3 (λ − λ X ( z )) + (1 − p ))] z P0 '( z ) = ⎜ v⎜ [S1 (λ − λ X ( z ))S2 (λ − λ X ( z ))( pS3 (λ − λ X ( z )) + q )] − z ⎝
(16)
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On integration, we get, ⎧ X (u ) ⎞ ⎫ ⎛ ⎪⎪ −λ 1 ⎜ 1 − [S1 (λ − λ X (u ))S2 (λ − λ X (u ))( pS3 (λ − λ X (u )) + q )] u ⎟ ⎪⎪ P0 ( z ) = P0 (1) exp ⎨ ⎟ du ⎬ . ∫⎜ ⎪ v z ⎜⎜ [S1 (λ − λ X (u ))S2 (λ − λ X (u ))( pS3 (λ − λ X (u )) + q )] − u ⎟⎟ ⎪ ⎪⎩ ⎝ ⎠ ⎪⎭ (17) Substituting the Equation (13) in Equation (10), we have, X (z ) ⎡ ⎤ (θ − λ + λ X ( z )) P1 ( z ,θ ) = (S1 (λ − λ X ( z )) − S1 (θ )) ⎢λ P0 ( z ) + vP0 '( z ) ⎥ . z ⎣ ⎦
(18)
Substituting the Equation (14) in Equation (11), we have,
(
)
(θ − λ + λ X ( z )) P2 ( z ,θ ) = P1 ( z ,0) S2 (λ − λ X ( z )) − S2 (θ ) .
(19)
Substituting the Equation (15) in Equation (12), we have,
(
)
(θ − λ + λ X ( z )) P3 ( z ,θ ) = pP1 ( z ,0)S2 (λ − λ X ( z )) S3 (λ − λ X ( z )) − S3 (θ ) .
(20)
Using the Equations (18), (19) and (20), the partial generating functions ∞
Pi ( z ,0) = ∑ Pij (0) z j ; i = 1, 2,3 j =0
are given by,
P1 ( z ,0) =
(S (λ − λ X ( z )) − 1) ⎡⎢⎣λ P ( z ) X z( z ) + vP '( z ) ⎤⎥⎦ 1
0
( −λ + λ X ( z ) )
P2 ( z ,0) =
P3 ( z ,0) =
0
,
(S (λ − λ X ( z )) − 1) P ( z,0) , 2
1
(22)
( −λ + λ X ( z ) )
(
)
S2 (λ − λ X ( z )) p S3 (λ − λ X ( z )) − 1 P1 ( z ,0)
( −λ + λ X ( z ) )
(21)
.
(23)
It should be noted that the probability generating function P(z) of number of customers in orbit at an arbitrary epoch can be expressed as follows, P ( z ) = P0 ( z ) + P1 ( z ,0) + P2 ( z ,0) + P3 ( z ,0) Under the normalizing condition P(1) =1, consider P0 (1) = (1 − ρ ) and lim( ∑ i3=1 Pi ( z ,0)) = z →1 where ρ is the traffic intensity. Using the Equations (21), (22) and (23), we have
On the Single Server Batch Arrival Retrial Queue
P ( z ) = P0 ( z ) +
+
151
(S (λ − λ X ( z )) − 1) ⎡⎢⎣λ P ( z ) X z( z ) + vP '( z ) ⎤⎥⎦ 1
0
0
( −λ + λ X ( z ) )
(S (λ − λ X ( z )) − 1) ⎛⎜⎝ λ X z( z ) P ( z ) + vP '( z ) ⎞⎟⎠ S (λ − λ X ( z )) 2
0
0
1
( −λ + λ X ( z ) )
(
)
⎛ X (z ) ⎞ S2 (λ − λ X ( z )) p S3 (λ − λ X ( z )) − 1 ⎜ λ P0 ( z ) + vP0 '( z ) ⎟ S1 (λ − λ X ( z )) z ⎝ ⎠ + − + ( ) X z λ λ ( )
P (z ) =
(1 − z ) P0 ( z ) [S1 (λ − λ X ( z ))S2 (λ − λ X ( z ))( pS3 (λ − λ X ( z )) + q )] − z
(24)
From the Equation (24), we have Lt z →1 P ( z ) = 1 . It immediately follows that the steady state condition is ρ = λ E ( X ) ( E (S1 ) + E (S2 ) + pE (S3 ) ) < 1 .
