The stationary distribution of the number of busy servers and the stability of the system are also considered

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MANAGEMENT OPTIMIZATION

Volume8, Number4, November2012 pp.939–968

M/M/C MULTIPLE SYNCHRONOUS VACATION MODEL WITH GATED DISCIPLINE

Zsolt Saffer

Department of Telecommunications

Budapest University of Technology and Economics, Budapest, Hungary

Wuyi Yue

Department of Intelligence and Informatics Konan University, Kobe 658-8501, Japan

Abstract. In this paper we present the analysis of an M/M/c multiple synchronous vacation model. In contrast to the previous works on synchronous vacation model we consider the model with gated service discipline and with independent and identically distributed vacation periods. The analysis of this model requires different methodology compared to those ones used for synchronous vacation model so far. We provide the probability-generating function and the mean of the stationary number of customers at an arbitrary epoch as well as the Laplace-Stieljes transform and the mean of the stationary waiting time.

The stationary distribution of the number of busy servers and the stability of the system are also considered. In the final part of the paper numerical examples illustrate the computational procedure.

This vacation queue is suitable to model a single operator controlled system consisting of more machines. Hence the provided analysis can be applied to study and optimize such systems.

1. Introduction. Vacation models are effective tools and hence widely applied in the performance analysis of modern telecommunication networks. In the regular vacation models the single server occasionally takes vacation, during which no customers are served. The start of the vacation period is governed by the applied service discipline, like e.g. the exhaustive or the gated ones. In case of exhaustive disciplinethe service period continues until the queue becomes empty. Under gated disciplineonly those customers are served during the service period which are already present in the queue at the end of the vacation period. In themultiple vacation model the server immediately takes another vacation, if the server finds the system empty upon returning from vacation. In the contrary in thesingle vacation model the server waits for the next arriving customer in the same situation. For the analysis of the regular vacation models we refer to the survey of Doshi [4] and to the book of Takagi [9].

The multi-server vacation model is the natural generalization of its single server counterpart. The generalization of the vacation model for multi-servers can be considered on more ways. In thesynchronous vacationmodel all servers take vacation

2000Mathematics Subject Classification. Primary: 60K25, 68M20; Secondary: 90B22.

Key words and phrases. Queueing theory, multi-server queue, synchronous vacation model, service discipline.

The reviewing process of the paper was handled by Yutaka Takahashi as Guest Editor.

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simultaneously, i.e all servers start the vacation at the same time and they return to the system at the expiry of the common vacation period. This is suitable to model systems, in which all machines are controlled by a single operator. Other practical cases can be described by other type of multi-server vacation models, in which only several servers take vacations. During their vacations they can perform secondary jobs, while the remaining servers are always available for the customers.

These vacation models are calledpartial server vacationmodels.

The multi-server vacation model was investigated first by Levy and Yechiali [6].

They investigated an M/M/c synchronous vacation model with exponentially distributed vacations. They provided the mean number of customers in the system and the distribution of the number of busy servers. Chao and Zhao [3] analyzed G/M/c multi-server models with exponentially distributed vacations and with two classes of vacation mechanisms. One of them is the synchronous vacation, which they calledstation vacation. The other vacation mechanism is called server vacation, in which each server can take vacation individually independently of the other servers. This can be applied to model e.g. the work in a post office, where a clerk immediately changes to perform other types of work (sorting, distributing, etc.) when e.g. observes an empty queue. In the above mentioned work they provided a numerical algorithm for computing the steady-state distribution of the number of customers. The synchronous vacation model with Phase (PH) type distributed vacation period was analyzed for M/M/c queue by Tian and Li [11] and for GI/M/c queue by Tian and Zhang [12].

The M/M/c partial server vacation model was studied by Begum and Nadarajan [1]. For a particular case they provided also the steady state system probabilities and the waiting time distribution of an arriving customer. A various kind of M/M/c partial server vacation model were analyzed in a series of papers by Zhang and Tian, e.g. [15], [16]. They made use of the quasi-birth-and-death (QBD) model formulation and applied the matrix analytic method to provide computational algo- rithms for the stationary distributions of the queue length and waiting time. They investigated also the optimization of a policy control problem related to the partial server vacation model.

For more details on the multi-server vacation models we refer to the books of Tian and Zhang [10] as well as of Yue, Takahashi and Takagi [14]. In all the above- mentioned multi-server vacation models the exhaustive discipline has been applied, and they have exponentially or Phase (PH) type distributed vacation times.

In this paper we analyze the M/M/c synchronous vacation model, but in contrast to the above references we consider the gated discipline and independent and identically distributed vacation periods. The methodology used in the above references is especially suitable for the exhaustive discipline and utilizes the exponential character of the distribution of the vacation period. The analysis of this model requires different methodology compared to those ones used in the above mentioned references. We apply a methodology, in which the problem is separated into two parts. This results in simplification in the overall analysis. This methodology is similar to the one used for polling models by Borst and Boxma [2].

In the first parts we derive the closed-form expression of the probability- generating function (PGF) of the stationary number of customers in terms of stationary quantities at the end of the vacation period. Due to the gated discipline this is achieved by the help of the joint PGF of two stationary quantities at customer departure epochs: the stationary number of customers in the system and the

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stationary number of customers to be served. In the second part the required stationary quantities at the end of the vacation period are determined. We establish a functional equation for the PGF of the stationary number of customers at the end of vacation by applying the buffer occupancy method (see e.g. in Takagi [8]). We solve this equation by using a convergence property of series of scalar functions (see in [5]). The required quantities are then derived from the solution of this equation.

