Note on a Binomial Schedule for an M^X/G/1 Queueing System with an Unreliable Server

Abstract

We consider in this paper a batch arrival queueing system with an unreliable server. If the queue is empty at a service completion, then the server becomes inactive and begins an idle period. However, if the queue is not empty, then the server will take at most 𝐾 vacation before serving the next customer. A linear cost structure is developed for the system and the optimal value of 𝐾 is obtained.

1. Introduction

As mentioned by Hlynka [1], “the use of queueing theory in the analysis and management of computer and telecommunication systems has by now a lengthy history, with many important contributions in the design and analysis of such systems. There appear to be no signs of this letting up opportunities abound for queueing to aid in developments in wireless networking, cloud computing, social networking, and many other modern application areas.” The part of queueing theory concerned with the optimization of a queueing system is called the optimal design and control of queues. For a comprehensive survey of the research on this topic, see the book by Stidham [2] and the survey by Tadj and Choudhury [3].

The current paper presents yet another model that combines many known features to accommodate the increasingly complex computer networks and telecommunication systems. Meaningful and systematic scheduling requires sophisticated models to allocate the sometimes scarce available resources. Since most of these traits are standard and can be found in many papers, our review of the literature will make do with citations of surveys where the reader can find the relevant references.

The first feature of our model is the batch arrival process. Batch arrival models have been extensively used in an uncountable number of papers. A topical textbook is that of Chaudhry and Templeton [4].

The second feature of the model considered in this paper is that of an unreliable server. Indeed, a server, such as a machine, may break down while providing service. The service of the customer being served is then interrupted and cannot resume until the server is repaired. This is a very realistic assumption that models real-life situations. For a review of the research on this topic, the reader is referred to the broad survey of Tadj et al. [5].

Finally, the last characteristic of the model under consideration is that of a binomial vacation schedule. For a review of vacation queueing systems, see the comprehensive surveys of Doshi [6, 7]. In this class of queueing models, we find the models with Bernoulli vacation schedule where if the queue is empty at a service completion, then the server becomes inactive and begins an idle period. However, if the queue is not empty, the server will choose randomly to either take a vacation or process the next customer. If a vacation is chosen, it is followed by the service of the customer at the head of the queue. Various aspects of Bernoulli vacation models have been discussed by a number of authors; see the survey of Ke et al. [8]. In real life, the vacation period may be used by the server to perform some other activities such as a maintenance operation or a quality control or even attend another queue of customers.

However, since there are different activities, we may have the server take more than one vacation, for a maximum number of, say, 𝐾 vacations, instead of just one. Thus, we use in this paper the notion of a binomial vacation schedule. Under this policy, at a service completion and before serving the next customer, the server takes a series of vacations. The number of vacations is a binomial random variable. Each vacation has a random duration and corresponds to some auxiliary activity.

Having in mind the optimization of the queueing system, the decision variable will be the maximum number of vacations (auxiliary works) that the server should take before processing the next customer. That is, since we allow the server to take many vacations, what would the ideal number of vacations that will optimize the operations of the system be?

The rest of the paper proceeds as follows. The next section reviews some relevant results from a well-studied queueing system, the M^X/G/1. Section 3 shows how to extend the results of Section 2 to our model. Section 4 is concerned with the design of the optimization of the system. Finally, Section 5 concludes the paper.

2. Review of M^X/G/1 Queueing Model

The M^X/G/1 is a well-studied queueing system and its analysis is available in classical queueing textbooks; see for example Medhi [9]. We just recall here some of the results relevant to our study. Let 𝜆 denote the arrival rate. Customers arrive to the system in batches of random size 𝑋. The batches are independent and identically distributed (iid) random variables with probability mass function 𝑎_𝑘=𝑃(𝑋=𝑘), 𝑘≥1, probability generating function (pgf) 𝑎(𝑧)=𝐸[𝑧^𝑋], and moments 𝑎_(𝑘)=𝐸(𝑋^𝑘), 𝑘≥1. The service times 𝜎 are iid random variables with common cumulative distribution function (CDF) 𝐵(𝑡)=𝑃(𝜎≤𝑡), 𝑡≥0(𝐵(0)=0), Laplace-Stieltjes Transform (LST) 𝐵^∗(𝜃)=𝐸[𝑒^−𝜃𝜎], Re(𝜃)≥0, and moments 𝑏_(𝑘)=𝐸(𝜎^𝑘), 𝑘≥1. We are assuming that the service process is independent of the arrival process. We use the following notation: 𝑄(𝑡): total number of customers in the system at any instance of time𝑡≥0, 𝑇₀=0,𝑇₁,𝑇₂,…: sequence of the successive departure times of individual units, 𝑄_𝑛=𝑄(𝑇⁺_𝑛), 𝑛=1,2,…: total number of customers in the system at a customer departure time.

