LNCS 8337 - Distributed Computing and Internet ...

Equilibrium Balking Strategy in an Unobservable GI/M/c Queue with Customers’ Impatience Dibyajyoti Guha1 , Abhijit Datta Banik1 , Veena Goswami2 , and Souvik Ghosh1 1

School of Basic Sciences, Indian Institute of Technology Samantapuri, Nandan Kanan Road, Bhubaneswar-751 013, India 2 School of Computer Application, KIIT University Bhubaneswar-751024, India {dg11,adattabanik}@iitbbs.ac.in, [email protected], [email protected]

Abstract. We consider an equilibrium threshold balking strategy in an unobservable GI/M/c queue with customers’ impatience. Upon arriving a customer decides whether to join or balk the queue based on random probability known as joining probability (f). Once a customer decides to join the system it initiates an impatient timer with random duration T, such that, if customers’ service is not completed before the timer expires, the customer abandons the system. The waiting time of a customer in system has been associated with a linear cost-reward structure for estimating the net benefit if a customer chooses to participate in the system. The study has been limited to unobservable queue where the information regarding system-length is unknown to the arriving customer. The proposed analysis is based on a root of the characteristic equation formed using the probability generating function of embedded pre-arrival epoch probabilities. Therefore, we obtain the stationary system-length distribution at pre-arrival and arbitrary epochs and thereby we obtain mean system sojourn time. Finally, we present numerical results in the form of graphs for observing net benefit against different model parameters. The proposed model has applications in the modeling of balking and impatient behavior of incoming calls in a call center, multi-core computing, multi-path routing in delay sensitive communications networks. Keywords: Renewal arrival, multi-server, balking strategy, reneging, roots, infinite-buffer, Pad´e-Laplace method.

1

Introduction

Many practical queueing systems are often encountered with the situation where customers are allowed to depart before joining (known as balking) the queueing system. In a reneging queueing system, the customers become impatient after joining the queue, i.e. the customers may depart the system after the expiration of the patience timer before completing service. Teletraffic analysis in a R. Natarajan (Ed.): ICDCIT 2014, LNCS 8337, pp. 188–199, 2014. c Springer International Publishing Switzerland 2014 

Equilibrium Balking Strategy in an Unobservable GI/M/c Queue

Fig. 1. GI/M/c model for multi-core computing

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Fig. 2. GI/M/c model for multi-path routing

call center or data center is an ideal example where the customers exhibits the balking and reneging behavior in a multi-server queueing system. The difference between balking and reneging is that the customers decide probabilistically to participate in the queueing system at the time of arrival where as in case of reneging the customer may abandon the system at the expiration of a timer after joining the system. There are two kinds of balking strategy: (1) Observable cases: The arriving customer has the information of the system-length before joining the system; (2) Unobservable cases: The arriving customers has no information about the system-length at the arrival epoch. The proposed model is also applicable in multi-core computing, multi-path routing in delay sensitive communications networks, as shown in Fig. 1 and Fig. 2. Balking strategy with various model parameters and their results are summarized in [11] with extensive bibliographical references. An M/M/1 queue with unreliable server have been analyzed by Economou et al. in [8]. In recent times, Economou and Kanta [7] have also considered an M/M/1 constant retrial queueing system. Balking strategy with setup/ closedown times for an M/M/1 queue have been considered by Sun et al. [18]. A balking strategy from reliability perspective for M/M/1 queue have been studied by Wang and Zhang [19] with unreliable server and delayed repair. Guo and Hassin [9] considered Markovian vacation queueing models with N -policy with and without the information about mean delay. Recently Liu et al. [14] have considered equilibrium threshold strategies for observable queueing system with single vacation policy. The assumption of vacation is implicit in the references mentioned [8,7,18,19,9,14]. An M/M/1/N queue with balking and reneging have been investigated by Ancker and Gafarian [1]. Abou-EI-Ata et al. [2] considered the multiple servers queueing system M/M/c/N by combining balking and reneging. Wang et al. [20] extended works of [2] to study an M/M/c/N queue with balking, reneging and server breakdowns. Queueing models with single and multiple servers with