Afterwards we provide the derivation of the Laplace-Stieljes transform (LST) and the mean of the stationary waiting time. In the next part we discuss the stability of the model and the number of busy servers. Finally the steps of the computational procedure and numerical examples are given to illustrate the numerical solution.

The contribution of this paper is the queueing theoretic analysis and the results for the M/M/c synchronous vacation model with gated discipline and independent and identically distributed vacation periods. The main results are the PGF and the mean of the stationary number of customers at an arbitrary epoch as well as the LST and the mean of the stationary waiting time.

This M/M/c synchronous vacation model is suitable to model a single operator controlled system consisting of more machines. Due to independent and identically distributed vacation periods it provides better modeling comparing to the previous works, in which the vacation model was modeled by either exponential or PH type distribution. The provided analysis enables to build an optimization framework to compute the optimal number of servers or the optimal customer service rates at given system parameters for the required range of the arrival rate.

The rest of the paper is organized as follows. In section 2 we give the model description. The expression of the joint PGF of the stationary number of customers in the system at customer departure epoch and the stationary number of customers to be served at the same customer departure epoch is established in Section3. In section 4 the PGF and the mean of the stationary number of customers in the system at an arbitrary epoch are derived. In section5we provide the expression of the stationary quantities at the end of the vacation period. In section 6 we derive the LST and the mean of the stationary waiting time. In section7 we provide the stability condition of the model and deal with the number of busy servers. The discussion of the numerical solution and illustrative numerical examples close the paper in section8.

2. Model description. We consider a multi-server queue with synchronous and multiple vacations. The queue has infinite buffer, i.e. all customers arriving to the system will be served. The customer arrival process is Poisson with rate λ. Due to the gated service only those customers are served, which are present at the start of the service period. There are c ≥1 servers in the system. At each server the distribution of the customer service time is exponential with the same parameter, which is denoted byµ. The service is non-preemptive and for the customers to be served in the actual service period it is also work conserving. In the multi-server queue each server starts to serve the next customer immediately upon finishing the service of a customer, if there is at least one customer to be served. After finishing the service according to the gated discipline all the servers go to vacation simultaneously. If there are no customers in the system at the end of the vacation, the servers immediately take another vacation further on simultaneously. Otherwise the next service period starts, in which at least one server starts its service. The consecutive vacation periods are independent and identically distributed. Let V,

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ev(s),v^[1] andv^[2]denote the vacation period r.v., its LST, its mean and its second moment, respectively. We define thecycle timeas a service period and a vacation period together. On this vacation model we set the following assumptions:

A.1 The arrival rate, the mean customer service time and the mean vacation time are positive and finite, i.e. 0< λ <∞, 0<1/µ <∞and 0< v^[1]<∞.

A.2 The arrival process, the customer service times and the sequence of the vacation periods are mutually independent.

A.3The customers are served in First-In-First-Out (FIFO) order.

We assume that the model is stable. In the stable system at least the arrival rate can not exceed the asymptotic maximum of the mean service rate as the traffic increases, which iscµ. Therefore the necessary condition of the stability isλ < cµ.

Introducingρ=_cµ^λ this can be expressed asρ <1.

In this paper we define several average limiting quantities. It can be shown that in the stable queueing model the “average limiting” and the corresponding

“stationary” quantities are the same. This is because the stability ensures the existence of the corresponding limiting distributions and that the mean cycle time is finite ([7]). Therefore throughout this paper we use the term “stationary” instead of “average limiting”.

For any y(z), which is a PGF of the random variableb Y, y^(k) denotes its k-th derivative atz= 1 fork≥1, i.e.,y^(k)= _dz^d^kkby(z)|z=1. Thusy^(k)is thek-th factorial moment of Y for k ≥ 1. Additionally y denotes the value of by(z) at z = 1, i.e.

y=y(1).

3. The stationary analysis at customer departure epochs. Throughout this paper the customer departure epoch stands for the epoch just after the customer departure.

According to the gated discipline the number of customers to be served at a customer departure epoch is the remaining number of customers among those, which are already present in the queue at the end of the previous vacation period.

In this paper the description of the evolution of the number of customers is based on the the number of customers to be served at a customer departure epoch. Due to the service rule of the gated discipline this equals to the remaining number of customers among those, which are already present in the queue at the end of the previous vacation period.

3.1. The PGF of the number of customers arriving between two consecutive departure epochs. Let baj(z) be the conditional PGF of the number of customers arriving between two consecutive customer departure epochs during the same service period, given that the number of customers to be served at the earlier customer departure epoch isjforj≥1. We remark here that this definition implies that there is customer service at least until the departure epoch, which is the later one of the consecutive customer departure epochs. Thus this definition implicitly includes this second condition.

We defineρj as

ρ_j = λ

j µ, 1≤j≤c.

Note thatρ_c=ρ.