The imbedded process {𝑄_𝑛;𝑛=0,1,…} is a homogeneous Markov chain. Its transition probability matrix 𝑃=(𝑝_𝑖𝑗) is a Δ₂ matrix; see Abolnikov and Dukhovny [10]. This fact can be used to derive the steady-state condition mentioned next. The process {𝑄(𝑡);𝑡≥0} is a semiregenerative process relative to the sequence {𝑇_𝑛;𝑛=0,1,…} of stopping times, and {(𝑄_𝑛,𝑇_𝑛)} is the corresponding imbedded Markov renewal process; see Çinlar [11]. The necessary and sufficient ergodicity condition is given by 𝜌∶=𝜆𝑎₍₁₎𝑏₍₁₎<1. (1)

Let the equilibrium probabilities be 𝑝_𝑖=lim_𝑛→∞𝑃(𝑄_𝑛=𝑖) and 𝜋_𝑖=lim_𝑡→∞𝑃(𝑄(𝑡)=𝑖). We are interested in the PGFs 𝑃(𝑧)=∑^∞_𝑖=0𝑝_𝑖𝑧^𝑖 and 𝜋(𝑧)=∑^∞_𝑖=0𝜋_𝑖𝑧^𝑖. In the M^X/G/1 queueing system, Kendall’s formula, most often referred to as Pollaczek-Khinchin formula, is 𝑃(𝑧)=𝐵^∗(𝜆−𝜆𝑎(𝑧))[𝑎(𝑧)−1]𝑝₀𝑧−𝐵^∗((𝑧,𝜆−𝜆𝑎)) (2) with 𝑝₀=1−𝜌𝑎₍₁₎. (3)

Also, 𝜋(𝑧)=𝑎₍₁₎1−𝑧(𝑧)𝑃(𝑧),1−𝑎 (4) which yields 𝜋₀=𝑎₍₁₎𝑝₀=1−𝜌. (5)

For this queueing system, some performance measures are as follows.(a)The expected number of customers in the system at an arbitrary instance of time: 𝐿₍₁₎=2𝜆𝑎²₁𝑏₍₁₎+𝑎₍₂₎−𝑎₍₁₎2𝑎₍₁₎+𝜆2𝑎1−𝜌⁽²⁾−𝑎₍₁₎𝑏₍₁₎+𝜆𝑎²₁𝑏₍₂₎. (6)(b)The expected length of an idle period of the server in the equilibrium: 𝐼₍₁₎=1𝜆. (7)(c)The expected length of the busy in the equilibrium: 𝐵₍₁₎=𝜌𝜆.1−𝜌 (8)(d)The expected length of the busy cycle: 𝐶₍₁₎=1𝜆.1−𝜌 (9)

3. Extension to M^X/G/1 Queueing System with an Unreliable Server and Binomial Vacation Schedule

It is interesting to note that, using a modified service time distribution, it is possible to obtain all the results of the M^X/G/1 queue with unreliable server and binomial vacation schedule (that we will call the new system) from the results of the classical M^X/G/1 queueing system. Similar change of variable is common and was recently used by Blanc [12].

Let us first describe the new system and define the notation used. While providing service, the server may break down and it is assumed that breakdowns occur according to a Poisson process with positive rate 𝛼. When service interruption occurs, repair is provided with a random time, and as soon as the server is repaired, it immediately returns to provide this service until the queue is empty again. Service time is cumulative. This means that, on repair after an interruption, the service of the customer which was interrupted earlier is resumed from where it got interrupted and is not repeated from the very beginning. Let 𝑅 denote the duration of the repair time. Then, 𝑅 is distributed according to the CDF 𝑅(𝑡) with LST 𝑅^∗(𝜃)=𝐸[𝑒^−𝜃𝑅] and finite moments 𝑟_(𝑗), 𝑗≥1.