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customers impatience have been analyzed by Altman and Yechiali in [3] and [4] when the server(s) is (are) on vacation and unavailable for service. An M/M/c queues in a 2-phase (fast and slow) Markovian random environment with impatient customers is reported in [15]. The system resides in the each phase with an exponentially distributed random time whose rate is slower in the slow phase and higher in the fast phase. The customers become impatient when the server is in slow phase. Shawky et al. [17] presented analysis on Hk /M/c/N queues with balking and reneging. An exact and approximation analysis based on virtual queueing time for M/G/c queues with balking and reneging can be found in [13]. Kumar et al. [16] discussed an M/M/1/N queueing model with balking and possibility of retaining reneged customers, where the reneged customer can be retained into system with probability p or it may abandon the system without receiving service with complementary probability. In this paper, we investigate an equilibrium threshold balking strategy in an unobservable GI/M/c queue with customers’ impatience. A reward-cost structure is assumed in the system which attracts the customer to join the queueing system. Each customer incurs a cost which is proportionate to the mean waiting time of the customer.

2

Description and Analysis of the Model

We consider an equilibrium threshold balking strategy in an unobservable GI/M/c queue with customers impatience wherein the inter-arrival times of successive arrivals are independent and identically distributed (i.i.d.) random variables with general distribution function A(u) (u ≥ 0), a probability density function (p.d.f.) a(u) (u ≥ 0), Laplace-Stieltjes (LST) transform a∗ (s) and mean 1/λ. The service discipline is first-come, first-served (FCFS). The service time follows exponential distribution with parameter μ . Consider the system just before an arrival which are taken as embedded points. Let t0 , t1 , t2 , ... be the time epochs at which successive arrivals occur and t− n denote the time epochs just before the arrival instant tn . The inter-arrival times Tn+1 = tn+1 − tn , n = 0, 1, 2, . . . are i.i.d.r.vs. with common distribution func− − tion A(x). The state of the system at t− i is defined as {Ns (ti )}, where Ns (ti ) is − the number of customers in the system. The process {Ns (ti )} is an embedded Markov chain with the state space Θ = {(k), k ≥ 0}. In limiting case let us − assume πn− = limi→∞ P (Ns (t− i ) = n), n ≥ 0, where πn represents the probability that there are n customers in the system just prior to an arrival epoch of a customer. We assume that the arriving customers have the options to decide whether to join or balk upon their arrival. We model this decision by assuming that each customer receives a reward of R units for completing service and is charged a cost of G units per time unit that he remains in the system (sojourn time). We also assume that customers are risk neutral and wish to maximize their net benefit. At the arrival instant the customer decides whether to join or balk the queue based on random probability known as joining probability (f ). Once a customer decides

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to join the system it initiates an impatient timer with random duration T, which follows an exponential distribution with parameter η, such that, if customers’ service is not completed before the timer expires, the customer abandons the system. The patience timer T is exponentially distributed whose density function is given by r(t) = ηe−ηt . The assumption regarding reneging is that the service times for those customers who eventually will renege without being served are not included in the calculation of the sojourn time. Thus, the effective departure rate of customers from the system due to reneging or completion of service is μ = μ + η. 2.1

System-Length Distribution at Pre-arrival Epoch

In this subsection, we analyze system-length distribution at the pre-arrival epoch. Let dk (k ≥ 0) represents the probability that k customers have been departed during an inter-arrival time given that all the c servers are busy during the interarrival time duration. Let ak+1,j be the probability that an arriving customer finds k (k ≤ c−1) customers in the system while the next arriving customer finds j (0 ≤ j ≤ k) customers in the system. As a result, exactly k + 1 − j customers have been departed during an inter-arrival time of a customer. Similarly, bk+1,j be the probability that an arriving customer finds k (k ≥ c) customers in the system while the next arriving customer finds j (0 ≤ j ≤ c − 1) customers in the system. Therefore, for all k ≥ 0, we have − ak,j = P (Ns (t− i ) = j|Ns (ti−1 ) = k − 1) where (0 ≤ k − 1 ≤ c − 1), (1 ≤ j ≤ k), ∞  = 0 kj e−μjt (1 − e−μt )k−j dA(t), − = P (Ns (t− i ) = j|Ns (ti−1 ) = k − 1) where (k − 1 ≥ c), (1 ≤ j ≤ c − 1),  ∞  t (μc)k−c uk−c−1 e−cμu c −μj(t−u) (1 − e−μ(t−u) )c−j dudA(t), = 0 0 (k−c−1)! j e ∞ k k c−1 k−c −cμt dk= 0 (cμt) dA(t), ak,0= 1 − j=1 ak,j , bk,0 = 1 − j=1 bk,j − j=0 dj . k! e

bk,j

We can simplify ak,j as  ∞  k −μjt ak,j = e (1 − e−μt )k−j dA(t) j 0  ∞  k −μjt (−1)k−j (e−μt − 1)k−j dA(t) e = j 0   ∞  k−j  k −μjt k − j −μt(k−j−l) k−j (−1) (−1)l dA(t) = e e j l 0 l=0    ∞  k−j k k−j+l k − j = (−1) e−μt(k−j−l+j) dA(t) j l 0 l=0   k−j   ∞ k k−j+l k − j = (−1) e−μt(k−l) dA(t) j l 0 l=0