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Proposition 1. In the stable M/M/c synchronous multiple vacation model with gated discipline satisfying assumptionsA.1-A.3the conditional PGF of the number of customers arriving between two consecutive customer departure epochs during the same service period, given that the number of customers to be served at the earlier customer departure epoch isj forj≥1, can be expressed as

baj(z) = 1

1 +ρ_j(1−z) for1≤j≤c−1 and ba_j(z) = ba_c(z) = 1

1 +ρ(1−z) forj≥c. (1)

Proof. The number of arriving customers between two consecutive customer departure epochs depends on the number of busy servers during that time period. This is because due to the exponentially distributed customer service times the distribution of time period between that two consecutive customer departure epochs depends only on the number of busy servers.

First we consider the case when the number of customers to be served at the earlier customer departure epoch is 1≤j≤c. In this case the number of busy servers until the next customer departure epoch is alsoj. Due to the memoryless property of the exponential distribution the time up to the next customer departure epoch is the minimum ofj simultaneous random time having exponential distribution with the same parameterµ. It follows that the distribution of this time period is exponential with parameterjµ. The customers arriving during this time period do not cause any change in the number of working servers, since they must wait for their service until the next service period according to the gated discipline. Therefore the conditional PGF of the number of customers arriving during this time period, given that the number of customers to be served at the earlier customer departure epoch isj, for 1≤j≤c, is given as

baj(z) = Z ∞

t=0

∞

X

k=0

z^k(λt)^k

k! e^−λtjµe^−jµtdt. (2) Rearranging (2) yields

baj(z) =jµ Z ∞

t=0

e−(jµ+λ(1−z))tdt= jµ

jµ+λ(1−z) = 1

1 +ρ_j(1−z). (3) If the number of customers to be served at the current customer departure epoch is j ≥ c then the number of busy servers at that customer departure epoch is c.

Hence for this case (3) can be applied by settingj=c.

Thek-th derivative ofba_j(z) atz= 1 is given by a^(k)_j =k!

λ j µ

k

=k! ρ^k_j, 1≤j ≤c, k≥1.

3.2. The joint PGF of the stationary number of customers in the system and the stationary number of customers to be served. In this paper the end of vacation stands for the epoch just after the end of vacation, which coincides with the epoch just after the start of the service period. Similarly the start of vacation stands for the epoch just after the start of vacation, which coincides with the epoch just after the end of the service period.

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We define G` as the number of customers served during a service period in the

`-th cycle for `∈ {1,2, . . .}. We define also the mean of the stationary number of customers served during a service period as

g= lim

m→∞

E[Pm

`=1G`]

m . (4)

Let N(t) be the number of customers in the system at time t for t ≥ 0. Fur- thermore letT`(k) be thek-th customer departure epoch during the`-th cycle for

`∈ {1,2, . . .}andk= 0,1, . . . , G`. Here the 0-th customer departure epoch,T`(0), is by definition the end of the previous vacation period, i.e. this epoch is already in the`-th cycle. The last departure epoch, T_`(G_`), coincides with the start of the next vacation period forG_` ≥1. By definitionT_`(G_`) means the start of the next vacation period also forG_`= 0. Additionally letI₍con)denote the indicator of condition “con”. We define the probability that the stationary number of customers at customer departure epochs isnas

q_n^d = lim

m→∞

Eh Pm

`=1

PG_`

k=1I_(N(T_`_(k))=n)i E[Pm

`=1G_`] , n≥0.

The corresponding PGF of the stationary number of customers at the customer departure epochs is defined as

qb^d(z) =

∞

X

n=0 m→∞lim

Eh Pm

`=1

PG_`

k=1I_(N_(T_`_(k))=n)i E[Pm

`=1G_`] zⁿ, |z| ≤1.

As it was already mentioned at the beginning of the section 3 the number of customers to be served at a customer departure epoch plays an important role in this analysis. Let Γ(t) be the number of customers to be served at time t for t≥0. We define the joint probability that the stationary number of customers at a customer departure epoch isnand the stationary number of customers to be served at the same customer departure epoch isj, forn≥0 andj≥0, as

q_n,j^d = lim

m→∞

Eh Pm

`=1

PG_`

k=1I_(N_(T_`_(k))=n) I_(Γ(T_`_(k))=j)i E[Pm

`=1G_`] .

We also define the corresponding PGFs as

qb_j^d(z) =

∞

X

n=0 m→∞lim

Eh Pm

`=1

PG_`

k=1I_(N_(T_`_(k))=n)I_(Γ(T_`_(k))=j)i E[Pm

`=1G_`] zⁿ, |z| ≤1,

qb^d(z1, z2) =

∞

X

j=0

∞

X

n=0 m→∞lim

Eh Pm

`=1

PG`

`=1G`] zⁿ₁z₂^j,

|z1| ≤1, |z2| ≤1.

We define the partial joint PGF of the the stationary number of customers at a customer departure epoch and the stationary number of customers to be served at the same customer departure epoch (j) forj≥c−1 as

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qb^d_(j≥c−1)(z1, z2) =

∞

X

j=c−1

∞

X

n=0 m→∞lim

Eh Pm

`=1

PG_`

`=1G_`] zⁿ₁z₂^j,

|z₁| ≤1, |z₂| ≤1.

We also define a partial joint PGF, which is related to the stationary number of customers at a customer service start epoch and the stationary number of customers to be served at the same customer service start epoch (j) forj≥cas

qb^s∗_(j≥c)(z1, z2) =

∞

X

j=c

∞

X

n=0 m→∞lim

Eh Pm

`=1

PG_`−c

k=0 I_(N_(T_`_(k))=n)I_(Γ(T_`_(k))=j)i E[Pm

`=1G_`] zⁿ₁z₂^j, (5)

|z₁| ≤1, |z₂| ≤1.