To analyze this model, we introduce the modified service time 𝐻, which includes the actual service time and possible repairs. Then, 𝐻 is distributed according to the CDF 𝐻(𝑡) with LST 𝐻^∗(𝜃)=𝐸[𝑒^−𝜃𝐻] and finite moments ℎ_(𝑗), 𝑗≥1. The modified service times, actual service times, and repair times are related through the following formula: 𝐻^∗(𝜃)=^∞_𝑛=0^∞₀𝑒^−𝜃𝑥𝑒^−𝛼𝑥(𝛼𝑥)^𝑛𝑅𝑛!^∗(𝜃)^𝑛𝑑𝐵(𝑥)=𝐵^∗𝜃+𝛼1−𝑅^∗(𝜃). (10)

We also assume that the server implements the binomial vacation schedule, so that, at the end of a service, the server has the option to take 𝑘(𝑘=0,…,𝐾) vacations of random length 𝑉_𝑘 with probability 𝑦_𝑘=𝐾𝑘𝑝^𝑘1−𝑝^𝐾−𝑘,𝑘=0,…,𝐾. (11)

Vacation times 𝑉 are distributed according to the CDF 𝑉(𝑡) with LST 𝑉^∗(𝜃)=𝐸[𝑒^−𝜃𝑉] and finite moments 𝑣_(𝑗), 𝑗≥1. Let 𝐺 denote the duration of the generalized service time of a customer. Then, 𝐺 is distributed according to the CDF 𝐺(𝑡) with LST 𝐺^∗(𝜃)=𝐸[𝑒^−𝜃𝐺] and finite moments 𝑔_(𝑗), 𝑗≥1. Now, since with probability 𝑦_𝑘 the service required for a customer is 𝐺=𝐵+𝑉₀+⋯+𝑉_𝑘, (12) then, we have 𝐺^∗(𝜃)=𝐻^∗(𝜃)^𝐾_𝑘=0𝑦_𝑘𝑉^∗(𝜃)^𝑘, (13) where 𝐻^∗(𝜃) is given by (10). We are assuming that the input process and service and repair times random variables are mutually independent of each other.

We now want to generalize the results (1)–(5) to the new system. Let 𝜌_𝐺=𝜆𝑎₍₁₎𝑔₍₁₎. Using (13), we find 𝑔₍₁₎=ℎ₍₁₎+𝐾𝑝𝑣₍₁₎. (14)

Now, using (10), we have ℎ₍₁₎=𝑏₍₁₎1+𝛼𝑟₍₁₎, (15) so that 𝜌_𝐺=𝜆𝑎₍₁₎[𝑏₍₁₎(1+𝛼𝑟₍₁₎)+𝐾𝑝𝑣₍₁₎]. Then, the necessary and sufficient ergodicity condition for the new system is given by 𝜌_𝐺∶=𝜌1+𝛼𝑟₍₁₎+𝜆𝐾𝑝𝑎₍₁₎𝑣₍₁₎<1. (16)

Let the equilibrium probabilities be 𝑝_𝑖=lim_𝑛→∞𝑃(𝑄_𝑛=𝑖) and 𝜋_𝑖=lim_𝑡→∞𝑃(𝑄(𝑡)=𝑖) and let the PGFs be 𝑃(𝑧)=∑^∞_𝑖=0𝑝_𝑖𝑧^𝑖 and 𝜋(𝑧)=∑^∞_𝑖=0𝜋_𝑖𝑧^𝑖. The results (2)–(5) are generalized to the new system as follows: 𝑃(𝑧)=𝐺^∗(𝜆−𝜆𝑎(𝑧))[𝑎(𝑧)−1]𝑝₀𝑧−𝐺^∗((𝑧,𝜆−𝜆𝑎)) (17) with 𝑝₀=1−𝜌_𝐺𝑎₍₁₎. (18)

Also, 𝜋(𝑧)=𝑎₍₁₎1−𝑧(𝑧)𝑃(𝑧),1−𝑎 (19) which yields 𝜋₀=𝑎₍₁₎𝑝₀=1−𝜌_𝐺. (20)

4. Optimal Management Policy

In order to design an optimal management policy for the service system, we first derive the relevant system characteristics.

4.1. System Characteristics

Using the results (6)–(9), we obtain the following performance measures.(a)The expected number of customers in the system at an arbitrary instance of time is given by 𝐿₍₁₎=2𝜆𝑎²₁𝑔₍₁₎+𝑎₍₂₎−𝑎₍₁₎2𝑎₍₁₎+𝜆21−𝜆𝑎₍₁₎𝑔₍₁₎×𝑎₍₂₎−𝑎₍₁₎𝑔₍₁₎+𝜆𝑎²₁𝑔₍₂₎, (21) where 𝑔₍₁₎ is found in (14), while 𝑔₍₂₎ is found from (13) as 𝑔₍₂₎=ℎ₍₂₎+2𝐾𝑝ℎ₍₁₎+𝐾(𝐾−1)𝑝², (22) where ℎ₍₁₎ is given by (15), while ℎ₍₂₎ is derived from (10) as ℎ₍₂₎=𝑏₍₂₎1+𝛼𝑟₍₁₎²+𝛼𝑏₍₁₎𝑟₍₂₎. (23)(b)The expected length of an idle period of the server in the equilibrium is given by 𝐼₍₁₎=1𝜆. (24)(c)The expected length of the busy in the equilibrium is given by 𝐵₍₁₎=𝜌_𝐺𝜆1−𝜌_𝐺. (25)(d)The expected length of the busy cycle is given by 𝐶₍₁₎=1𝜆1−𝜌_𝐺. (26) Using (24)–(26), one can derive the following probabilities.(e)The probability that the server is idle, as expected: 𝑃_𝐼=𝐼₍₁₎𝐶₍₁₎=1−𝜌_𝐺. (27)(f)The probability that the server is busy, as expected: 𝑃_𝐵=𝐵₍₁₎𝐶₍₁₎=𝜌_𝐺. (28)