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  k−j   k k−j+l k − j = (−1) a∗ (μ(k − l)). j l l=0

Similarly bk,j can be simplified as   (μc)k−c uk−c−1 e−cμu c −μj(t−u) (1 − e−μ(t−u) )c−j dudA(t) e (k − c − 1)! j 0 0   ∞ t (μc)k−c c e−cμu uk−c−1 e−μj(t−u) (1 − e−μ(t−u) )c−j dudA(t) = (k − c − 1)! j 0 0   ∞ t c (μc)k−c = g(u)h(t − u)dudA(t), (1) (k − c − 1)! j 0 0 

∞



t

bk,j =

where g(u) = e−cμu uk−c−1 and h(t − u) = e−μj(t−u) (1 − e−μ(t−u) )c−j . Thus, the second integral is the convolution of g(u) and h(t − u), so the whole integral is the LST of the convolution of these two functions. The LST of g(t) can be computed as  ∞ (k − c − 1)! e−st e−cμt tk−c−1 dt = . (2) (s + cμ)k−c 0 Similarly the LST of h(t) can be obtained as  ∞  ∞ e−st h(t)dt = e−st e−μjt (1 − e−μt )c−j dt 0 0 ∞ dz = μe−μt ] e−μt(s/μ+j) (1 − e−μt )c−j dt [if z = 1 − e−μt , ∴ = dt 0 1  1 −μt(s/μ+j−1) c−j e z dz (1 − z)−μt(s/μ+j−1) z c−j dz 0 = 0 = μ μ Γ (j + s/μ)Γ (c − j + 1) . (3) = μΓ (c + s/μ + 1) The literature on queueing theory shows that distributions having LaplaceStieltjes transform as a rational function cover a wide range of distributions that arise in applications, see Botta et al. [5]. In view of this, we consider those distributions that have rational Laplace-Stieltjes transform of the form v(s) = P (s)/Q(s), where degree of the polynomial Q(s) is n and that of the polynomial P (s) is at most n. Thus, the convolution of (2) and (3) can be expressed as (k − c − 1)!(c − j)! P (s) (k − c − 1)! Γ (j + s/μ)Γ (c − j + 1)  , (s + cμ)k−c μΓ (c + s/μ + 1) μ Q(s) P (s) (j+s/μ) where Q(s)  (s+cμ)Γk−c Γ (c+s/μ+1) . We consider that inter-arrival time is following a P H-type distribution having parameters (α, B) with density a(t) =

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αeB t B 0 . By replacing a(t) = αeB t B 0 in (1), bk,j can be written as   ∞ t c (μc)k−c bk,j = e−cμu uk−c−1 e−μj(t−u) (1 − e−μ(t−u) )c−j dudA(t) (k − c − 1)! j 0 0   ∞  t c (μc)k−c = αeB t e−cμu uk−c−1 e−μj(t−u)(1−e−μ(t−u) )c−j dudtB 0 (k − c − 1)! j 0 0   ∞  t k−c (μc) c = αeB t g(u)h(t − u)dudtB 0 (k − c − 1)! j 0 0 

P (s) (μc)k−c c! α.[ | ].B 0 . j!μ Q(s) s=−B

(4)

We use Pad´e-Laplace method in (4) (see Harris and Marchal [10]) to approximate bk,j by a rational function of order (m, n). Hence bk,j can be expressed as bk,j 

P (s) (μc)k−c c! α.[ | ].B 0 . j!μ Q(s) s=−B

(5)

Let us define f be the joining probability when the arriving customer chooses to join the system. The probability generating function (p.g.f.) of dk is given by D(z) =

∞

dk z k = a∗ (cμ − cμz).