If Γ (T`(0))≥c thenc customer services start together at the first start epoch, i.e. at the same time. Thus in this case the number of customer service start epochs is c−1 less than the number of customer departures, i.e. it equals to G`−c+ 1.

Furthermore these customer service start epochs coincide with theT`(k) epochs for k= 0, . . . , G`−c. We also remark here thatbq_(j≥c)^s∗ (z1, z2) is not a partial joint PGF at customer service start epoch, sinceG`in the denominator of (5) does not equal the number of start epochs during the`-th cycle.

We define also the probability that the stationary number of customers at the 0-th customer departure epoch, i.e. at the end of the vacation period isnas

fn= lim

m→∞

EPm

`=1I(N(T_`(0))=n)

m , n≥0.

The corresponding PGF is given as

fb(z) =

∞

X

n=0 m→∞lim

EPm

`=1I(N(T_`(0))=n)

m zⁿ, |z| ≤1.

Similarly we define the PGF of the stationary number of customers at the start of the vacation period as

m(z) =b

∞

X

n=0 m→∞lim

EPm

`=1I_(N_(T_`_(G_`_))=n)

m zⁿ, |z| ≤1.

We define also probabilities, which are related to the stationary number of customers at the 0-th customer departure epoch as

f_n^∗= lim

m→∞

EPm

`=1I_(N_(T_`_(0))=n) E[Pm

`=1G_`] , n≥0.

Lemma 3.1. In the stable M/M/c synchronous multiple vacation model with gated discipline satisfying assumptions A.1 - A.3, the expressions of the partial joint PGF qb_(j≥c−1)^d (z1, z2)and the partial PGF qb₀^d(z1) are given as

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qb^d₀(z₁) = 1

g(m(zb ₁)−f₀), (6) bq_(j≥c−1)^d (z₁, z₂) = qb^d(z₁, z₂)−

c−2

X

j=0

z₂^jqb^d_j(z₁).

Proof. The proof of the lemma can be found in AppendixA.

Lemma 3.2. In the stable M/M/c synchronous multiple vacation model with gated discipline satisfying assumptions A.1 - A.3, the probabilities f_n^∗ and the partial joint PGFqb_(j≥c)^s∗ (z₁, z₂)can be expressed as

f_n^∗ = 1

gfn, n≥0, (7)

qb^s∗_(j≥c)(z1, z2) = 1 g



 bf(z1z2)−

c−1

X

j=0

z₁^jz₂^jfj



+qb^d(z1, z2)−

c−1

X

j=0

z₂^jqb^d_j(z1).

Proof. The proof of the lemma can be found in AppendixB.

Theorem 3.3. In the stable M/M/c synchronous multiple vacation model with gated discipline satisfying assumptionsA.1- A.3, the expression of the joint PGF of the stationary number of customers at a customer departure epoch and the number of customers to be served at the same customer departure epoch is given as

qb^d(z1, z2) = 1 f⁽¹⁾

ba_c(z₁) bac(z1)−z1z2

fb(z₁)

ev(λ−λz1)−fb(z1z2)

!

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+ 1

f⁽¹⁾

c−1

X

j=1

z₁^j z₂^j bac(z1)−baj(z1) ba_c(z₁)−z₁z₂ fj

+

c−1

X

j=1

z₂^j bac(z1)−baj(z1)

ba_c(z₁)−z₁z₂ qb_j^d(z₁).

Proof. The number of customers from thek−1-th to the k-th customer departure epoch, for k = 1,2, . . . , G_`, increases by the number of customers arriving during that period and it decrements by one due to the departing customer, whose service was completed. The number of customers to be served from the k−1-th to the k-th customer departure epoch, again fork= 1,2, . . . , G_`, decreases by one due to the actually completed customer service. Hence the simultaneous evolution of the number of customers in the system and the number of customers to be served from the k−1-th to the k-th customer departure epoch can be given on PGF level as multiplication by

baj+1(z1) z₁z₂ ,

wherejis the number of customers to be served at thek−1-th customer departure epoch.

The transition to a customer departure epoch, in which the number of customers to be served is 0≤j≤c−2, can happen either from the first start epoch or from

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busy servers number of

start of vacation end of

vacation

t service start epoch t

departure epochs j < c

0

,,,

...

t t

0 c

end of

departure epochs service start epochs j >= c

j >= c−1

start of busy servers

number of

vacation

Figure 1. Evolution of number of busy servers forj < c(left side) and forj≥c(right side).

the previous customer departure epoch, at which epochs the number of customers to be served isj+ 1 (see both sides of figure1). Hence in this case the simultaneous evolution of the number of customers in the system and the number of customers to be served can be given on PGF level as

z₂^jqb^d_j(z₁) =baj+1(z1) z₁z₂

z₁^j+1z₂^j+1f_j+1^∗ +z₂^j+1qb^d_j+1(z₁)

, 0≤j≤c−2. (9) We point out here that f_j+1^∗ arises in (9) instead of fj+1. This is because the definition of the related quantities must include averaging over the same set of epochs, i.e. in this case over the customer departure epochs, in order to result in a valid relation.