Once all the performance measures of the system are available, it is possible to use them in order to design an optimal management policy. The goal is to derive the threshold level that yields the minimum system cost. We derive next an expression for the cost function, and then illustrate on a numerical example how the management policy is designed.

4.2. Total Expected Cost per Unit of Time

In order to develop an optimal management policy for the system, it is customary to write an expression for the total expected cost per unit of time incurred by the system and then obtain the optimal value of the threshold parameter. In our case, the decision variable sought to minimize the cost function is 𝐾, the maximum number of the server vacations. The total expected cost function per unit time is given by TC(𝐾)=𝑐_ℎ𝐿₍₁₎+𝑐_𝑜𝐵₍₁₎𝐶₍₁₎+𝑐_𝑎𝐼₍₁₎𝐶₍₁₎+𝑐_𝑠1𝐶₍₁₎, (29) where 𝑐_ℎ is the holding cost per unit for each customer present in the system, 𝑐_𝑜 is the cost per unit time for keeping the server on and in operation, 𝑐_𝑎 is the startup cost per unit time for the preparatory work of the server before starting, and 𝑐_𝑠 is the setup cost per busy cycle. Upon substitution using (21) and (24)–(26), we get TC(𝐾)=𝑐_ℎ2𝜆𝑎²₁𝑔₍₁₎+𝑎₍₂₎−𝑎₍₁₎2𝑎₍₁₎+𝜆21−𝜆𝑎₍₁₎𝑔₍₁₎×𝑎₍₂₎−𝑎₍₁₎𝑔₍₁₎+𝜆𝑎²₁𝑔₍₂₎+𝑐_𝑜−𝑐_𝑎−𝜆𝑐_𝑠𝜆𝑎₍₁₎𝑔₍₁₎+𝑐_𝑎+𝜆𝑐_𝑠, (30) where 𝑔₍₁₎ and 𝑔₍₂₎ are given by (14) and (22), respectively, and are functions of the decision variable 𝐾. Let 𝐴₁=(𝜆𝑝𝑎₍₁₎)²𝑐_ℎ(1−𝑣²₍₁₎), 𝐵₁=𝜆𝑝𝑐_ℎ[(𝑎₍₁₎+𝑎₍₂₎)𝑣₍₁₎+𝜆𝑎²₍₁₎(2ℎ₍₁₎−4ℎ₍₁₎𝑣₍₁₎−𝑝)],  𝐵₂=2𝜆𝑝𝑎₍₁₎𝑣₍₁₎, 𝐵₃=𝜆𝑝𝑎₍₁₎𝑣₍₁₎(𝑐_𝑜−𝑐_𝑎−𝜆𝑐_𝑠), 𝐶₁=𝜆𝑐_ℎ[(𝑎₍₁₎+𝑎₍₂₎)ℎ₍₁₎+𝜆𝑎²₍₁₎(ℎ₍₂₎−2ℎ²₍₁₎)], 𝐶₂=2(1−𝜆𝑎₍₁₎ℎ₍₁₎), and 𝐶₃=𝜆𝑎₍₁₎ℎ₍₁₎(𝑐_𝑜−𝑐_𝑎−𝜆𝑐_𝑠)+(𝑐_ℎ/2)(𝑎₍₂₎/𝑎₍₁₎−1)+𝑐_𝑎+𝜆𝑐_𝑠. Then, (30) is rewritten as (𝐾)=𝐴TC¹𝐾²+𝐵₁𝐾+𝐶₁𝐶₂−𝐵₂𝐾+𝐵₃𝐾+𝐶₃. (31)