(6)

k=0

Observing the state of the system at two consecutive pre-arrival epochs, we get the following difference equations: c−1 ∞ π0− = f ( πk− ak+1,0 + πk− bk+1,0 ) k=0

k=c

+(1 − f )(π0− +

c−1

πk− ak,0 +

k=1

πi− = f (

c−1

πk− ak+1,i +

k=i−1

∞

πk− bk,0 ),

(7)

k=c ∞

πk− bk+1,i )

k=c

c−1 ∞ − +(1 − f )( πk ak,i + πk− bk,i ), 1 ≤ i ≤ c − 1,

πi− = f

∞

k=i

k=c

πk− dk+1−i + (1 − f )

k=i−1

∞

πk− dk−i ,

(8)

i ≥ c.

(9)

k=i

We probability generating function (p.g.f.) of πi− (i ≥ c) as πc −∗ (z) = ∞define − i−c . Multiplying (9) by z i−c , summing them and using the definition i=c πi z − of p.g.f. of πi (i ≥ c), πc −∗ (z) can be expressed as πc −∗ (z) =

πc− − f

∞  i=1

di z i−1

i−1  j=0

− πc+j z j − (1 − f )

∞  i=0

1 − (1 − f + f z)D( 1z )

− πc+i zi

∞  k=i

dk zk

.

(10)

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The expression in (10) is analytic and convergent in |z| ≤ 1. We need to calculate zeros in the denominator of (10). We know that the equation 1 − (1 − f + f z)D( 1z ) = 0 has exactly one root in the region |z| > 1. Since the equation 1 − (1 − f + f z)D( 1z ) = 0 has one root outside the unit circle, the function 1 − (1 − f + fz )D(z) = 0 has one zero ω inside the unit circle |z| < 1. As πc −∗ (z) is an analytic function of z for |z| ≤ 1, applying the partial-fraction method, we obtain K1 πc −∗ (z) = , (11) 1 − ωz where K1 is the constant to be determined. Now, equating the coefficient of z i−c from both sides of (11), the pre-arrival epoch probabilities can be obtained as πi − = K1 ω i−c , i ≥ c.

(12)

Let us compute the pre-arrival epoch  probabilities of πi− for i ≥ c. The constant ∞ K1 can be determined by considering i=c πi− = 1. Hence, K1 = 1−ω and πi− = i−c (1 − ω)ω . At this point, we have c unknown probabilities πi− for i = 0 · · · c − 1. One can obtain c equations for solving c unknown probabilities πi− (∀i = 0 · · · c − 1) by considering the Equations (7) and (8) where we have c unknown probabilities πi− (∀i = 0 · · · c − 1) and πi− (∀i ≥ c) known probabilities. The desired probabilities of system-length at pre-arrival epoch can be obtained by ∞ normalization condition i=0 πi− = 1. 2.2

System-Length Distribution at Arbitrary Epoch

In this subsection, we derive the expression for steady-state system-length distribution at arbitrary epochs. We have applied the classical argument based on renewal theory (see Chaudhury, Templeton [6]) which relates the steady-state system-length distribution at an arbitrary epoch to that at the corresponding pre-arrival epoch. Let d k (k ≥ 1) represents the probability that k customers have been departed during residual inter-arrival time given that all the c servers are busy during the residual inter-arrival time duration. Let

ak,j and bk,j be the corresponding probability of ak,j and bk,j , respectively while considering residual inter-arrival time instead of inter-arrival time. Hence for all k ≥ 0, we have  ∞ (cμt)k −cμt e d k = λ(1 − A(t))dt, (k ≥ 0), k! 0  ∞  k −μjt

ak,j = (1 − e−μt )k−j λ(1 − A(t))dt, e j 0 = where (0 ≤ k − 1 ≤ c − 1), (1 ≤ j ≤ k),    ∞ t (μc)k−c uk−c−1 e−cμu c −μj(t−u)

bk,j = (1−e−μ(t−u) )c−j duλ(1−A(t))dt, e (k − c − 1)! j 0 0 = where (k − 1 ≥ c), (1 ≤ j ≤ c − 1),

ak,0 = 1 −

k j=1

ak,j , and bk,0 = 1 −

c−1 j=1

bk,j −

k−c j=0

d j .

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k which is denoted by D(z).