The transition to a customer departure epoch, in which the number of customers to be served isj≥c−1, happens always from the preceding start epoch (see right side of figure1). Therefore in this case the simultaneous evolution of the number of customers in the system and the number of customers to be served can be given on PGF level as

qb_(j≥c−1)^d (z1, z2) = bac(z1)

z₁z₂ qb^s∗_(j≥c)(z1, z2). (10) The statement of the theorem can be derived by applying (9), (10) and the lemmas3.1and3.2.

Applying the first statement of lemma 3.2 in (9), taking the sum Pc−2 j=0 and rearranging it yields

qb^d₀(z₁) +

c−1

X

j=1

z₂^jqb^d_j(z₁)−z₂^c−1qb^d_c−1(z₁)

= 1

z₁z₂ 1

g

c−2

X

j=0

z₁^j+1z₂^j+1ba_j+1(z₁)f_j+1+

c−2

X

j=0

z^j+1₂ ba_j+1(z₁)qb_j+1^d (z₁)

= 1

z1z2

1 g

c−1

X

j=1

z₁^jz^j₂ba_j(z₁)f_j+

c−1

X

j=1

z₂^jba_j(z₁)bq_j^d(z₁)

.

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Further rearranging the above relationship results in

z₂^c−1z₁z₂qb_c−1^d (z₁)=z₁z₂qb₀^d(z₁)−1 g

c−1

X

j=1

z₁^jz₂^jba_j(z₁)f_j+

c−1

X

j=1

z^j₂(z₁z₂−ba_j(z₁))qb^d_j(z₁).(11)

Applying the second statements of lemmas3.1 and 3.2in (10) and rearranging it yields

z1z2qb^d(z1, z2) −

c−1

X

j=1

z^j₂z1z2qb^d_j(z1)−z1z2qb^d₀(z1) +z^c−1₂ z1z2qb^d_c−1(z1) (12)

= 1

gbac(z1)fb(z1z2)−1 g

c−1

X

j=1

z₁^jz₂^jbac(z1)fj−1

gbac(z1)f0+bac(z1)qb^d(z1, z2)

−

c−1

X

j=1

z^j₂bac(z1)qb_j^d(z1)−bac(z1)bq₀^d(z1).

Applying (11), the first statement of lemma 3.1 in (12) and further rearrange- ments leads to

bq^d(z₁, z₂) =1 g

ba_c(z₁) bac(z1)−z1z2

m(zb ₁)−fb(z₁z₂) +1

g

c−1

X

j=1

z^j₁z^j₂ba_c(z₁)−ba_j(z₁) bac(z1)−z1z2

f_j(13)

+

c−1

X

j=1

z₂^jba_c(z₁)−ba_j(z₁) bac(z1)−z1z2 qb_j^d(z₁).

Due to the service rule of the gated discipline the mean number of customers served during a service period equals to the mean number of customers, which are present at the end of the vacation period. This leads to

g=f⁽¹⁾. (14)

In the vacation model the number of customers at the end of the vacation period is the sum of two independent random variables. One of them is the number of customers at the start of that vacation period and the other one is the customers arriving during that vacation, whose PGF is given asev(λ−λz) due to assumption A.2. Hence the number of customers at the end of the vacation period can be expressed on PGF level as

fb(z) =m(z)b ev(λ−λz). (15)

The theorem comes by applying (14) and (15) in (13).

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4. The stationary number of customers at an arbitrary epoch.

4.1. The PGF of the stationary number of customers. We define the PGF of the stationary number of customers in an arbitrary epoch as

q(z) = limb

t→∞

∞

X

n=0

P{N(t) =n}zⁿ, |z| ≤1.

Theorem 4.1. In the stable M/M/c synchronous multiple vacation model with gated discipline satisfying assumptionsA.1-A.3, the PGF of the stationary number of customers in an arbitrary epoch can be expressed as

q(z) =b 1 f⁽¹⁾

1 (1−z)(1−ρz)

1−ev(λ−λz)

ev(λ−λz) fb(z) (16)

+ 1

f⁽¹⁾

c−1

X

j=1

z^j ρj−ρ

(1−ρz) (1 +ρ_j(1−z)) fj

+

c−1

X

j=1

ρj−ρ

(1−ρz) (1 +ρ_j(1−z)) qb^d_j(z), where the partial PGFsqb^d_j(z)are given as

qb^d_j(z) = z^j f⁽¹⁾

fb(z)

ev(λ−λz)−f0

! _j Y

k=1

(1 +ρk(1−z)) (17)

− z^j f⁽¹⁾

j

X

k=1

f_k

j

Y

i=k+1

(1 +ρ_i(1−z)), j= 1,2, . . . , c−1

and an empty product is 1.

Proof. Based on the definitionsbq^d(z) can be expressed fromqb^d(z₁, z₂) as

bq^d(z) =qb^d(z,1). (18)

The PASTA property [13] and the unit step change of the number of customers in the system together ensures that the stationary number of customers at customer arrival, at customer departure and at arbitrary epochs are all the same. Hencebq(z) can be expressed as

q(z) =b qb^d(z). (19)

Settingz₂= 1 andz₁=zin (8), using (18) and (19) and rearrangement leads to

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q(z)b = 1 f⁽¹⁾

bac(z) ba_c(z)−z

1−ev(λ−λz)

ev(λ−λz) f(z)b (20)

+ 1

f⁽¹⁾

c−1

X

j=1

z^jbac(z)−baj(z) bac(z)−z fj

+

c−1

X

j=1

bac(z)−baj(z) bac(z)−z qb^d_j(z).