4. Performance Characteristics Some useful results of our model are listed below. (a) The traffic intensity is given by
ρ = λ E ( X ) ( E (S1 ) + E (S2 ) + pE (S3 ) ) ,
(25)
(b) The mean number of customers in the orbit ‘L’ is derived using the Equation (24) ∞
L = ∑ nPn = Lt n =0
d
z →1 dz
P (z ) ,
Using L’Hospital rule, we have,
L=
λ 2 ( E ( X )) 2 β 2 + ρ ( X ''(1))/ E ( X ) λ ( ρ + E ( X ) − 1) + , 2(1 − ρ ) v (1 − ρ )
(26)
where
β 2 = E [S12 ] + E [S22 ] + pE [S32 ] + 2 E [S1 ]E [S2 ] + 2 E [S1 ] pE [S3 ] + 2 E [S2 ] pE [S3 ] . (c) Mean waiting time in retrial queue Using Little’s formula, the mean waiting time in the retrial queue (W) is obtained as,
E (W ) =
L λE (X )
(27)
5. Particular Cases Case I: In case, if there is no second phase of essential service (ie. S2 (λ − λ X ( z )) =1) and no vacation (ie. S3 (λ − λ X ( z )) = 1 ), then
(i) The PGF of distribution of number of customers in the orbit is
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P (z ) =
(
(1 − z ) P0 ( z ) , S1 (λ − λ X ( z )) − z
(28)
)
where ⎧ X (u ) ⎞ ⎫ ⎛ ⎪⎪ −λ 1 ⎜ 1 − [S1 (λ − λ X (u ))] u ⎟ ⎪⎪ P0 ( z ) = P0 (1) exp ⎨ ⎟ du ⎬ . ∫⎜ ⎪ v z ⎜⎜ [S1 (λ − λ X (u )) − u ] ⎟⎟ ⎪ ⎪⎩ ⎝ ⎠ ⎭⎪ Equation (28) agrees the PGF of the distribution of number of customers in the orbit of M/G/1 batch arrival retrial queue in the steady state obtained by Falin [8]. Further, by specifying vacation time random variable as well as service time random variables as Deterministic, Erlang and exponential distribution, we will discuss some other particular cases of this model. Case II: Single server retrial queue with batch arrivals and Erlangian vacation time under Bernoulli’s schedule and two phases of heterogeneous services.
We assume that vacation time is k Erlang with probability density function, s3 =
( k μ ) k x k −1e − k μ x , k > 0 , where μ is the parameter. ( k − 1)!
Hence, the PGF of the retrial queue size distribution is as follows when k
⎛ ⎞ uk S3 (λ − λ X ( z )) = ⎜ ⎟ , ⎝ uk + λ (1 − X ( z )) ⎠ P (z ) =
(1 − z ) P0 ( z ) k ⎡ ⎛ ⎛ ⎞⎤ ⎞ uk ⎢S1 (λ − λ X ( z ))S2 (λ − λ X ( z )) ⎜ p ⎜ ⎟⎥ − z q + ⎜ ⎝ uk + λ (1 − X ( z )) ⎟⎠ ⎟⎥ ⎢ ⎝ ⎠⎦ ⎣
(29)
,
(30)
where k ⎧ ⎛ ⎡ ⎛ ⎞ ⎤ X (t ) ⎞ uk ⎪ ⎜ 1 − ⎢S (λ − λ X (t ))S (λ − λ X (t )) ⎜ p ⎛ ⎟⎥ + q 1 2 ⎪ ⎜ ⎜ ⎜⎝ uk + λ (1 − X (t )) ⎟⎠ ⎟⎥ t ⎢ ⎪ −λ 1 ⎜ ⎝ ⎠⎦ ⎣ P0 ( z ) = P0 (1) exp ⎨ ∫⎜ k ⎡ ⎛ ⎛ ⎞⎤ ⎪ v z ⎞ uk ⎜ ⎢S1 (λ − λ X (t ))S2 (λ − λ X (t )) ⎜ p ⎜ ⎟⎥ − t q + ⎪ ⎜ ⎝ uk + λ (1 − X (t )) ⎟⎠ ⎟⎥ ⎜ ⎢ ⎪⎩ ⎝ ⎠⎦ ⎝ ⎣
⎞ ⎫ ⎟ ⎪ ⎟ ⎪ ⎟ dt ⎬⎪ . ⎟ ⎪ ⎟ ⎪ ⎟ ⎪ ⎠ ⎭
The steady state condition is obtained as ⎡ ⎣
p⎤
ρ = λ E ( X ) ⎢ E ( S1 ) + E ( S 2 ) + ⎥ < 1 . u ⎦
Case III: Single server retrial queue with batch arrivals and Deterministic vacation time under Bernoulli’s schedule and two phases of heterogeneous services.
On the Single Server Batch Arrival Retrial Queue
153
It is well known that as k → ∞ , the whole mass of the distribution E k tends to concentrate at a point d = (1/ μ ) . Then we have S3 ( x ) → e − dx . Hence, the PGF of queue size distribution P(z) is as follows when
S3 (λ − λ X ( z )) = e − d λ (1− X ( z )) , P (z ) =
(1 − z ) P0 ( z ) , ⎡S1 (λ − λ X ( z ))S2 (λ − λ X ( z )) pe − d λ (1− x ( z )) + q ⎤ − z ⎣ ⎦
(
(31)
)
where
⎧ ⎛ − d λ (1− X ( t )) ⎡ +q ⎪⎪ −λ 1 ⎜ 1 − ⎣S1 (λ − λ X (t ))S2 (λ − λ X (t )) pe P0 ( z ) = P0 (1) exp ⎨ ∫⎜ − d λ (1− X ( t )) +q ⎪ v z ⎜⎜ ⎡⎣S1 (λ − λ X (t ))S2 (λ − λ X (t )) pe ⎝ ⎩⎪
( (
)⎦⎤ Xt(t ) ⎞⎟ ⎫⎪⎪ ⎟ dt ⎬ , ⎤ − t ⎟ )⎦ ⎟ ⎪ ⎪ ⎠
⎭
The steady state condition is obtained as, ρ = λE( X )[E(S1 ) + E(S 2 ) + pd ] (