Let us derive the p.g.f. of d We use the LST of probability density function (pdf) of residual inter-arrival time as given in Kleinrock ∞ ∗ [12], 0 e−st λ(1 − A(t))dt = λ 1−as (s) ,  ∞ ∞ (cμt)k −cμt k

D(z) = e z λ(1 − A(t))dt k! 0 k=0  ∞ 1 − a∗ (cμ − cμz) . e−(cμ−cμz)t λ(1 − A(t))dt = λ = cμ − cμz 0 Similarly,

ak,j can be expressed as  ∞  k −μjt

ak,j = (1 − e−μt )k−j λ(1 − A(t))dt e j 0   k−j   ∞ k k−j = (−1)k−j+l e−μt(k−l) λ(1 − A(t))dt j l 0 l=0   k−j   k k − j 1 − a∗ (μ(k − l)) . (−1)k−j+l = λ j l μ(k − l) l=0

As the inter-arrival time is following a P H-type distribution with parameters (α, B) then the residual inter-arrival time follows a P H-type distribution with parameters (β, B) where the stationary probability vector β satisfies β(B + B 0 α)=0. System length at an arbitrary epoch πi can be obtained from pre-arrival epoch probability πi− (∀i ≥ 0) by employing d k ,

ak,j and bk,j which is given below c−1 ∞ πk−

ak+1,0 + πk− bk+1,0 ) π0 = f ( k=0

k=c

+(1 − f )(π0 +

c−1

πk−

ak,0 +

k=1

πi = f (

c−1

πk−

ak+1,i +

k=i−1

+(1 − f )( ∞

3.1

(13)

k=c ∞

πk−

ak,i +

k=i

πk− bk+1,i )

∞

πk− bk,i ), 1 ≤ i ≤ c − 1,

k=c

πk− d k+1−i + (1 − f )

k=i−1

3

πk− bk,0 ),

k=c c−1

πi = f

∞

∞

πk− d k−i ,

i ≥ c.

(14) (15)

k=i

Performance Measures Mean System-Length and Sojourn Time Analysis

Computation of mean system-length and mean sojourn time are the key aspects of performance measures for queueing system. Steady-state system-length

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distribution at arbitrary epochs obtained in (13) - (15) is used to compute ∞ the mean system-length. Thus, the mean system-length is given as L = k=0 kπk . Waiting time in the system can be obtained from the pre-arrival epoch probabilities πi− that we have derived in Section 2.1. Let Wa ∗ (s) denotes the LaplaceStieltjes transform of the distribution function of the sojourn time of an arriving customer. Hence, Wa ∗ (s) can be given by ∗

Wa (s) = 1 − f + f

c−1

πj−

j=0



μ s+μ

 +f

∞ j=c

πj−



cμ s + cμ

j+1−c .

(16)

Let W be the r.v. denoting the sojourn time of an arriving customer. Thus, the mean of the sojourn time of an arriving customer is given by E[W ] = f

c−1 − πj j=0

μ

+f

∞

πj−

j=c

j+1−c . cμ

(17)

By the help of Little’s law, mean system sojourn time E[W ] can be obtained from mean system-length E[W ] = L λ . The compliance of Little’s law has been verified numerically for moderate values of ρ and c. 3.2

An Equilibrium Balking Strategy for Unobservable Queue

There is a finite service charge G that has to be paid by every customer who passes through the system after completing service. We denote it as Δ = R − G.E[W ], the net benefit of a customer who has been served. The basic assumption about reward-cost structure is that system attracts a customer to participate even when the system is empty at the arrival instant of the customer. Using similar derivations as in Equation (17), the mean sojourn time of an arriving cus πj− tomer in an empty system (denoted by Swait ) is given by Swait = c−1 j=1 μ . In other words, the reward for service is bigger than the cost of an arriving customer who finds the system empty, which is given as follows R > G.Swait = G

c−1 − πj j=1

μ

.

The particular value of f which produces Δ closest to zero is the desired equilibrium joining probability. Let us denote the equilibrium joining probability as fΔ . It is observable that the reward R is proportionate to the mean waiting time in system. Hence, the net benefit Δ = R − G.E[W ] is also depending on mean waiting time in the system.

4

Numerical Results

This section presents numerical results to demonstrate how the joining probability influences net benefit with considering two aspects: Case 1: Different traffic

Equilibrium Balking Strategy in an Unobservable GI/M/c Queue

5 Net benefit of an arriving customer

Net benefit of an arriving customer

5 0 −5 −10

ρ=0.93 ρ=0.82 ρ=0.74 ρ=0.67

−15 −20 −25 −30 −35 0.4

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0.5 0.6 0.7 0.8 0.9 Probability of joining in the system when c=4

1

Fig. 3. f vs Δ for different values of μ when c=4

0 −5 −10 −15 −20 −25

c=4, ρ=0.93 c=5, ρ=0.74 c=6, ρ=0.62 c=7, ρ=0.53

−30 −35 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability of joining in the system when μ=0.7, η=0.1