Applying the expressions ofbaj(z) andbac(z) from (1) in (20) and rearrangement gives the first statement of the theorem.

Setting z₂ = 0 and applying qb^d(z₁,0) = bq₀^d(z₁) and fb(0) = f₀ in (8) gives the expression ofqb^d₀(z1) as

bq₀^d(z1) = 1 f⁽¹⁾

fb(z1) ev(λ−λz₁)−f0

!

. (21)

qb^d_k(z₁) can be related toqb^d(z₁, z₂) as

qb_k^d(z1) = 1 k!

d^k qb^d(z1, z2) dz2k

_z

2=0

, k≥1.

In order to get a recursive relation forqb_k^d(z1) we multiply (8) by bac(z1)−z1 z2

and take itsk-th derivative atz2= 0 for 1≤k≤c−1. This leads to

k!qb_k^d(z1)bac(z1) − k

1

(k−1)! bq_k−1^d (z1)z1

= − 1

f⁽¹⁾bac(z1)z₁^k k!fk

+ 1

f⁽¹⁾ z₁^k k! (ba_c(z₁)−ba_k(z₁))f_k + k! (bac(z1)−bak(z1))bq_k^d(z1).

We rearrange this relationship as

bq_k^d(z1) = z1

bak(z1)qb^d_k−1(z1)− 1

f⁽¹⁾ z^k₁ fk.

Solving these equations by recursive substitution fork= 1,2, . . . , c−1 leads to

qb^d_k(z₁) =

k

Y

j=1

z1

ba_j(z₁)bq₀^d(z₁)− 1 f⁽¹⁾

k

X

j=1

z^j₁f_j

k

Y

i=j+1

z1

ba_i(z₁). (22) Applying (21) in (22) yields

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qb_k^d(z₁) = z₁^k f⁽¹⁾

fb(z₁) ev(λ−λz1)−f₀

! _k Y

j=1

1

baj(z1) (23)

− z₁^k f⁽¹⁾

k

X

j=1

fj k

Y

i=j+1

1 ba_i(z₁).

The second part of the theorem comes by settingz1=z, applying the expressions ofba_j(z) andba_c(z) in (23) followed by exchanging the indicesj andk.

4.2. The mean of the stationary number of customers.

Corollary 1. In the stable M/M/c synchronous multiple vacation model with gated discipline satisfying assumptions A.1 -A.3, the mean of the stationary number of customers at arbitrary epoch is given by

q⁽¹⁾ = ρ+ ρ²

1−ρ+λ²v^[2]+ 2 λv^[1] (f⁽¹⁾−λv^[1])

2(1−ρ)f⁽¹⁾ (24)

+ 1

(1−ρ)f⁽¹⁾

c−1

X

j=1

(j+ρ_j)(ρ_j−ρ)f_j

− 1 1−ρ

c−1

X

j=1

(ρ−ρ_j)(q^d(1)_j +ρ_jq_j^d), wheref⁽¹⁾,q^d_j andq_j^d(1) are given as

f⁽¹⁾ =

λv^[1]+Pc−1

j=1(ρ_j−ρ)

1−Pj−1 k=0f_k

1−ρ , (25)

q^d_j = 1−Pj k=0fk

f⁽¹⁾ , j= 1,2, . . . , c−1 (26) and

q^d(1)_j = f⁽¹⁾−λv^[1]+ (1−f0)(j−Pj k=1ρk)

f⁽¹⁾ (27)

− Pj

k=1fk(j−Pj

i=k+1ρi)

f⁽¹⁾ , j= 1,2, . . . , c−1.

Proof. The statements of the corollary can be derived from (17) and (20). The detailed proof of the corollary can be found in AppendixC.

Remark 1. If there is only one server then the mean of the stationary number of customers at arbitrary epoch can be given by setting c= 1 in (24) and (25). This results in an explicit form as

q⁽¹⁾=ρ+ ρ²

1−ρ+ ρ

1−ρ λv^[1]+λv^[2]

2v^[1]. (28)

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5. The stationary number of customers at end of the vacation period.

Theorem 5.1. In the M/M/c synchronous multiple vacation model with gated discipline satisfying assumptions A.1 - A.3, the governing equation of the system at the end of the vacation can be expressed as

fb(z) = fb(bac(z))ev(λ−λz)

c−1

Y

j=1

baj(z)

bac(z) (29) +

c−1

X

k=0

f_k ev(λ−λz) 1−

c−1

Y

i=k+1

ba_i(z) bac(z)

! _k Y

j=1

ba_j(z).

Proof. According to the service rule of the gated discipline the number of customers at the start of the vacation period equals to the number of customers arriving during the service period.