4.3. Optimal Threshold Level 𝐾^∗

The first-order optimality condition yields the critical value. Taking 𝐾 as a continuous variable and calculating the first derivative of the expected cost per unit of time with respect to 𝐾, we find 𝑑TC(𝐾)=𝐵𝑑𝐾²𝐵₂𝐵₃−𝐴₁𝐾²+2𝐶₂𝐴₁−𝐵₂𝐵₃𝐾+𝐵₁𝐶₂+𝐵₂𝐶₁+𝐵₃𝐶²₂×𝐶₂−𝐵₂𝐾²⁻¹. (32)

Setting this expression to zero and solving for 𝐾 yields the two roots 𝐾₁=𝐶₂𝐵₂+𝐴₁𝐶²₂+𝐵₁𝐵₂𝐶₂+𝐵²₂𝐶₁𝐵₂𝐵₂𝐵₃−𝐴₁,𝐾₂=𝐶₂𝐵₂−𝐴₁𝐶²₂+𝐵₁𝐵₂𝐶₂+𝐵²₂𝐶₁𝐵₂𝐵₂𝐵₃−𝐴₁. (33)

The second optimality condition allows deciding which value should be chosen. The second derivative of the expected cost per unit of time with respect to 𝐾 is given by 𝑑²TC(𝐾)𝑑𝐾²=2𝐴₁𝐶²₂+𝐵₁𝐵₂𝐶₂+𝐶₁𝐵²₂𝐶₂−𝐵₂𝐾³. (34)

Note that the sign of the second derivative is the same as the sign of 𝐶₂−𝐵₂𝐾. Direct substitution shows that 𝐶₂−𝐵₂𝐾₁<0 yielding the maximum cost value, while 𝐶₂−𝐵₂𝐾₂>0 yielding the minimum cost value. Therefore, the optimal threshold level is the integer closest to 𝐾^∗=𝐶₂𝐵₂−𝐴₁𝐶²₂+𝐵₁𝐵₂𝐶₂+𝐵²₂𝐶₁𝐵₂𝐵₂𝐵₃−𝐴₁. (35)

4.4. Numerical Illustration

We present numerical computations to illustrate the analytical results obtained. We also perform some sensitivity analyses to assess the effect of some system parameters on the optimal values of the threshold and cost function. Let us assume that the distributions involved are exponential. We recall that, in this case, the second moment is equal to twice the square of the first moment. For the size of the batch arrival, let us assume that it follows the shifted geometric distribution 𝑎_𝑘=𝑃(𝑋=𝑘)=(1−𝜀)𝜀^𝑛−1, 𝑛≥1. Then, 𝑎(𝑧)=𝐸[𝑧^𝑋]=(1−𝜀)𝑧/(1−𝜀𝑧), 𝑎₍₁₎=1/(1−𝜀), and 𝑎₍₂₎=(1+𝜀)/(1−𝜀)². The illustrative system parameters chosen are listed in Table 1.

Table 1 

System parameters.

The variations of the total cost per unit of time as 𝐾 changes are decpited in Figure 1. The curve is perfectly convex. The optimal value of 𝐾 is 𝐾^∗=16 and the optimal value of the total expected cost per unit of time is TC(𝐾^∗)=53.49. Therefore, in order to optimize the system operations, for this specific set of values of the system parameters, the server should take at most 16 vacations to perform different auxiliary works before serving the next available customer.

Figure 1 

Plot of the total expected cost per unit of time as a function of the threshold level 𝐾.

The expressions of the optimal threshold level and the total expected cost per unit of time are quite complex, and it is difficult to assess analytically the effect of the system parameter on these quantities. However, sensitivity analysis can be performed for a specific set of values of the system parameters. For example, suppose that we are interested in the effect of the arrival process on the optimal management policy. This can be done numerically by changing the values of the arrival process parameter and keeping all other parameters at the base values listed in Table 1. Tables 2 and 3 show that the optimal expected cost increases as the batch arrival size parameter 𝜀 or the arrival rate 𝜆 increases; however, the optimal number of vacations 𝐾^∗ decreases when 𝜀 increases and increases when 𝜆 increases.

Table 2 

Effect of the batch arrival size parameter 𝜀 on the optimal threshold level 𝐾^∗ and the optimal expected cost TC(𝐾^∗).

Table 3 

Effect of the arrival rate 𝜆 on the optimal threshold level 𝐾^∗ and the optimal expected cost TC(𝐾^∗).

5. Conclusion

We have considered in this paper a variant of the M^X/G/1 queueing system where the server may break down while providing service. Also, the server implements a binomial vacation policy. Using a suitable change of variable, we obtain from the M^X/G/1 performance measures for our queueing system. We then developed a linear cost structure and derived an expression for the optimal value of the maximum number of vacations the server should take at a service completion epoch before processing the next customer in line.