Fig. 4. f vs Δ for different values of c when μ = 0.7, η = 0.1

intensity by changing the service rate (μ) for fixed value of c; Case 2: Different traffic intensity for different values of c but μ remains constant. Let us consider P H/M/c model where inter-arrival ⎛ time is a P H-type distribution having the ⎞ −5 0 3 representation (α, B) where B = ⎝ 1 −5 2 ⎠ and α = (0.35 0.45 0.20), 1 3 −10 with λ = 2.978723. The reneging parameter remained constant η = 0.1. By following the cost-reward constraint, we have kept R = 8, C = 6 to be constant throughout the experiment. The net benefit for various joining probabilities for different values of traffic intensity while c remains constant (case 1) and c kept on increasing (case 2), is represented in Fig. 3 and Fig. 4, respectively. From Fig. 3, it can be seen that the net benefit is decreasing when the joining probabilities are increasing for each of the traffic intensity taking the value of ρ = 0.67, 0.74, 0.82, 0.93, while c = 4 through out the experiment. The equilibrium joining probability corresponding to case 1 is fΔ = 1, 0.9, 0.9 for ρ = 0.74, 0.82, 0.93, respectively. The net benefit never becomes zero when ρ = 0.67. Fig. 4 shows similar kind of decreasing net benefit when the joining probabilities are increasing for each of values of c = 4, 5, 6, 7, when μ = 0.7, η = 0.1 The equilibrium joining probability corresponding to case 2 is fΔ = 1, 1, 0.9 for ρ = 0.62 (c = 6), 0.74 (c = 5), 0.93 (c = 4), respectively. The net benefit is always non-negative when ρ = 0.53 (c = 7). A tabular representation of Fig. 3 and Fig. 4 is shown in Table 1 and Table 2, respectively. We have presented a scenario where ρ = 0.74 remains same for different combinations of c, μ, e.g. μ = 0.8 and c = 4 (see row 2, Table 1) and μ = 0.7 and c = 5 (see row 3, Table 2). It is visible that μ = 0.8 and c = 4 produces less sojourn time compared to μ = 0.7 and c = 5. It is an indication that an increase of service rate by (0.8 − 0.7)/0.7 proportion has more impact on reducing mean sojourn time in comparison with an increase of the number of servers by (5 − 4)/4 proportion.

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μ μ=1 μ=0.9 μ=0.8 μ=0.7

E[W ] E[W ]ρ=0.67 E[W ]ρ=0.74 E[W ]ρ=0.82 E[W ]ρ=0.93

f = 0.4 3.640551 3.521595 3.387136 3.228818

f = 0.5 3.483346 3.325253 3.138518 2.904274

f = 0.6 3.289418 3.075633 2.809666 2.449008

f = 0.7 3.042016 2.742324 2.343724 1.742697

f = 0.8 2.702272 2.252745 1.596378 0.433512

f = 0.9 2.160621 1.391438 0.093999 -2.990116

f =1 0.959224 -0.844460 -4.866042 -30.24920

Table 2. An experiment to observe E[W ] for c = 4, 5, 6, 7 when μ = 0.7, η = 0.1 c c=7 c=6 c=5 c=4

5

E[W ] E[W ]ρ=0.53 E[W ]ρ=0.62 E[W ]ρ=0.74 E[W ]ρ=0.93

f = 0.4 0.532693 1.471397 2.385867 3.228818

f = 0.5 0.458900 1.336969 2.179768 2.904274

f = 0.6 0.361475 1.164467 1.913846 2.449008

f = 0.7 0.242898 0.947626 1.561243 1.742697

f = 0.8 0.119455 0.678269 1.058076 0.433512

f = 0.9 0.050295 0.336339 0.204006 -2.990116

f =1 0.029993 -0.178311 -2.071733 -30.24920

Conclusion

In this paper, we have investigated an equilibrium threshold balking strategy in an unobservable GI/M/c queue with customers’ impatience based on a root of the characteristic equation formed using the probability generating function of embedded pre-arrival epoch probabilities. We have obtained the steady-state distributions of the number of customers in the system at pre-arrival and arbitrary epochs. On the similar direction, the analysis of renewal input batch arrival queue with multi-server with balking and reneging is an interesting problem for future work. Acknowledgements. The second and fourth author acknowledges partial financial support from the Department of Science and Technology, New Delhi, India research grant SR/FTP/MS-003/2012.

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