If the number of customers at the end of the vacation period is 1≤k ≤c−1 then the service period is the sum of k exponentially distributed intervals, where the parameters of these distributions arekµ,(k−1)µ, . . . , µ. Hence the PGF of the number of arriving customers during such a kind of service period can be expressed as

k

Y

j=1

baj(z). (30) If the number of customers at the end of the vacation period is k≥c then the service period is the sum ofk−(c−1) times exponentially distributed intervals with parametercµand additionalc−1 exponentially distributed intervals, for which the parameters of their distributions are (c−1)µ,(c−2)µ, . . . , µ. Therefore the PGF of the number of arriving customers during such service period is given as

(bac(z))^k−(c−1)

c−1

Y

j=1

baj(z). (31) In case ofk= 0 there is no service period. Thus the number of customers at the start of the next vacation period is also 0, i.e. its PGF is 1. Using this and (30) and (31) the PGF of the number of customers at the start of the vacation period can be established as

m(z)b =

c−1

X

k=0

fk k

Y

j=1

baj(z) +

c−1

Y

j=1

baj(z)

∞

X

k=c

fk(bac(z))^k−(c−1). (32)

Rearranging (32) leads to

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m(z) =b

c−1

X

k=0

f_k ^k

Y

j=1

ba_j(z)−(bac(z))^kQc−1 i=1bai(z) (bac(z))^c−1

(33)

+ Qc−1

j=1baj(z) (bac(z))^c−1

∞

X

k=0

fk(bac(z))^k. Taking into accountfb(z) =P∞

k=0f_kz^k and rearranging (33) yields

m(z) =b f(bba_c(z))

c−1

Y

j=1

baj(z)

ba_c(z) (34) +

c−1

X

k=0

fk

1−

c−1

Y

i=k+1

bai(z) bac(z)

^k Y

j=1

baj(z).

The theorem is completed by applying (15) in (34).

A series of functions is defined recursively as α0(z) = z, |z| ≤1,

α_r+1(z) = ba_c(α_r(z)), r≥0. (35) Theorem 5.2. In the M/M/c synchronous multiple vacation model with gated discipline satisfying assumptions A.1 - A.3, the PGF of the stationary number of customers at the end of the vacation period can be given as

fb(z) =

∞

Y

r=0

ξ_r(z) +

∞

X

r=0 c−1

X

k=0

f_kθ_k,r(z)

r−1

Y

`=0

ξ_`(z), (36)

where

ξr(z) =ev(λ−λαr(z))

c−1

Y

j=1

baj(αr(z))

ba_c(α_r(z)), r≥0 and (37) θk,r(z) =ev(λ−λαr(z)) 1−

c−1

Y

i=k+1

bai(αr(z)) ba_c(α_r(z))

!

×

k

Y

j=1

ba_j(α_r(z)), k= 0,1, . . . , c−1, r≥0.

Proof. Replacing z byα_r(z) in (29) forr≥0 and using (35) yields

f(αb r(z)) =f(αb r+1(z))v(λe −λαr(z))

c−1

Y

j=1

baj(αr(z)) bac(αr(z))

+

c−1

X

k=0

fkv(λe −λαr(z)) 1−

c−1

Y

i=k+1

bai(αr(z)) bac(αr(z))

! _k Y

j=1

baj(αr(z)).

These equations can be solved by means of recursive substitution forr≥0, which leads to

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fb(z) =fb

r→∞lim αr(z)Y^∞

r=0

v(λe −λαr(z))

c−1

Y

j=1

ba_j(α_r(z))

bac(αr(z)) (38) +

∞

X

r=0 c−1

X

k=0

fk ev(λ−λαr(z)) 1−

c−1

Y

i=k+1

ba_i(α_r(z)) bac(αr(z))

!

×

k

Y

j=1

baj(αr(z))

r−1

Y

`=0

ev(λ−λα`(z))

c−1

Y

j=1

baj(α`(z)) bac(α`(z)).

It can be shown that fora⁽¹⁾c =ρ <1, which is the necessary condition of stability, lim_r→∞α_r(z) = 1 for any |z| ≤1 ([5]). Using this in (38) leads to

fb(z) =

∞

Y

r=0

ev(λ−λαr(z))

c−1

Y

j=1

baj(αr(z))

bac(αr(z)) (39) +

∞

X

r=0 c−1

X

k=0

fk ev(λ−λαr(z)) 1−

c−1

Y

i=k+1

bai(αr(z)) ba_c(α_r(z))

!

×

k

Y

j=1

baj(αr(z))

r−1

Y

`=0

ev(λ−λα`(z))

c−1

Y

j=1

baj(α`(z)) ba_c(α_`(z)).

The final expression offb(z) can be obtained from (39) by rewriting it using the notations, which are introduced in (37).

The unknown probabilitiesfk, fork= 0,1, . . . , c−1 in (36) are determined from a system ofc linear equations, which is derived from (36). One equation comes by setting z = 0 in (36), while the remaining c−1 equations are obtained by taking thex-th derivative of (36) atz= 0 forx= 1,2, . . . , c−1. Applying alsofb(0) =f₀ and ^d^x_dz^f(z)^b_x

_z=0=x!f_xthis results in the system of linear equations as f0=

∞

Y

r=0

ξr(0) +

c−1

X

k=0

fk

∞

X

r=0

θk,r(0)

r−1

Y

`=0

ξ`(0), (40)

x!f_x= d^x (Q∞

r=0ξr(z)) dz^x

_z=0

+

c−1

X

k=0

f_k

×

∞

X

r=0

d^x

θk,r(z)Qr−1

`=0ξ`(z) dz^x

_z=0

, x= 1, . . . , c−1.

6. The stationary waiting time. We determine the LST of the stationary waiting time by the help of the PGF of the stationary number of customers in the queue at the epochs when the customers move from the queue to the servers. From the point of view of the queue these epochs are “departure” epochs.

These epochs coincide with the customer service start epochs. However in case of a first customer service start epoch of a service period more customers can move from the queue to the servers at the same time. In order to make tractable the number of customers in the queue just after the “departure” of each of these customers

(17)

moving simultaneously from the queue, we apply the Orderliness Convention (Wolff [17] p. 388). According to it we suppose that these customers make a line - in the same order as they arrived - to move instantaneously from the queue to the servers one after the other. By applying this convention the “departure” epoch of each of these customers moving simultaneously from the queue can be seen as individual epoch. Hence the number of such “departure” epochs are the same as the number of customers, which get service during a service period.

Taking into account the above considerations we define the PGF of the stationary number of customers in the queue at the individual “departure” epochs defined by means of the Orderliness Convention as

br^d(z) =

∞

X

n=c m→∞lim

Eh Pm

`=1

PG_`−c

k=1 I_(N_(T_`_(k))=n)i E[Pm

`=1G`] z^n−c +

∞

X

n=c m→∞lim

EPm

`=1I_(N_(T_`_(0))=n) E[Pm

`=1G_`] z^n−c

c−1

X

k=0

z^k

+

c−1

X

n=1 m→∞lim

EPm

`=1I_(N_(T_`_(0))=n) E[Pm

`=1G`]

n−1

X

k=0

z^k, |z| ≤1.

The first term on the r.h.s. of the above definition stands for the non-first customer service start epochs, when c customers are under service. The second term on the r.h.s. of this definition stands for the first customer service start epochs, at which the number of customers in the system is at least c. In this case theccustomers getting service simultaneously see different number of customers in the queue at the individual “departure” epochs. This is taken into account by the term Pc−1

k=0z^k. Similarly in the third term on the r.h.s. of this definition, which stands for the first customer service start epochs, at which the number of customers in the system is at least 1 and at mostc−1, the above explained difference is taken into account by the termPn−1

k=0z^k.

We also define the joint PGF of the stationary number of customers in the queue and the stationary number of customers to be served at the individual “departure”

epochs defined by means of the Orderliness Convention as

br^d(z1, z2) =

∞

X

j=c

∞

X

n=0 m→∞lim

Eh Pm

`=1

PG`−c

k=1 I_(N(T_`_(k))=n)I_(Γ(T_`_(k))=j)i E[Pm

`=1G`] z₁^n−c z₂^j +

∞

X

j=c m→∞lim

EPm

`=1I_(N(T_`_(0))=j)I_(Γ(T_`_(0))=j) E[Pm

`=1G`] z₁^j−c

c−1

X

k=0

z₁^k z₂^j

+

c−1

X

j=1 m→∞lim

EPm

`=1I_(N(T_`_(0))=j)I_(Γ(T_`_(0))=j) E[Pm

`=1G`]

j−1

X

k=0

z^k₁ z^j₂,

|z1| ≤1, |z2| ≤1.

Proposition 2. In the M/M/c synchronous multiple vacation model with gated discipline satisfying assumptions A.1- A.3, the PGF br^d(z)can be expressed as

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br^d(z) =q(z) +b _f₍₁₎¹ ^1−z_1−z^cfb(z) +_f¹₍₁₎Pc−1 j=1

z^c−z^j

1−z fj−_f¹₍₁₎^1−z_1−z^cf0−Pc−1 j=0qb^d_j(z)

z^c , (41)

where

bq₀^d(z) = 1 f⁽¹⁾

fb(z) ev(λ−λz)−f₀

!

. (42)

Proof. The proof of the proposition can be found in AppendixD.

LetW_τ be the waiting time in the system at time τ. We define the distribution function of the stationary waiting time,W(t), as

W(t) = lim

τ→∞P{W_τ≤t}. The LST of the stationary waiting time is defined as

we(s) = Z ∞

t=0

e^−stdW(t), Re(s)≥0.

Theorem 6.1. In the M/M/c synchronous multiple vacation model with gated discipline satisfying assumptionsA.1- A.3the LST of the stationary waiting time is given as

w(s)e = 1 f⁽¹⁾

λ s

λ λ−s

^c λ λ−ρ(λ−s)

1−ev(s) ev(s) + 1−

λ−s λ

^c

fb(1−s λ) + λ²

f⁽¹⁾ λ

λ−s ^{c c−1}

X

j=1

λ−s λ

^j (ρj−ρ)

(λ−ρ(λ−s))(λ+ρ_js)fj

+ λ

f⁽¹⁾ λ

λ−s ^{c c−1}

X

j=1 λ−s

λ

^c

− ^λ−s_λ ^j

s fj

− λ

λ−s ^{c c−1}

X

j=1

λ(λ−ρ_j(λ−s)) +ρ(λ−ρ_j(λ−s))s (λ−ρ(λ−s)) (λ+ρjs) bq_j^d

1− s λ

− λ

λ−s ^c

bq₀^d 1− s

λ

+ λ f⁽¹⁾

1− ^λ−s_λ ^c s f0

!

. (43)

Proof. The PASTA property ensures that an arriving customer sees the stationary waiting time. Due to the FIFO service order the customers arriving during the waiting time of a selected customer are the ones which are present in the queue at the individual “departure” epoch of that selected customer, which is defined by means of the Orderliness Convention. The model assumptions ensure that the arrival process and the waiting time are mutually independent. Thus the above argument can be formulated on transform level as

rb^d(z) =w(λe −λz). (44)

Applying (41) in (44) yields